Advanced Methods for Decision Making and Risk ...

Advanced Methods for Decision Making and Risk Management in Sustainability Science

Edited By J. P. Kropp and J. Scheffran

NOVA Science Publishers

Contents Part I. Uncertainty and Viability 1 Climate Protection Strategies under Ambiguity about Catastrophic Consequences

11

By E. Kriegler, H. Held, and T. Bruckner (With 11 Figures) 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Decision Making under Ambiguity . . . . . . . . . . . . . . Decision Making in the Face of Catastrophic Consequences Setup of a Prototypical Climate Policy Analysis . . . . . . Acting upon the Imprecise Probability of Ruin . . . . . . . Representations of Imprecise Probability . . . . . . . . . . The Imprecise Probability of Climate Sensitivity . . . . . . 1.7.1 Random Set Approximation . . . . . . . . . . . . . . 1.7.2 Possibilistic Decomposition . . . . . . . . . . . . . . 1.8 Estimating Global Mean Temperature Change . . . . . . . 1.8.1 Possibilistic Extension . . . . . . . . . . . . . . . . . 1.8.2 Random Set Extension . . . . . . . . . . . . . . . . . 1.9 Estimating Mitigation Costs . . . . . . . . . . . . . . . . . 1.10 Ranking Stabilization Policies under Ambiguity . . . . . . . 1.11 Summary and Conclusion . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

2 An Introduction to Viability Theory and Management of Renewable Resources

11 12 15 17 20 24 28 30 32 34 36 37 40 43 47 50 55

By J.-P. Aubin and P. Saint-Pierre (With 9 Figures) 2.1

2.2

Introduction . . . . . . . . . . . . . . . . . . 2.1.1 From Malthus to Verhulst and Beyond 2.1.2 Purpose of this Chapter . . . . . . . . The Mathematical Framework . . . . . . . . 2.2.1 Viability and Capturability . . . . . . 2.2.2 The Evolutionary System . . . . . . . v

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

55 55 57 60 60 61

vi

Contents 2.2.3 Viability Kernels and Capture Basins . . . . . . . . . . . . . 2.3 Characterization of Viability and/or Capturability . . . . . . . . . . 2.3.1 Tangent Directions . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Adaptive Map . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Viability Theorem . . . . . . . . . . . . . . . . . . . . . . 2.3.4 The Adaptation Law . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Planning Tasks: Qualitative Dynamics . . . . . . . . . . . . . 2.3.6 The Meta-System . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Selecting Viable Feedbacks . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Static Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Dynamic Feedbacks . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Heavy Evolutions and the Inertia Principle . . . . . . . . . . 2.5 Management of Renewable Resources . . . . . . . . . . . . . . . . . 2.5.1 Discrete Versus Continuous Time . . . . . . . . . . . . . . . . 2.5.2 Introducing Inertia Bounds . . . . . . . . . . . . . . . . . . . 2.5.2.1 Affine Feedbacks and the Verhulst Logistic Equation 2.5.2.2 Inert Evolutions . . . . . . . . . . . . . . . . . . . . 2.5.2.3 Heavy Evolutions . . . . . . . . . . . . . . . . . . . 2.5.2.4 The Heavy Hysteresis Cycle . . . . . . . . . . . . . 2.5.3 Verhulst and Graham . . . . . . . . . . . . . . . . . . . . . . 2.5.4 The Inert-Schaeffer Meta–System . . . . . . . . . . . . . . . . 2.5.5 The Crisis Function . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Towards Dynamical Games . . . . . . . . . . . . . . . . . . . 2.6 Viability and Optimality . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Restoring Viability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Designing Regulons . . . . . . . . . . . . . . . . . . . . . . . 2.7.1.1 Viability Multipliers . . . . . . . . . . . . . . . . . . 2.7.1.2 Connection Matrices . . . . . . . . . . . . . . . . . . 2.7.1.3 Hierarchical Organization . . . . . . . . . . . . . . . 2.7.1.4 Evolution of the Architecture of a Network . . . . . 2.7.2 Impulse Systems . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Mutational Equations Governing the Evolution of the Constrained Sets . . . . . . . . . . . . . . . . . . . . . 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II. Qualitative Methods

62 63 63 64 64 64 65 65 66 66 66 66 67 68 69 70 71 72 74 76 79 80 82 84 85 86 86 86 87 88 90 91 92 93

Contents

vii

3 Qualitative Modeling Techniques to Assess Patterns of Global Change

99

By K. Eisenack, M.K.B. L¨ udeke, G. Petschel-Held, J. Scheffran, and J.P. Kropp (With 13 Figures) 3.1 3.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Describing and Analyzing Archetypes as Model Ensembles . . . . . 3.2.1 Model Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Qualitative Differential Equations . . . . . . . . . . . . . . . 3.2.3 Viability Concepts . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Syndromes of Global Change . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Traditional Approach . . . . . . . . . . . . . . . . . . . . 3.3.2 The Overexploitation Syndrome: Terrestrial and Marine Overexploitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1 Basic Interactions . . . . . . . . . . . . . . . . . . . 3.3.2.2 Model Ensemble . . . . . . . . . . . . . . . . . . . . 3.3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.4 A Refined Model Ensemble . . . . . . . . . . . . . . 3.3.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 The Dust-Bowl Syndrome: Consequence of a Non-adapted Industrial Agriculture . . . . . . . . . . . . . . . . . . . . . . 3.3.3.1 Basic Interactions . . . . . . . . . . . . . . . . . . . 3.3.3.2 Model Ensemble . . . . . . . . . . . . . . . . . . . . 3.3.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 The Sahel Syndrome . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.1 Basic Interactions . . . . . . . . . . . . . . . . . . . 3.3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 From Competition to Cooperation in Sustainable Resource Management - A Multi-Actor Approach . . . . . . . . . . . . . . . . . . . . 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Analyzing a Sustainability Indicator by Means of Qualitative Kernels

99 101 101 104 105 107 107 109 112 114 117 119 120 121 122 124 126 128 128 131 132 139 141 142 147

C. Angulo, L. González, F. Ruiz, A. Català, F. Velasco, and N. Agell (With 2 Figures) 4.1 4.2 4.3 4.4

Introduction . . . . . . . . . . . . . . . . . . . . . . Design of the Survey . . . . . . . . . . . . . . . . . . Artificial Intelligence for the Qualitative Treatment Interval Kernel for Qualitative Processing . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

147 148 149 151

viii

Contents 4.5

Kernel for Qualitative Orders of Magnitude Spaces . . . . 4.5.1 An Explicit Feature Mapping from (OM (n))k . . . 4.5.2 Building a Kernel on an Order of Magnitude Space 4.6 Application of the Kernel Distances . . . . . . . . . . . . 4.6.1 Application of the Interval Distance . . . . . . . . 4.6.2 Application of the Absolute OM Distance . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

5 Constraint Satisfaction Problems for Modeling and Explanation in Sustainability Science

153 154 154 155 156 158 160 161

163

By F. Wotawa (With 6 Figures) 5.1 Introduction . . . . . . . . 5.2 Basic Definitions . . . . . . 5.3 Computing Solutions . . . 5.4 Decomposition Methods . . 5.5 Searching for Explanations 5.6 Conclusion . . . . . . . . . References . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

163 164 167 170 176 178 178

Part III. Stochastic and Fuzzy Approaches 6 Catastrophic Risk Management using Spatial Adaptive Monte Carlo Optimization

183

By T. Ermolieva, G. Fischer, and M. Obersteiner (With 10 Figures) 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Simple Risk Management Model . . . . . . . . . . . . . . . 6.3 Stochastic Integrated Catastrophic Risk Management Model 6.4 Spatial Adaptive Monte Carlo Optimization . . . . . . . . . . 6.5 Case Studies: Earthquake Risks Management . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Genetic Algorithms and Their Applications in Environmental Sciences

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

183 186 188 190 195 202 203

205

By S. E. Haupt (With 4 Figures) 7.1 7.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Introduction to Genetic Algorithms . . . . . . . . . . . . . . . . . . 206 7.2.1 Creating the Population . . . . . . . . . . . . . . . . . . . . . 207

ix

Contents 7.2.2 Choosing the Mates . . . . . . . . . . . . . . . . . . . . . 7.2.3 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Uses of Genetic Algorithms in Environmental Science Problems . 7.4 Example 1 – Fitting a Non-linear Inverse Model . . . . . . . . . 7.5 Example 2 – Inverse Modeling of Air Pollution Data . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

8 Using Fuzzy Logic to Quantify Climate Change Impacts on Spawner–Recruitment Relationships for Fish from the North–Eastern Pacific Ocean

207 208 209 209 211 213 216 217

221

By D. G. Chen and J. R. Irvine (With 3 Figures) 8.1 8.2 8.3 8.4

Introduction . . . . . . . . . . . . . . . . Ricker Stock–Recruitment Model . . . . . Crisp Regime Stock–Recruitment Model . Fuzzy Logic Stock–Recruitment Model . 8.4.1 Fuzzy Knowledge Base . . . . . . . 8.4.2 The Fuzzy Reasoning . . . . . . . 8.4.3 Parameter Estimation . . . . . . . 8.5 Data Analyses and Model Comparison . . 8.5.1 Data Description . . . . . . . . . . 8.5.2 Model Comparison . . . . . . . . . 8.6 Conclusion . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

221 222 223 224 226 226 227 228 228 228 230 232

Part IV. Practical Decision Support 9 A Hybrid Case-Based Reasoning System for Supporting the Modeling of Estuaries

237

By S. Passone, P.W.H. Chung, and V. Nassehi (With 14 Figures) 9.1

9.2 9.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Background: Status and Threats to Estuaries and their Environment . . . . . . . . 9.1.2 Estuarine Management . . . . . . . . . . . . . 9.1.3 Estuarine Modeling . . . . . . . . . . . . . . . 9.1.4 Water Modeling and Artificial Intelligence . . . 9.1.5 Case-Based Reasoning for Estuarine Modeling The CBEM Architecture . . . . . . . . . . . . . . . . The CB Module . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . 237 . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

238 239 240 241 241 242 243

x

Contents 9.3.1 9.3.2

Description Component . . . . . . . Retrieval Component . . . . . . . . . 9.3.2.1 Similarity Score . . . . . . 9.3.2.2 Similarity Measurements . 9.4 The GA Module . . . . . . . . . . . . . . . 9.4.1 Chromosome Representation . . . . 9.4.2 Fitness Function . . . . . . . . . . . 9.4.3 Initial Population . . . . . . . . . . . 9.4.4 GA Operators . . . . . . . . . . . . 9.4.5 Selection . . . . . . . . . . . . . . . 9.4.6 Crossover and Mutation . . . . . . . 9.4.7 Model Results with GA Calibration - Upper Milford Haven Estuary . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

243 245 247 249 251 252 254 255 256 256 256

. . . . . . . . . . . . . . 256 . . . . . . . . . . . . . . 258 . . . . . . . . . . . . . . 260

10 Credal Networks for Hazard Assessment of Debris Flows

263

By A. Antonucci, A. Salvetti, and M. Zaffalon (With 1 Figure) 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Debris Flows: Triggering Factors and Mobilization . . 10.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2.1 Credal Sets and Probability Intervals . . . . 10.2.2.2 The Imprecise Dirichlet Model . . . . . . . . 10.2.2.3 Credal Networks . . . . . . . . . . . . . . . . 10.3 The Credal Network for Hazard Assessment of Debris Flows 10.3.1 Network Description . . . . . . . . . . . . . . . . . . . 10.3.2 Network Quantification . . . . . . . . . . . . . . . . . 10.4 Using the Model to Support Domain Experts . . . . . . . . . 10.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Description . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Results and Discussion . . . . . . . . . . . . . . . . . . 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Multi-Criteria Decision Support for Integrated Technique Assessment

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

263 264 264 266 266 267 268 269 269 273 275 277 277 277 280 281 285

By J. Geldermann and 0. Rentz (With 6 Figures) 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.2 Determination of BAT . . . . . . . . . . . . . . . . . . . . . . . . . . 286 11.3 Case Study in the Sector of Industrial Paint Application . . . . . . . 288

xi

Contents 11.3.1 Approaches for Multi-Criteria Decision Making based on the Ranking of Discrete Alternatives . . . . . . . . 11.3.2 Simple Additive Ranking . . . . . . . . . . . . . . . . 11.3.3 Application of Simple Additive Weighting (SAW) . . . 11.3.4 Application of PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation) . . . 11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 289 . . . . 292 . . . . 293 . . . .

. . . .

. . . .

. . . .

295 299 301 302

Glossary

305

Subject Index

307

List of Contributors Jean Pierre Aubin Réseau de Recherche Viabilité, Jeux, Contrôle Université Paris-Dauphine, 75005 Paris, France [email protected] N´ uria Agell ESADE University Ramon Llull 08034 Barcelona, Spain [email protected] Cecilio Angulo Technical University of Catalonia 08800 Vilanova i la Geltr´ u, Spain [email protected] Alessandro Antonucci Istituto Dalle Molle di Studi sull’Intelligenza Artificiale 6928 Manno (Lugano), Switzerland [email protected] Thomas Bruckner Technical University of Berlin 10587 Berlin, Germany [email protected] Andreu Catal` a Technical University of Catalonia 08800 Vilanova i la Geltr´ u, Spain [email protected]

Ding-Geng Chen Department of Mathematics and Statistics South Dakota State University Brookings, South Dakota 57007, USA [email protected] Paul W. H. Chung Loughborough University Loughborough LE11 3TU, United Kingdom [email protected] Klaus Eisenack Potsdam Institute for Climate Impact Research 14412 Potsdam, Germany [email protected] Tatiana Ermolieva International Institute for Applied Systems Analysis 2361 Laxenburg, Austria [email protected] G¨ unther Fischer International Institute for Applied Systems Analysis 2361 Laxenburg, Austria [email protected]

xiv Jutta Geldermann French-German Institute for Environmental Research University of Karlsruhe 76187 Karlsruhe, Germany [email protected] Luis Gonz´ alez University of Sevilla 41005 Sevilla, Spain [email protected] Sue Ellen Haupt Pennsylvania State University Applied Research Laboratory and Meteorology Department State College PA 16804, USA [email protected] Hermann Held Potsdam Institute for Climate Impact Research 14412 Potsdam, Germany [email protected] James R. Irvine Fisheries and Oceans Canada Nanaimo BC V9T 6N7, Canada [email protected] Elmar Kriegler Potsdam Institute for Climate Impact Research 14412 Potsdam, Germany [email protected] and Department of Engineering and Public Policy Carnegie Mellon University Pittsburgh, PA 15213, USA [email protected]

List of Contributors J¨ urgen Kropp Potsdam Institute for Climate Impact Research 14412 Potsdam, Germany [email protected] Matthias K. B. L¨ udeke Potsdam Institute for Climate Impact Research 14412 Potsdam, Germany [email protected] Vahid Nassehi Loughborough University Loughborough LE11 3TU, United Kingdom [email protected] Michael Obersteiner International Institute for Applied Systems Analysis 2361 Laxenburg, Austria [email protected] and Institute for Advanced Studies 1060 Vienna, Austria [email protected] Sara Passone Loughborough University Loughborough LE11 3TU, United Kingdom [email protected] Gerhard Petschel-Held Potsdam Institute for Climate Impact Research 14412 Potsdam, Germany [email protected] Otto Rentz French-German Institute for Environmental Research

xv

List of Contributors University of Karlsruhe 76187 Karlsruhe, Germany [email protected] Francisco Ruiz ESADE University Ramon Llull 08034 Barcelona, Spain [email protected] Patrick Saint-Pierre Laboratoire d’Applications des Systèmes Tychastiques Régulés Université Paris-Dauphine 75116 Paris, France [email protected] Andrea Salvetti Andrea Salvetti Canton Ticino Land Department 6500 Bellinzona, Switzerland [email protected]

J¨ urgen Scheffran University of Illinois, ACDIS Urbana-Champaign, USA [email protected] Francisco Velasco University of Sevilla 41005 Sevilla, Spain [email protected] Franz Wotava Graz University of Technology Institute for Software Technology 8010 Graz, Austria [email protected] Marco Zaffalon Istituto Dalle Molle di Studi sull’Intelligenza Artificiale 6928 Manno (Lugano), Switzerland [email protected]

Introductory Remarks Human-induced environmental change and the degradation of natural resources affect the quality of life and the viability of social and natural systems in many regions of the world. The impacts of climate change, overfishing, deforestation, scarcity of water, food and energy create hazards and threaten the security of individuals and states. To prevent and manage these risks requires cross-cutting and novel approaches that deepen the understanding of the roots and impacts and support sustainable decision making. In fact, a so-emerging transdisciplinary enterprise will be increasingly interested in all aspects of systems dynamics, i.e. primary causes and precursor signals, as well as generic repercussions, for instance, the consequences of ill-managements Understanding sustainability is vital to resolve and manage many of today’s problems, on global as well as local scales. Sustainability science is an emerging field of research that comprises concepts and methodologies from different disciplines in a problem-oriented manner. Research efforts are often concentrated in a variety of sectoral domains. The heterogeneity of scientific tasks involved here and the complexity of environmental and social systems call for specific research strategies which are generally a compromise between high-precision analysis and educated guesswork. For understanding of global change, which embraces a variety of processes on several scales, information needs to be refined and compressed rather than amplified. This enterprise will benefit from existing methodologies developed over recent decades in theoretical physics, complex systems analysis, operations research, artificial intelligence, stochastics and computer sciences. Increasingly these techniques are used in the domain of global change and sustainability science, and appropriate tools are applied or potentially applicable to stakeholder assessment in environmental management. Whether the methods are appropriate to assess and address the transdisciplinary problems are addressed, depends on the following questions: • Are these techniques suitable to open new ways of integration? • Which modeling strategies are appropriate to cope with complexity and uncertainty? • Are new model concepts able to supply knowledge for practical applications?

2

Introductory Remarks

After this short introduction, let us now briefly discuss how this book is organized. Structured in four parts the book selects several advanced methods with applications in relevant fields of sustainability science which are summarized as follows. Uncertainty and Viability In the first part uncertainty and viability are discussed, which are both cross-cutting issues in sustainability science. The lack of knowledge about quantities or causal relationships and the difficulty in quantifying epistemic uncertainty is common to many aspects of global environmental change. In situations of deep uncertainty it is often impossible to fulfill the rather demanding informational requirements for assessing probabilities on the state of the world, for projecting possible future consequences and for identifying favorable actions under uncertainty. Probability assessments which are based on incomplete and poor information might lead to a controversial policy advice. In many situations, indecision, arbitrariness, or precaution arise as a reality of everyday life, but alien to conventional decision theories based on probabilistic information. Elmar Kriegler, Hermann Held and Thomas Bruckner argue that a probabilitybased uncertainty assessment of global environmental change can be inadequate (Chapter 1). To assess climate protection strategies under ambiguity about catastrophic consequences, they address uncertainty that cannot be reduced to a single probability measure, and discuss the challenge of decision situations entailing potentially catastrophic consequences. They employ decision criteria under ambiguity about catastrophic consequences and apply imprecise probability concepts for quantifying ambiguity. Based on the quantitative assessment of the ambiguity about climate sensitivity they estimate the uncertainty about global warming and economic welfare losses entailed by four different CO2 stabilization policies. Another innovative approach is presented by Jean-Pierre Aubin and Patrick Saint-Pierre who provide an introduction to viability theory and the management of renewable resources, with a focus on fishery (Chapter 2). Viability theory explains the evolution of a control system, governed by a non-deterministic dynamics which are subject to viability constraints and regulated by feedbacks that provide selection mechanisms for implementation. The theory of viable control is a useful instrument to design and control the complex interaction between the economic, environmental and political spheres in natural resource management. The impact of changing crucial couplings is studied to improve viability in resource networks, resolving conflict between environmental damage and expected gains from resource use. The question is raised how to modify a given dynamical system governing the evolution of the signals, the connectionist tensors and the coalitions in such a way that the architecture remains viable. This would allow stakeholders to identify controls necessary to stay within sustainable limits, e.g. for fish catch, and to avoid non-viable regions in which fish resources decline or fishery becomes unprofitable, taking into account uncertainties about fish stocks and catch efficiency.


3

Qualitative Methods Complexity is an ingredient aspect of sustainability science. Many systems are characterized by an exceptional dynamics involving a large number of system parts and a multitude of (non-linear) interrelations between the socio-economic and the natural sphere. Thus it is essential to understand the most relevant mechanisms, driving forces, and/or feedback loops of systems, at least on an intermediate scale of complexity. Qualitative methods are appropriate to represent heterogeneous data and dynamic behavior under uncertainty, integrating knowledge from different disciplines on an aggregated level. Qualitative differential equations (Qdes) represent the qualitative system properties in dynamic systems under uncertainty. Instead of an exact functional and numerical specification, it is only necessary to formulate qualitative if-then relationships to classify dynamical systems and solutions with similar properties and formulate rules about the interrelationship of nature and human action which are robust to uncertainties and parameter changes. Chapter 3 by Klaus Eisenack, Matthias K.B. L¨ udecke, Gerhard Petschel-Held, J¨ urgen Scheffran, and J¨ urgen P. Kropp introduces qualitative modeling techniques to assess patterns of global change. Deriving archetypes of social-ecological systems as building blocks of society-nature interaction, the authors demonstrate how these can be used in multiple case studies to pose relevant questions across different domains of sustainability science. Describing “typical patterns” (syndromes) is an essential technique to cope with complex situations and is also part of the learning process in human-environment interaction. The syndrome approach describes patterns of global environmental change, based on qualities and dynamic interactions, perceived as relevant by stakeholders and decision-makers. In a more general context of global change research the syndrome approach was suggested as an instrument to analyze complex transsectoral phenomena such as “dynamical degradation patterns” that characterize contemporary human–environment interactions across the planet. Examples are the so-called Sahel, Dust-Bowl, and Overexploitation Syndromes. An example is the overexploitation of marine resources, a pattern associated with the loss of marine biodiversity, overcapitalization, and declining coastal economies. The syndrome approach can help to identify hot spots and key mechanisms as a precondition to design successful management regimes. By clustering system patterns and policy options, qualitative approaches can be useful in stakeholder dialogues. New methods for identifying classes of similar cases are also valuable to characterize patterns of global environmental change and to develop sustainability indicators. Cecilio Angulo, Luis González, Francisco Ruiz, Andreu Català, Francisco Velasco, and N´ uria Agell analyze a sustainability indicator by means of qualitative kernels (Chapter 4). They use new inference techniques based on kernel and statistical learning methods, shown to be effective in machine learning theory, and analyze this highly uncertain and subjective information using new artificial intelligence techniques based on collective and interval reasoning. These make it possible to work on any data space and seek similarities in data by using kernels, cluster-

4


ing of data and the distances between clusters. This topic is of high relevance for sustainability issues, especially when data are heterogeneous (e.g. categorical, ordinal and numerical). Either qualitative variables are converted to interval values, or intervals obtained from numerical variables are converted to qualitative variables. Both approaches are studied for several examples, and an initial application to the sustainability data of the Town Council of Vilanova i la Geltr´ u is performed. Feasibility analysis uses a soft-computing tool which allows to qualitatively assess the degree of citizen satisfaction. In the following Chapter 5 a qualitative approach for the natural representation of models is given by the methodology of constraint satisfaction problems (Csps), which in computer science are used to represent the knowledge for solving different tasks including configuration, diagnosis, and the computation of explanations. Csps comprise a set of variables and a set of constraints where the scope of each constraint is given by a subset of the variable set and a solution is an assignment of values to the variables such that no constraint is violated. Csps are in wide-spread use in many fields of applications, including bioinformatics, knowledge-based configuration, scheduling, diagnosis, user-and graphical interfaces, and intelligent information systems. Franz Wotawa introduces the basic concepts behind Csps and their efficient solution with regard to modeling and explanation in sustainability science. Explanations can be used both for deriving decisions and explaining solutions, for example, in tutoring systems which are important in sustainability science in order to ease the understanding of complex processes and relationships. The introduced concepts and definitions are illustrated for a small solar-panel water-pump system. Stochastic and Fuzzy Approaches The losses from natural and man-made catastrophes are rapidly increasing and will likely be further exacerbated by the increasing frequency and severity of weather– related extreme events due to global climate change which is expected to lead to more frequent economic and social shocks at national and regional levels with consequences to the global economy. The impacts of disasters have spatial profiles and determine prospects for sustainable regional agricultural and rural development. Catastrophic risks create new scientific problems requiring integrated approaches, new concepts and tools for land use planning. Catastrophic risk management is a complex transdisciplinary task requiring knowledge of environmental, natural, financial and social systems and a wider range of methodologies, including stochastic and statistical approaches. Models enable the simulation of possible catastrophes and estimation of potential damages to facilitate mitigation and adaptation programs. In Chapter 6 Tatiana Ermolieva, G¨ unther Fischer and Michael Obersteiner present an integrated approach of catastrophic risk management based on data from geographic information systems (Gis). The methodology enables the analysis of the spatial and temporal heterogeneity of various agents (stakeholders) and offers solutions for coherent, comprehensive and robust policy responses. Using the


5

spatial adaptive Monte Carlo (stochastic) optimization procedure allows to analyze a variety of complex interactions between decisions and risks in order to find robust optimal portfolios of risk management measures for decreasing regional vulnerability with regard to economic, financial, and human losses. The model addresses the specifics of catastrophic risks, including the lack of information, the need for longterm perspectives and geographically explicit models, and a multi-agent decision making structure. The approach is illustrated with a case study of earthquake risk management in the Tuscany region of Italy. Genetic algorithms (Gas) that mimic natural selection and genetics are a useful tool for solving problems requiring search and optimization. Sue Ellen Haupt introduces Gas (Chapter 7) and their applications in the environmental sciences. The purpose oft his paper is to introduce the elements of Gas and present a couple of simple examples of how a Ga might be applied to a problem. The key to using Gas is to pose the problem as one in optimization, such as the inverse models used in the environmental sciences. Other problems can be manipulated into optimization form by careful definition of the cost function, so that even nonlinear differential equations can be approached using Gas. Global climate variation poses enormous challenges to the assessment of marine ecosystem changes and to the management of fishery resources. The successful implementation of sustainable fisheries policies requires a good understanding of relationships between numbers of spawners and subsequent recruitment (cf. Chapter 8). Non-traditional analytical techniques, including artificial intelligence methodologies, such as fuzzy logic, offer significant advantages over traditional statistical techniques in analyzing these relationships. Din G. Chen and James R. Irvine use fuzzy logic to quantify climate change impacts on spawner-recruitment relationships for fish from the North-Eastern Pacific Ocean. They utilizes a continuous membership function to provide a rational basis to categorize spawner-recruitment data of Pacific salmon, herring, and halibut. Because fuzzy logic models address uncertainty better than traditional approaches, they help to understand factors influencing spawner-recruitment relationships, and thereby manage fisheries more effectively. Practical Decision Support The fourth part of the book is directly devoted to decision makers. They are confronted with difficult problems of environmental change across different spatial and temporal scales. Decision analysis provides a wealth of methods and tools to support the ranking and selection of actions from a set of options, following certain rules, preferences and criteria. The scope and scale of many problems require the involvement of various experts with different backgrounds and communication between modelers and policy makers in designing different natural resource management schemes.

6


Sara Passone, Paul W. H. Chung and Vahid Nassehi present a hybrid case-based reasoning system for supporting the modeling of estuaries that can be utilized by users from different backgrounds (Chapter 9). The system links a case-based reasoning scheme, a genetic algorithm and a library of numerical estuarine models, guiding a general user in selecting the model that best matches his/her goals and the nature of the problem to be solved. The case-based module returns possible solutions from the system’s memory, adapting past experience to new problems. The genetic algorithm component estimates a valid set of hundreds of model parameters to suit the particular estuarine environment. A case study is given to demonstrate the system’s ability to provide the user with an appropriate estimation of the available model strategies and the robustness of the designed routine for adjustment of the model parameters. This system facilitates understanding of complex concepts and communication between the users so that time and effort needed in a multidisciplinary work of this nature is significantly reduced. In order to close the gap between theory and practice it seems necessary to merge theoretical models with qualitative human knowledge. It is particularly important to identify a framework for credibly dealing with uncertainty that is flexible enough to cope with complex scenarios and different sources of information, such as theoretical models, historical data, and expert assessment. Such a framework is given by credal networks, an imprecise probability model that extends Bayesian networks to manage sets of probability mass functions. Alessandro Antonucci, Andrea Salvetti and Marco Zaffalon present a credal network model developed to characterize and assess the debris flow hazard in the Ticino canton, southern Switzerland (Chapter 10). Debris flows are among the most dangerous natural hazards, worldwide claiming hundreds of lives and millions of dollars in property loss every year. Significant scientific and engineering advances in understanding the processes and in the recognition of debris flow potential have been achieved, several experiments and field investigations have been conducted to derive threshold values for debris flow initiation. Many aspects are still poorly understood and rely on human expertise. The introduced model aims at supporting experts in the prediction of dangerous events of debris flow. It is based on the qualitative representation of causal knowledge and on imprecise probability quantifications of uncertainty. The causal structure is represented by a directed graph, connecting the triggering factors for debris flows. Uncertainty is quantized by probability intervals, showing how intervals are obtained from historical data, expert knowledge, and physical theories about debris flows. Credal networks allow to represent theoretical and empirical knowledge coherently as a single model of debris flow hazard. Finally, Jutta Geldermann and Otto Rentz introduce methods of multi-criteria decision support for integrated technique assessment (Chapter 11). In the European Union, the Integrated Pollution Prevention and Control Directive (IPPC-Directive 96/61/EC) obliges the member states to take an integrated approach to the licensing of environmentally relevant installations. The “best available techniques” (Bat)

7


serve as a basis for the determination of reference values for emission limits and for the granting of permits for installations. A case study from the sector of industrial coating is presented, and the multi-criteria approaches Maut (multi-attribute utility theory) and Promethee (outranking approach) are applied exemplarily. The results are discussed and conclusions are derived. We also thank first and foremost the authors for their cooperation. We also to thank Nova Science Publishers for their help during the preparation of this book. This volume aims at presenting advanced methods and techniques to make them available to a wider scientific community involved in global change and sustainability research. We hope that this monograph can be used as a textbook or reference source for graduate students, teachers at universities, and experts in authorities facing the challenge to develop problem-oriented solutions for environmental problems. The contributions describe novel schemes to study the relationship between the socioeconomic and the natural sphere and/or the social dimensions of climate and global change. The methodological approaches can be useful in the design and management of environmental systems, for policy development, environmental risk reduction, and prevention/mitigation strategies. In this context a variety of environmental and sustainability aspects could be addressed, e.g. changes in the natural environment and land use, environmental impacts on human health, economics and technology, institutional interactions, human activities and behavior. We hope that this book can encourage rethinking of common practices in environmental sciences and provide a transdisciplinary and comprehensive surplus. *

Potsdam, Germany Urbana-Champaign, USA May 2006

*

*

J.P. Kropp J. Scheffran

Part I

Uncertainty and Viability

CHAPTER 1

Climate Protection Strategies under Ambiguity about Catastrophic Consequences Elmar Kriegler, Hermann Held, and Thomas Bruckner

1.1

Introduction

Human-induced environmental change is altering the flow of natural services and hazards to societies in many regions of the world. Meanwhile, the human impact on the environment continues to grow on the global scale, as the increasing land and water use, and the rising emissions of greenhouse gases clearly show. Decision makers are confronted with the difficult problem to address environmental change on spatial scales from local to global and temporal scales from years to centuries. As the spatial and temporal scales increase, potentially catastrophic consequences, represented here by fundamental changes to the flow of natural services and hazards, have to be considered for a growing number of economic sectors and societies, while at the same time the scientific uncertainty about the impact chain grows considerably. Epistemic uncertainty, such as lack of knowledge about a quantity or causal relationship, is pervasive in the assessment of global environmental change. Probability theory faces some severe problems in quantifying epistemic uncertainty and deriving policy implications from it. This becomes particularly apparent in situations with very poor information. In these situations it is often impossible to fulfill the rather demanding informational requirements for assessing probabilities on the actual state of the world. If one maintains to use probabilities for projecting possible future consequences and identifying favorable actions under uncertainty, additional information that is not backed by evidence has to be added deliberately or – which is worse – unwittingly. Such probability assessments might lead to a controversial policy advice, if stakeholders felt that the risk of recommended actions and the opportunity of discarded actions is undervalued. At the root of this controversy is often a discomfort with the underlying probability assessments as accurate representations of the state of information. Different stakeholders may tend to different probability assessments. Others may reject the use of probabilities in light of the poor state of information. In such situations, indecision, arbitrariness, or precaution frequently arise. These types of behavior are a reality of everyday life, but alien to conventional decision theories based on probabilistic information. In such situations, as we will argue in this Chapter, a probability-based uncertainty assessment can already miss the basic structure of the environmental decision making problem in question. Two characteristics of global environmental change

12

E. Kriegler et al.

work hand in hand to impair the applicability of conventional probability-based policy analyses: • Global environmental change ranges across a multitude of sub-systems, where some sub-systems and particularly the interactions between the sub-systems are poorly understood. Thus, the uncertainty about the subsystems and their interactions will be diverse, e.g., depending on whether natural or socioeconomic systems are concerned, whether the uncertainty is aleatory or epistemic in nature, and whether the scientific knowledge base about a particular subsystem is weak or strong. It seems hardly feasible at current to conduct an environmental policy analysis solely on the basis of a single probability distribution over the set of possible hypotheses about global environmental change. • The range of possible paths global environmental change might take and possible consequences environmental policies might bring about is very large. If utility differences among consequences are on the order of magnitude that is handled by humans on a daily basis, informational inaccuracies in probabilitybased assessments might be accepted with ease (“best-guessing”). However, if consequences entail catastrophic losses in excess of the usual range of losses experienced by individuals, societies, and economies during their life span, this will no longer be the case. Guided by these intuitions, we assess decision situations under ambiguity,, i.e., uncertainty that cannot be reduced to a single probability measure, from a rather general point of view in the next section. Section 1.3 discusses the additional challenge that comes with decision situations entailing potentially catastrophic consequences. Section 1.4 outlines a prototypical analysis of climate protection strategies to which the insights from the general discussion in the previous sections will be applied. The decision criteria under ambiguity about catastrophic consequences that we will employ in the course of the analysis are discussed in Section 1.5. Section 1.6 recaptures some imprecise probability concepts that we employ for quantifying ambiguity. In Section 1.7, we detail the quantitative assessment of the ambiguity about climate sensitivity. In Section 1.8 and 1.9, we estimate the uncertainty about global warming and economic welfare losses entailed by four different CO2 stabilization policies. Section 1.10 integrates these results into a policy analysis of the four stabilization policies under consideration.

1.2

Decision Making under Ambiguity

Conventional policy analyses under uncertainty employ an expected utility criterion to discriminate between alternative courses of action. This decision-analytical framework is justified by Savage’s theory of choice under uncertainty (Savage 1954).

Climate Protection Strategies under Ambiguity

13

In a nutshell, Savage’s theory states the following. For any preference order over acts f : S → X from the set of possible states of the world S to the set of possible consequences X which fulfills seven “rationality” axioms, there exists a unique probability measure P over events A ⊆ S and a utility function u : X → R unique up to an affine transformation, so that Rthe preference ordering can be represented by the expected utility criterion E[f ] = S u(f (s))p(s)ds. Thus, if we want to argue against the applicability of the conventional approach to particular cases of prescriptive policy analysis, we first have to dispute the general claim that the seven Savage axioms provide a necessary condition for “rational” behavior. Ellsberg (1961) has shown in a famous experiment that ambiguity, i.e., lack of probabilistic information, can lead to behavior contradicting Savage’s theory of choice. If ambiguity comes into play, the decision maker might exhibit a preference order over R acts f : S → X that is not compatible with an expected utility criterion E[f ] = S u(f (s))p(s)ds under any combination of a probability distribution p over S and a utility function u : X → R. To see this let us consider an Ellsberg-type experiment in detail. It consists in presenting an urn with 90 colored balls to a set of subjects. 30 balls are known to be red, while the other balls are colored black and yellow in unknown proportions. The subjects are offered a prize if they manage to obtain the color of their choice with a random draw of a single ball from the urn. It turns out that the majority of subjects will pick the color red, where they know the probability of winning the prize beforehand. They avoid the other two colors whose proportion of balls are surrounded by ambiguity. If this preference was to be represented by an expected utility criterion, the subjective probabilities of drawing a black ball and of drawing a yellow ball, respectively, would need to be lower than the probability of drawing a red ball from the urn (p(red) = 1/3). Obviously, this is impossible. There exists no pair of subjective probability weights for obtaining a black and yellow ball, respectively, that would make both colors less attractive than red. Hence, this instance of ambiguity aversion is incompatible with the decision theory presented by Savage (1954). However, it can hardly be maintained that the revealed preference of the subjects is “irrational”. Ellsberg’s experiment shows that it can be misleading to eliminate the presence of ambiguity by adding information that is not backed by evidence, and then to apply a conventional decision analysis based on classical probabilities to the problem in question. It can be argued that concepts like the precautionary principle are motivated by the presence of ambiguity. The European Commission, for instance, states in a guideline on the applicability of the principle that a recourse to it presupposes “a scientific evaluation of the risk which because of the insufficiency of the data, their inconclusive or imprecise nature, makes it impossible to determine with sufficient certainty the risk in question” (European Commission 2000, p. 15). In the literature, various attempts have been made to formalize the precautionary principle by drawing on the difference between ambiguous uncertainty and probabilistic uncertainty (see, e.g., Chevé and Congar 2003).

14

E. Kriegler et al.

In the wake of Ellsberg’s experiment, several axiomatic decision theories were presented that can accommodate ambiguity aversion (e.g., Gilboa and Schmeidler 1989). These theories establish uncertainty representations that generalize classical probability1 . It has been shown, for instance, that preference orders fulfilling some weaker requirements than in Savage’s theory or the theory of horse-lotteries by Anscombe and Aumann (1963) can be represented by a utility function u : X → R unique up to an affine transformation, a convex set of probabilities P, R and a decision criterion based on the minimum expected utility E[f ] = minp∈P S u(f (s))p(s)ds (Savage framework: Casadesus-Masanell et al. 2000; Anscombe-Aumann framework: Gilboa and Schmeidler 1989). In order to see the ability of the generalized uncertainty representations to capture ambiguity, consider again the Ellsberg experiment as described above. In contrast to Savage’s theory, the subjects are not required to base their decision about the winning color on a single assumption about the proportion of black and yellow balls in the urn. Instead, they can account for all possibilities between the two extremes that no yellow balls or no black balls are contained in the urn. This gives rise to a convex set of probabilities p ∈ [0, 2/3] for obtaining a yellow and black ball, respectively, from a random draw. In this situation, the minimum expected utility criterion recommends to choose the color red as winning color, since the chance of winning the prize can be no lower than a third (in contrast to yellow and black, where it can be zero). This result agrees with the behavior of the majority of subjects in the Ellsberg experiment. Hence, decision theories based on a generalization of classical probability can accommodate ambiguity aversion. Such generalized uncertainty representations come in several forms, e.g. convex sets of probabilities (Levi 1980), lower previsions (Walley 1991), and interval probabilities (Kuznetsov 1991; Weichselberger 2000). The various representations are summarized under the label imprecise probabilities. Imprecise probabilities have become a subject of active research during the last decade. It is beyond the scope of this Chapter to give an overview over this young and rapidly developing field (for an introduction, see the web-site of the Society of Imprecise Probabilities and Their Applications at http://www.sipta.org). In Section 1.6, we will only recapture the basic notions that we employ in our analysis. For the time being, it will suffice to note that imprecise probabilities of whatever nature share the following characteristics: they yield a lower and upper bound on the probability that an event A ⊂ S in the set S of possible states of the world will occur, and a lower and upper bound on the expected value of a gamble Z := u ◦ f : S → R over S. When lower and upper bounds fall onto each other, the classical case of probability theory obtains. When the lower bound is strictly smaller than the upper bound, the state of information is fraught with ambiguity. Besides the provision of richer decision theories, a particularly important asset of 1 In the following, we will call a probability measure in the usual sense, i.e., a set function P (.) satisfying Kolmogorov’s axioms, a classical probability.


15

imprecise probability lies in the fact that it allows to capture the state of complete ignorance in a satisfactory manner. It is modeled by the vacuous lower and upper / (Walley 1991, Chapter probability P (A) = 0 and P (A) = 1 , ∀ A ⊂ S , A 6= ◦ 2.9.1) (regarding further applications of Walley’s concept, cf. Chapter 10). A satisfactory model of complete ignorance is an important prerequisite for quantifying poor states of information without having to add additional information not backed by evidence. As pointed out in the introduction, probability theory can face severe difficulties here. Specifying classical probabilities requires a minimum level of information that is often not available in many problems of global environmental change. These difficulties surface, e.g., when constructing Bayesian priors, or when deriving an uncertainty representation from (in)equality constraints on the probability of a selected set of events (Walley 1991, Chapter 5). Moreover, imprecise probabilities are flexible enough to encompass not only classical probabilities as a special case, but also interval estimates. Such estimates are widely used in situations with large uncertainty. In its Third Assessment Report (Tar), the Intergovernmental Panel on Climate Change (Ipcc) has frequently assessed uncertainty in terms of intervals and scenarios. There is an ongoing debate whether the Ipcc should be more outspoken, and provide probability assessments for, e.g., future emissions trajectories. While some argue that this is what policy analysis currently needs (Schneider 2001; Dessai and Hulme 2003), others emphasize that this is not what a scientific inquiry can currently provide (Gr¨ ubler and Nakicenovic 2001). Imprecise probabilities might help to resolve the debate. On the one hand, they offer an opportunity to account for the poor state of information in quantitative terms. On the other hand, they can be utilized for policy analyses by drawing on formalized decision models like the one of Gilboa and Schmeidler. The Ellsberg experiment indicates that the admission of ambiguity can fundamentally alter the nature of the policy recommendation. Therefore, the presence of ambiguity should not be neglected in policy analyses of global environmental change.

1.3

Decision Making in the Face of Catastrophic Consequences

The ambiguity in our understanding of global environmental change is not the only challenge that associated policy analyses have to face. As pointed out in the introduction, they also have to take into account the possibility of catastrophic consequences, i.e., fundamental changes to the flow of natural services and hazards, that could put the very survival of societies and economic sectors into question. Climate change serves as a good example (see, e.g., Smith et al. 2001 and Alley et al. 2003 for overviews on potentially catastrophic climate change impacts). For instance, sea level rise due to thermal expansion of the world’s oceans and land ice melt threatens the existence of Pacific and Caribbean island states, and coastal communities

16

E. Kriegler et al.

throughout the world (Parry et al. 2001). On the other hand, stringent mitigation policies might tip the world economy into a prolonged recession aggravating distributional conflicts throughout the globe. Moreover, geo-engineering solutions for mitigating climate change might open another Pandora’s box. In summary: the stakes for climate policy are high. What does the possibility of such catastrophic “high impact” events imply for the application of decision-analytical frameworks to questions of global environmental change? Usually, policy analyses on the basis of expected utility theory either neglect high impact events entailing irreversible losses of existential services, or discount them with a very low probability. Independently of the fact that such probabilities in the tails of a distribution are frequently fraught with ambiguity, it is important to note that the expected utility criterion allows to compensate catastrophic risks with “every-day” benefits. This can lead to an insensitivity of the policy analysis to low-probability, high-impact events, which might miss a crucial aspect of the environmental policy situation (Chichilnisky 2000). Hence, the challenge posed by the possibility of catastrophic consequences concerns not only the representation of the uncertainty, but also the nature of the decision criterion. Assume that the uncertainty on the set S of possible states of the world is captured by a probability. Then, every act f : S → X can be represented by a probability distribution p : X → [0, 1] on the set of consequences X. Let there be a sub-set C ⊂ X of catastrophic consequences that cannot be compensated for. Then, it can be reasonable to prefer every act f (characterized by probability distribution p) that avoids the domain C with certainty (supp(p) ∩ C = ◦ / ) over every act g (characterized by q) that entails the possibility of catastrophic consequences (supp(q) ∩ C 6= ◦ / ). Such a preference order violates the continuity axiom that underlies most of the standard and non-standard decision frameworks. It has been shown already by Debreu (1954) that the continuity axiom is a necessary condition for preference orders to be represented by a real-valued functional. Thus, the possibility of catastrophic consequences does not only challenge the expected utility representation. It challenges any representation in terms of a real-valued functional. If the continuity axiom is completely dropped, lexicographic preferences emerge. So-called non-Archimedian decision theories, that drop or weaken the continuity axiom, are relatively rare in the literature. A rather general model termed expected utility on restricted sets (Eurs) has been proposed by Schmidt (1998). Eurs entails the existence of a real-valued indicator function for the probability distributions p : X → [0, 1] that are associated with the acts f . This indicator function constitutes the basis of a first-order decision criterion, while expected utility acts only as secondorder criterion in the lexicographic preference structure. Eurs can accommodate a variety of approaches for dealing with catastrophic consequences. One sensible choice for the first-order criterion is constituted by the probability P (C) of running into the catastrophic domain C ⊂ X, which we will call the probability of ruin in the following. This leads to a decision theory that advocates policies minimizing the


17

probability of ruin. Expected utility plays only a role in discriminating between the remaining minimal ruinous policies. If we intended to give more weight to expected utility considerations, we could choose a different first-order criterion constituted by an indicator function for the violation of a constraint on the maximum tolerable probability of ruin. This generates a model which discards all acts that lead to a violation of the constraint on the probability of ruin, and employs an expected utility criterion to rank the remaining acts. The discussions shows that a whole host of possible decision criteria unravels, once the continuity axiom is dropped in the presence of catastrophic events that cannot be compensated for.

1.4

Setup of a Prototypical Climate Policy Analysis

The rather theoretical discussion of suitable decision models under ambiguity about catastrophic consequences has major implications for applied policy analyses in the field of global environmental change. The expected utility model is not an appropriate choice in general. The insights from Eurs combined with decision models under ambiguity open up a host of alternative choices. Nevertheless, any decision model however adequate for the problem under consideration is useless to the practitioner, if methods are lacking to apply it to concrete policy problems. As imprecise probabilities are mathematically more involved than classical probabilities, and optimization algorithms for minimizing such things like the probability of ruin are far less developed than, e.g., stochastic expected value programming, this proves a major challenge. It is the purpose of this paper to piece together a conceptual and methodological framework enabling us to perform a prototypical assessment of climate protection strategies that accounts for ambiguity about the probability of ruin. As pointed out above, climate change constitutes a paramount aspect of global environmental change that requires us to act under both ambiguity and the possibility of catastrophic consequences. It is widely accepted by now that a discernible influence of anthropogenic emissions of greenhouse gases (Ghg) on the earth’s climate exists. Predominantly due to the combustion of fossil fuels, the concentration of carbon dioxide in the atmosphere, which is the largest contributor to the anthropogenic greenhouse effect, has risen from a value of 280 ppm (parts per million molecules) in the late 18th century to 372 ppm in the year 2002. In this analysis, we consider four so-called “Wigley-Richels-Edmonds (Wre)”scenarios for stabilizing the atmospheric CO2 concentration at levels of 450 ppm, 550 ppm, 650 ppm and 750 ppm, respectively (see Fig. 1.1a; Wigley et al. 1996). The Wre scenarios have been proposed on the basis of cost-effectiveness considerations. They constitute our set of potential mitigation policies to be decided upon, i.e., the set of acts f : S → X. Following Cubasch and Meehl (2001), we assume that the emissions trajectories of aerosols and Ghg other than CO2 are identical to the Sres A1B scenario until 2100 (Nakićenović and Swart 2000), and stay constant

18

E. Kriegler et al.

Figure 1.1: The four Wre policies considered in this analysis: (a) atmospheric CO2 trajectories, (b) radiative forcing trajectories.

thereafter. The resulting radiative forcing of the earth system was derived from standard parameterizations employed by the Intergovernmental Panel on Climate Change (Ipcc) in its Third Assessment Report (Tar) (Fig. 1.1b). The Wre mitigation policies f : S → X play out in an enormously complex set S of possible states of the world. For the sake of simplicity, we will consider only two important dimensions of S. Concerning the consequence for climate, the amount of future warming will depend strongly on climate sensitivity, which is a measure of the response of global mean temperature to a change in the radiation balance of the earth system (see Section 1.7). Concerning the consequence for the economy, the mitigation costs of the Wre policies will be influenced by a myriad of uncertain factors, e.g., the efficiency of international climate policy regimes, the rate of technological progress in the renewable energy sector, the costs of carbon-capturing and sequestration technologies, to name just a few. In this analysis, we will not try to explicitly resolve these factors and the associated uncertainties. Instead, we will perform a literature survey in Section 1.9 to assess the range of estimates about mitigation costs. The set of consequences X of the Wre mitigation policies is of similar complexity as the set S of possible states of the world. In our decision analysis, we will account only for two categories of consequences. 1. The rise in global mean temperature in the year 2300 relative to the preindustrial value: T (2300) provides an indication of the amount of global mean warming that will be inflicted by the Wre policies. The stabilization of the atmospheric CO2 concentration is achieved as late as 2200. Due to the slow response time of the world’s oceans, the temperature will continue to rise thereafter for centuries. Thus, T (2300) does not represent the total amount of warming that will be realized. However, it will incorporate the major fraction of the overall warming commitment. An even longer time horizon would be hardly compatible with the time scales that are relevant for socio-economic systems.


19

We delimit the onset of the catastrophic domain C by a guardrail T † (2300), beyond which it is assumed that climate change reaches a dimension threatening the very survival of a significant portion of present-day societies, economies, and ecosystems, respectively. We consider two choices T † (2300) = {2 ◦ C, 4 ◦ C}. Restricting the total amount of climate change to 2 ◦ C corresponds to a recommendation of the German Advisory Council on Global Change (cf. WBGU 1995). Beyond 2 ◦ C (and to a limited extent also below 2 ◦ C) extreme weather events are very likely to cause severe droughts and floods with higher frequency, a portion of unique systems might suffer extinction, and intolerable net negative impacts on market sectors in many developing countries are likely to occur. The high value of 4 ◦ C marks a climate change, where the uncertainty in climate predictions becomes overwhelming, and large climate discontinuities are increasingly plausible (Smith et al. 2001). 2. The economic costs of mitigation in terms of percentage losses of gross world product (Gwp): ∆Gwp usually captures the time-discounted present value economic loss over the 21st century. Another option would be to consider the maximum percentage loss of Gwp inflicted during some decade in the 21st century. Both measures may provide a rough estimate of the global economic losses inflicted by the requirement to mitigate greenhouse gases. Due to the large uncertainty about the multi-faceted consequences for regional and sectoral consumption that are associated with global Gwp losses, there exists no consensus about what should be regarded as a catastrophic Gwp loss. As part of a sensitivity analysis, we will consider a maximum and a present value Gwp loss ∆Gwp† of more than 2% and, for comparison, twice this value as unbearable. These constraints are purely illustrative and should not be interpreted as a policy proposal by the authors. In order to put these values into context, one may compare them to the current maximum environmental-related expenditures that amount to 2% of gross domestic product (Gdp) in the most active Oecd countries’ (OECD 2001). Note that the worst case cost for the Us Iraq engagement is assessed to be on the same order of magnitude (Kaysen et al. 2002). In contrast, the annual cost of World War II to the United States amounted to more than 30% of Gdp. The entire catastrophic domain C in the consequence space X is defined by a logical disjunction (OR) of the two guardrail constraints for the individual categories, i.e., (T (2300), ∆Gwp) ∈ C ⇔ T (2300) > T † (2300) ∨ ∆Gwp > ∆Gwp†

(1.1)

This specification means in particular that the decision maker expresses no preference of environmental over economic catastrophe, and vice versa. The guardrails are assumed to be chosen in such a way that the sense of disaster does not depend on whether an environmental or economic constraint has been violated. Thus, we

20

E. Kriegler et al.

can characterize various types of decision makers, which differ in their emphasis that they place on the economy vs. the environment, by different combinations of guardrails. In this analysis, we consider four alternative “cartoons” of potential decision makers: “Environmentalist”: ∆Gwp† = 4% , T † (2300) = 2 ◦ C †

T † (2300)

†

T † (2300)

(1.2)

=

4 ◦C

(1.3)

=

4 ◦C

(1.4)

“Alarmist”: ∆Gwp = 2% , T † (2300) = 2 ◦ C

(1.5)

“Capitalist”: ∆Gwp = 2% , “Easy rider”: ∆Gwp = 4% , †

It is interesting to note that this treatment of controversy about the catastrophic domain resembles the treatment of general system uncertainty in the cultural theory approach proposed by van Asselt and Rotmans (1996). In the context of our analysis, however, the cultural interpretation should not be taken too seriously. The consideration of four different guardrail combinations is a mere sensitivity study amounting, at best, to a scenario analysis.

1.5

Acting upon the Imprecise Probability of Ruin

In the spirit of the discussion in Section 1.2 and 1.3, we wish to perform a policy analysis of the alternative Wre mitigation strategies that accounts for ambiguity about the actual state of the world and the existence of catastrophic consequences. Consider a situation in which an imprecise probability on the set S of possible states of the world is specified, and a catastrophic domain C ⊂ X has been identified. Every measurable2 act f : S → X implies an imprecise probability of ruin, Ipr, which is bracketed by a lower and upper probability of running into the catastrophic domain C, i.e., Ipr(f ) = [ P (f −1 (C)), P (f −1 (C))]. Note that f −1 (C) ⊂ S is the sub-set of possible states of the world that entail catastrophic consequences, if action f is taken. Equivalently, the implications of an act f can be captured in terms of the imprecise probability to avoid ruin, Ipar(f ), i.e., ¯ = 1 − P (f −1 (C)) , P (f −1 (C)) ¯ = 1 − P (f −1 (C)) ] , (1.6) Ipar(f ) = [ P (f −1 (C)) where C¯ = S/C denotes the domain of non-catastrophic consequences. Since Ipr and Ipar are conjugate to each other, we restrict the discussion to Ipar in the following, with the analogous relationships for Ipr being implied by conjugation. Ipar provides a natural way to characterize the role of both ambiguity and the possibility of catastrophic consequences in the decision situation. We choose this indicator as the basis of our decision model. As pointed out in Section 1.3, 2

A function f : S → X is measurable FS /FX , i.e., on fields FS of S and FX of X, if for all B ∈ FX it is f −1 (B) = { x ∈ S : f (x) ∈ B } ∈ FS (Billingsley 1995). Every continuous function f : Rn → Rm , for instance, is measurable on the Borel field of the Euclidean space.

21


however, there exists a host of other choices that we will not explore any further here. We are not aware of any other application of the imprecise probability of ruin as basic decision indicator in the literature to date. In this sense, the setup of the prototypical climate policy analysis presented here provides a fresh perspective. It is important to note that Ipar is a set-valued quantity, and therefore does not imply a strict ordering of acts f : S → X. Several decision criteria under ambiguity have been discussed in the literature (see Troffaes 2004 for a review). They induce a (partial) ordering on a set of gambles Gf = u ◦ f : S → R, with u : X → R being a utility function, by judging the imprecise expectation about the outcome of the gambles. If the uncertainty is described by a closed convex set of probabilities3 P on the set S of possible states of the world, the imprecise expected utility of Gf constitutes the interval [E(Gf ), E(Gf )], with Z Z Gf (s) p(s) ds , Gf (s) p(s) ds (1.7) E(Gf ) = max E(Gf ) = min P ∈P

P ∈P

S

S

the lower and upper expected utility, respectively. The decision criteria can be readily transferred to the case of Ipar by making use −1 (C) ¯ = {s ∈ of the concept of an indicator gamble If −1 (C) ¯ : S → {0, 1} for the set f S|f (s) 6∈ C} of all ‘good’ states of the world in which the act f can avoid the catas−1 (C) ¯ and nothing in the trophic domain. If −1 (C) ¯ pays one unit of utility if s ∈ f −1 (C)) ¯ and E(I −1 ¯ ) = P (f −1 (C)) ¯ remaining case. Since it is E(If −1 (C) ¯ ) = P (f f (C) (Walley 1991, Section 2.7), the use of indicator gambles transfers decision criteria judging the imprecise expected utility of an act to judgments that are based on its imprecise probability to avoid ruin. In the following, we discuss several criteria that may be used to establish a (partial) order on a set of policies under ambiguity about catastrophic consequences. Interval dominance: A natural choice for ordering interval-valued quantities is the interval order. An interval dominates another interval iff the lower bound of the former is strictly larger than the upper bound of the latter. The application of interval dominance as a decision criterion under imprecise information was discussed in Zaffalon et al. 2003, Section 2.3.3, pp. 68-69 (and references therein). In our context, an act g is called undominated in the interval sense, iff the associated upper probability of avoiding ruin is larger or equal the lower probability of avoiding ruin under any of the remaining acts f , i.e., ∀f ⇔ ∀f 3

¯ ≥ P (f −1 (C)) ¯ P (g−1 (C)) ∃ P1 , P2 ∈ P

(1.8)

¯ ≥ P2 (f −1 (C)) ¯ . P1 (g−1 (C))

We restrict the discussion to convex sets of probabilities P that are closed in order to avoid the complication of boundary cases, where the lower (upper) expected value of a gamble is reached only as an infimum (supremum) over the set of probabilities, but not realized by any probability P ∈ P.

22

E. Kriegler et al.

where P is the closed convex set of probabilities that describes the uncertainty on the set S of possible states of the world (cf. Eq. 1.7). It is important to note that interval dominance constitutes only a partial ordering criterion. In general, there will be many acts that are undominated in the interval sense. Interval dominance is the weakest criterion for decisions under imprecise probability. Maximality: The criterion of maximality was proposed by Walley (1991), Chapter 3.9, and received an axiomatic foundation by Seidenfeld et al. (1995). A gamble G : S → R is called maximal in a set of available gamble F , iff ∀F

E(F − G) ≤ 0 .

(1.9)

The criterion can be motivated on the basis of a lower and upper betting rate interpretation of imprecise probabilities (Walley 1991, Section 2.3). The lower expectation E(F ), for instance, is associated with the notion of a supremum buying price below which the decision maker is inclined to buy gamble F (Walley 1991, Section 2.3). Since F − G describes the exchange of the gamble G with F , maximality means that the decision maker is not willing to pay a positive price for exchanging G with any other gamble F . In order to transfer the concept of maximality to the case of Ipar, we need to insert the indicator gambles for avoiding the catastrophic domain into expression 1.9. In this context, an act g is maximal among the available acts f , iff ∀f ⇔ ∀f

E(If −1 (C) ¯ − Ig −1 (C) ¯ )≤0 ∃P ∈P

¯ ≥ P (f −1 (C)) ¯ , P (g−1 (C))

(1.10)

where P is the closed convex set of probabilities on the set S. Obviously, maximality strengthens interval dominance. Every maximal act is undominated in the interval sense, but not every such undominated act needs to be maximal. E-admissibility: This decision criterion was formulated by Levi (1980). It directly utilizes the description of the uncertainty in terms of a convex set of probabilities P on the set S of possible states of the world. In our context, an act g : S → X is E-admissible, iff there exists a classical probability P ∈ P, for which g maximizes the probability to avoid ruin over all acts f under consideration, i.e., g E-admissible

⇔

∃P ∈P

∀f

¯ ≥ P (f −1 (C)) ¯ . P (g−1 (C))

(1.11)

Evidently, every E-admissible act is also maximal, but not every maximal act needs to be E-admissible. Hence, E-admissibility strengthens the maximality criterion. The underlying rationale of the E-admissibility criterion is a sensitivity analysis interpretation of the decision situation. A group of Savage-type decision makers can be imagined that hold subjective probabilities compatible with the ambiguous


23

state of information as described by the convex set of probabilities P. For each E-admissible act, a decision maker might be found in the group who supports the act. Thus, the divergence of E-admissible acts indicates the scope for controversy that might arise in a group of expected utility maximizers, or probability to avoid ruin maximizers, due to the presence of ambiguity. Bivalent dominance: A bivalent dominance criterion can be constructed by focusing exclusively on the lower and upper extreme of the expected outcome and the probability to avoid ruin, respectively, of an act. The rationale of this criterion is that the decision maker, devoid of any information about the distribution of probabilities between the lower and upper extreme, will focus her attention on the bounds of the probability interval (see also Arrow and Hurwicz 1972). Although such an approach to judging imprecise information is intuitively obvious, it has not been endorsed in the literature to our knowledge. Nevertheless, we will consider a bivalent dominance criterion in this analysis due to its intuitive appeal. We call an act g bivalently undominated, iff there exists no other act f , for which ¯ ≤ P (f −1 (C)) ¯ P (g−1 (C))

¯ ≤ P (f −1 (C)) ¯ , and P (g−1 (C))

(1.12)

where at least one inequality is strict. Γ-maximin: The Γ-maximin criterion is maybe the most prominent decision criterion under ambiguity that can be found in the economic literature. It was established by Gilboa and Schmeidler (1989) from a set of axioms for binary preference relations between acts (cf. Section 1.3). In our context, an act g is strictly preferred to an ¯ to avoid ruin is strictly act f under Γ-maximin, iff its lower probability P (g−1 (C)) −1 ¯ larger than P (f (C)), i.e., ¯ > P (f −1 (C)) ¯ . g ≻ f ⇔ P (g−1 (C))

(1.13)

Unlike the aforementioned decision criteria, Γ-maximin yields a weak ordering over acts. The optimal choice is constituted by those acts that maximize the lower probability to avoid ruin. In this sense, the Γ-maximin criterion can be linked to a precautionary attitude (Chevé and Congar 2003). Hurwicz Criterion: The Γ-maximin criterion can be extended by incorporating information on the upper bound of the expected utility interval, or probability to avoid ruin interval, into the decision rule. Such a criterion was already proposed by Hurwicz (1951) for the case of complete ignorance over a set of possible outcomes. Arrow and Hurwicz (1972) derived the criterion axiomatically for this particular case. Later on, Jaffray (1989) adopted the Hurwicz criterion in the context of socalled belief functions, a special class of imprecise probabilities (see Section 1.6). An axiomatic foundation can be found in Jaffray (1994).

24

E. Kriegler et al.

In our context, the Hurwicz criterion ranks the available acts on the basis of a linear combination of the lower and upper probability to avoid ruin, i.e., ¯ + (1 − λ) P (g−1 (C)) ¯ > g ≻ f ⇔ λ P (g−1 (C)) ¯ + (1 − λ) P (f −1 (C)) ¯ , λ P (f −1 (C))

λ ∈ [0, 1] . (1.14)

λ is a security index that determines the proportion in which considerations of pre¯ ¯ and opportunity, related to P (g−1 (C)), influence caution, related to P (g−1 (C)), the decision criterion. The security index adds an additional degree of freedom to the policy analysis. It allows to explore the continuum between completely precautionary (λ = 1) and completely opportunity-based decision rules (λ = 0). The Γ-maximin criterion is recovered for λ = 1. The other extreme of focusing exclusively on the upper probability to avoid ruin (λ = 0) has been frequently called Γ-maximax criterion in the literature.

1.6

Representations of Imprecise Probability

In order to calculate the imprecise probability to avoid ruin (Ipar) for each Wre mitigation policy, we need to apply methods that allow us • to represent an imprecise probability on the set S of possible states of the world, i.e., in our particular case on the domain ST2x ⊂ R of plausible values for climate sensitivity T2x , and • to project this information onto the range X of possible consequences for a given mitigation policy, i.e., in our case onto the range XT (2300) ⊂ R for the global mean warming in the year 2300. The consequences of an environmental policy are usually calculated on the basis of dynamic models that generate a transfer function x = f (xo , a, c(·)) which depends on the initial state xo , a set of model parameters a, and the given policy c(·). In this context, the set S of possible states of the world can be identified with the domain for plausible parameters and initial values. Although this analysis focuses on decision criteria for climate policy analyses under imprecise information (see Section 1.5 and Section 1.10), we will have to address the task of estimating an imprecise probability on S (Section 1.7) and transferring it onto X (Section 1.8). For reasons of mathematical tractability, we have chosen to model the uncertainty about S, i.e., climate sensitivity in our prototypical case, by two special classes of imprecise probability: belief functions and possibility measures. Both allow for an efficient representation of the information that can be easily transfered to estimates of global mean temperature change. In the following, we will briefly introduce these two classes, and discuss how they fit into the hierarchy of imprecise probability models. In so doing, we will restrict ourselves to the concepts

25


and relationships that are employed later on. Those readers focusing on the climate change application may skip this section, and return to it whenever they wish to understand the underlying rationale of the mathematical procedures utilized in Section 1.7 and 1.8. We have investigated the methodological challenges of applying belief functions to the estimation of future climate change in detail in Kriegler and Held (2005) and Kriegler (2005). We begin our methodological discussion by highlighting the relationship between convex set of probabilitiesAn equivalent choice are coherent lower and upper previsions (Walley 1991). P and more specialized imprecise probability models like belief functions. Important characteristics of P are its lower envelope P : F → [0, 1] and upper envelope P : F → [0, 1], respectively, i.e., ∀A ∈ F

P (A) := inf P (A) , P ∈P

P (A) := sup P (A) ,

(1.15)

P ∈P

where F is a field of subsets of S. P and P are called coherent lower and upper probabilities. Since lower and upper probability are conjugate to each other, i.e., ∀ A ∈ F P (A) = 1 − P (A), it suffices to consider only one of them. The preferable choice will depend on the application. It is important to note that different convex sets of probabilities P can have the same envelope. Hence, coherent lower (upper) probabilities can represent only a subclass of convex sets of probabilities. A coherent lower (upper) probability is a monotone, in general super-additive (sub-additive) set function (Walley 1991, Section 2.7). The mathematical tractability of these set functions is greatly enhanced, if they can be defined on a finite field, ˜ of a finite partition S˜ = {S1 , ..., Sn } of the set of possie.g., the power set ℘(S) ble states of the world S. In this case, there exists an additive representation of ˜ →R a coherent lower (upper) probability in terms of its M¨ obius inverse ν : ℘(S) (Chateauneuf and Jaffray 1989) defined by X ˜ ∀ A ∈ ℘(S) ν(A) := (−1)|A−B| P (B). B⊆A

˜ with non-zero Möbius assignment ν(A) 6= 0 are called focal The sets E ∈ ℘(S) elements. Knowledge of the Möbius assignments ν(E) to the focal elements E suffices to fully characterize a coherent lower (upper) probability: X X P (A) := ν(B) . ν(B) , ∀A ∈ F P (A) := ˜ ˜ | B∩A6= ◦ B∈℘(S) | B⊆A B∈℘(S) /

It is interesting to note that the Möbius inverse coincides with a probability mass distribution, if P (P ) constitutes a simple additive probability measure. There exists a special class of coherent lower probabilities, called belief functions, whose Möbius inverses fulfill the nice property that all focal elements E1 , ..., Ek carry a positive Möbius assignment ν(Ei ) > 0. Then, the Möbius inverse can be

26

E. Kriegler et al.

˜ The so-called interpreted as a probability mass assignment over the power set ℘(S). random set {(E1 , ν1 ), ..., (Ek , νk )} carries the full information that is contained in the belief function bel : F → [0, 1], and its conjugate upper probability, the plausibility function pl : F → [0, 1]. We have ∀A ∈ F

bel(A) =

X

νi ,

pl(A) =

i | Ei ⊆A

X

i | Ei ∩A6=

νi

(1.16)

◦/

Belief functions and random sets have been investigated in the context of the Dempster-Shafer theory of evidence, which encompasses a variety of semantic uncertainty models that all draw on the same mathematical primitives (Dempster 1967; Shafer 1976; Smets and Kennes 1994). In the context of this analysis, belief functions are interpreted solely as a special case of coherent lower probability. Equation (1.16) can be given an intuitive interpretation. Assume that the imprecision in the information has not allowed us to locate the distribution of probability masses below the level of focal elements. Then, every focal element intersecting an arbitrary subset A ⊂ S might have its probability mass located entirely in the intersection (upper probability). However, it is also possible that every focal element not fully enclosed by A will have its mass situated outside the intersection (lower probability). The usefulness of the random set representation becomes evident when considering the projection of a coherent lower probability P S for the possible states of the world S onto the range of consequences X through a dynamically generated transfer function f : S → X. Given that f is measurable FS /FX , the transfered lower probability on the consequences X is defined by (see, e.g., Kriegler and Held 2003, 2005) ∀ B ∈ FX

P X (B) := P S (f −1 (B)) .

(1.17)

If P S is a belief function belS described by a random set {(E1 , ν1 ), ..., (Ek , νk )}, the random set can be transfered directly onto the field FX by applying the extension principle for random set-valued variables (Dubois and Prade 1991): f (Ei ) := { x | ∃ s ∈ Ei x = f (s) } ,

∀ B ∈ FX

ν f (B) :=

X

νi

(1.18)

f (Ei )=B

It can be easily checked that Definition (1.18) yields a random set {(f (E1 ), ν1f ), ..., (f (El ), νlf ), l ≤ k} generating a belief function belX that fulfills Condition (1.17), i.e., ∀ B ∈ FX belX (B) = belS (f −1 (B)) (Kriegler and Held 2005). Belief functions are the simplest type of imprecise probability that incorporates classical probability as special case. There exists another special case, which is rather complementary to classical probability theory. If the random set is constituted by a chain of focal elements E1 ⊆ ... ⊆ Ek , the plausibility function becomes a possibility

27

Climate Protection Strategies under Ambiguity measure Π : ℘(S) → [0, 1]4 (Dubois and Prade 1990), defined by Π(∪i∈I Ai ) = sup Π(Ai ) i∈I

for any index set I and any family of sub-sets Ai ⊆ S. Such set functions are the basic primitive of possibility theory (Zadeh 1978; Dubois and Prade 1988). The ˜ → [0, 1]. conjugate set function of possibility is called necessity N : ℘(S) Possibility theory is convenient in the sense that the set functions Π and N can be represented by a single possibility distribution π : S → [0, 1], somewhat reminiscent of the probability density function characterizing a probability measure. ∀ A ∈ ℘(S)

Π(A) = sup π(s)

(1.19)

s∈A

Here, the convention sups∈ / π(s) := 0 has been silently assumed. The possibility ◦ distribution π is normalized by ∃ s ∈ S with π(s) = 1. Thus, formally it represents a normal fuzzy set (regarding some further applications of fuzzy sets, cf. also Chapter 8). Possibility theory has been proposed by Zadeh (1978) in order to provide an uncertainty model for fuzzy sets. A few studies have explored possibility in the context of an imprecise probability interpretation (Dubois and Prade 1992; de Cooman and Aeyels 1999; Kriegler and Held 2003). They show that possibility measures Π represent the upper envelope of only a very special type of convex sets of probabilities PΠ := {P | ∀A ∈ ℘(S) P (A) ≤ Π(A)} .

(1.20)

Due to relationship (1.19) in combination with the normalization of possibility distributions, convex sets PΠ have to contain at least one Dirac measure which represents complete information about the true state of the world. Nevertheless, we will consider possibility ΠS : ℘(S) → [0, 1] as a special form of upper probability in this analysis. Due to its representation by a possibility distribution πS : S → [0, 1], we can apply the extension principle of fuzzy set theory, πX (x) := sup πS (s) , (1.21) s∈f −1 (x)

to project ΠS over ℘(S) onto a possibility measure ΠX over ℘(X). Among others, it has been shown by Kriegler and Held (2003) that ΠX fulfills condition (1.17), i.e., ∀ B ∈ ℘(X) ΠX (B) = ΠS (f −1 (B)). Thus, it is rather straightforward to project an upper probability over a field of subsets of S onto events B ⊂ X through a transfer function f : S → X, if this upper probability can be described by a possibility measure ΠS . 4 A possibility measure can be defined on the power set ℘(S) of the generally uncountable domain S of possible initial state and parameter values.

28

1.7

E. Kriegler et al.

The Imprecise Probability of Climate Sensitivity

In our prototypical climate policy analysis, the set S of possible states of the climate system is characterized by the range of possible values for climate sensitivity T2x (see Section 1.4). Climate sensitivity T2x is a crucial parameter to characterize the response of the climate system to an increase of Ghg concentrations in the atmosphere. Due to complex feedbacks within the climate system, its value is clouded by uncertainty. Model-based estimates of climate sensitivity differ greatly. Moreover, the empirical evidence at this stage is too weak to determine climate sensitivity directly from the data, since the runtime of the unintentional greenhouse experiment of humankind, i.e. the industrial era, has been relatively short compared to the time scales of heat distribution in the world’s oceans. As a result, estimates of T2x differ greatly. The Ipcc specifies an interval T2x = [1.5 ◦ C, 4.5 ◦ C] based on model results and expert opinions (Cubasch and Meehl 2001). Recently, several probabilistic estimates of climate sensitivity have been derived from a comparison of model simulations with the instrumental temperature record (Andronova and Schlesinger 2001; Forest et al. 2002; Knutti et al. 2002; Murphy et al. 2004). Figure 1.2 depicts those probability estimates that we will consider in our analysis. Forest et al. (2002) used Bayesian statistics to update a prior probability for T2x with a likelihood function from a comparison of the temperature record with the output of a statistical-dynamical atmosphere-ocean model. They produced posterior probabilities for two different choices of prior distribution: a uniform prior over the domain T2x ∈ [0.5 ◦ C, 10 ◦ C] and an expert prior based on the expert elicitation of Morgan and Keith (1995). Knutti et al. (2002) used a climate model of similar complexity to estimate a probability distribution for climate sensitivity. In contrast to Forest et al. (2002), Knutti et al. (2002) introduced a Bayesian flavor rather implicitly by assuming subjective sampling distributions for the model parameters (leading to a prior for climate sensitivity) and removing those model responses that did not accord with observations (0-1 likelihood formulation). Andronova and Schlesinger (2001) compared the historical temperature record with the output of an energy balance model, and estimated probability distributions for climate sensitivity by a bootstrapping technique applied to the residual between model output and data. Various probability distributions were generated under different assumptions about the historical radiative forcing. Here, we consider the probability estimate for climate sensitivity under the forcing scenario Gtas of Andronova and Schlesinger (2001) that includes the contribution from greenhouse gases (G), aerosols (A), tropospheric ozone (T) and solar activity (S) (see Kriegler 2005, Chapter 3.2.3 for a motivation of this choice). Recently, the first probability estimate of climate sensitivity from ensemble simulations with an atmospheric general circulation model became available (Murphy et al. 2004). Due to the high computational costs, the model parameters were perturbed from their standard values one at a time and later

29


1 0.9

Cumulative Probability

0.8 0.7 0.6 0.5 0.4 0.3

Forest et al. (2002) − expert prior Forest et al. (2002) − uniform prior Andronova et al. (2001) − GTAS Murphy et al. (2004) − weighted Murphy et al. (2004) − unweighted Knutti et al. (2002)

0.2 0.1 0 0

1

2

3

4 5 6 7 Climate Sensitivity [K]

8

9

10

Figure 1.2: Cumulative probability distributions for climate sensitivity from the literature.

on extended to an ensemble of multi-parameter variations under a linearity assumption. From the extended ensemble, Murphy et al. (2004) generated two probability estimates for climate sensitivity with and without the discounting of ensemble members in terms of their root mean square difference between model response pattern and observations. It is evident from Fig. 1.2 that the probability estimates for climate sensitivity differ significantly. Hence, the uncertainty about climate sensitivity is fraught with ambiguity and should be represented by an imprecise probability. We construct a convex set of probabilities P2x for climate sensitivity T2x from the family of probability estimates depicted in Fig. 1.2 by making two assumptions: (A) Inclusion of the plausible: We assume that every probability estimate whose cumulative distribution function Cdf lies between the lower and upper envelope F : S → [0, 1] and F : S → [0, 1] of the family of six Cdfs, with S = [0 ◦ C, 10 ◦ C] ⊂ R, is compatible with the current state of information. (B) Exclusion of the implausible: We assume that every estimate for climate sensitivity, whose Cdf is not fully enclosed by the lower and upper envelope F and F is not compatible with the current state of information. Assumption (B) is rather reasonable, if the family of estimates includes all estimates that are “scientifically accepted”5 . The situation is more difficult with Assumption 5 We acknowledge that “scientifically accepted” is a difficult notion to work with, since it is both fuzzy and not necessarily related to truth. However, we try to define the current state of information

30

E. Kriegler et al.

(A). It basically says that every estimate should be considered plausible which cannot be excluded on the basis of Assumption (B). This is a very conservative statement, because we can imagine Dirac δ-measures with their masses concentrated at point values that are rather implausible but are still fully contained in the distribution band. In this sense, a convex set of probabilities P[F , F ] := { P | ∀ s ∈ R F (s) ≤ P (−∞, s] ≤ F (s) }

(1.22)

constitutes a conservative choice. It contains too much rather than too few probabilities, and therefore overstates rather than understates the uncertainty about climate sensitivity. P[F , F ] has been called a distribution band in the literature (Basu and DasGupta 1990). In the following, we will compress the information content of the distribution band in two different ways, i.e., in terms of a random set of a belief function and alternatively in terms of two possibility distributions, and compare both approaches with each other.

1.7.1

Random Set Approximation

In order to approximate a distribution band by a belief function associated with a finite random set (E, ν) = {(E1 , ν1 ), ..., (Ek , νk )}, we need to enclose the continuous band by a right-continuous step function SF from below and a left-continuous step function SF from above. Let s∗1 < ... < s∗n be the n points of the real line at which SF : R → [0, 1] exhibits steps in the cumulative probability, and s∗1 < ... < s∗m the m points associated with the steps of SF : R → [0, 1]. Then,   F (s∗i ) s∗i ≤ s < s∗i+1 SF (s) = 0 s < s∗1  1 s∗n ≤ s

  F (s∗j+1 ) s∗j < s ≤ s∗j+1 SF (s) = 0 s ≤ s∗1  1 s∗m < s

.

The enlarged convex set of probabilities P[SF , SF ] is called a p(robability)-box in the literature (see Ferson et al. 2002 for a good overview about p-boxes). P-boxes have emerged in different contexts, in particular in the calculation of bounds for the convolution of two random variables with unknown dependency (Williamson and Downs 1990; Regan et al. 2004). It has been already noted by Yager (1986) that p-boxes can be related to belief functions and their associated random sets. The following algorithm constructs a random set (E, ν) directly from knowledge of the lower envelope SF and upper envelope SF of the p-box P[SF , SF ] (Ferson et al. 2002; Regan et al. 2004; Kriegler and Held 2005). The algorithm is illustrated in Fig. 1.3. here. From a practical point of view, “scientifically accepted” could be defined as every estimate published in the peer-reviewed literature and not considered outdated by the community.

31


1

s1*

s*2

s*3

s*4

(a)

s*5

1 s*k=1 s*k=2,3 s*k=4,5 s*k=6,7 s*k=8,9 pk=9 pk=8 pk=7 pk=6

(b)

pk=5 pk=4 pk=3 0

s 1 *

s 2 *

s 3 *

s 4 *

pk=2 x pk=1 0 s 5 s 6 * *

X s

*k=1,2

s

*k=3,4

s

*k=5

s k=6 s k=7,8 s k=9 * * *

Figure 1.3: Illustration of the p-box approximation of a distribution band (a), and the construction of a random set from a p-box by use of Algorithm 1 (b).

Algorithm 1 1. Initialize indices k = 1 (running over the focal elements of the random set to be constructed), i = 1 (running over s∗i ), j = 1 (running over s∗j ). Let pk denote the cumulative probability already accounted for in step k. Assign p0 = 0. 2. Construct random set Ek = (s∗j , s∗i ]. νk = SF (s∗i ) − pk−1 , 3. (a) SF (s∗i ) < SF (s∗j+1 ): indices k → k + 1, i → i + 1. Return to step 2. (b) SF (s∗i ) > SF (s∗j+1 ): νk = SF (s∗j+1 ) − pk−1 , indices k → k + 1, j → j + 1. Return to step 2.

pk = SF (s∗i ). Raise pk = SF (s∗j+1 ). Raise

(c) SF (s∗i ) = SF (s∗j+1 ): νk = SF (s∗j+1 )−pk−1 . If SF (s∗i ) = SF (s∗j+1 ) = 1, stop. If SF (s∗i ) = SF (s∗j+1 ) < 1, set pk = SF (s∗j+1 ). Raise indices k → k + 1, i → i + 1, j → j + 1. Return to step 2.

For each step k, it is s∗j ≤ s∗i (since SF ≥ SF ), and νk > 0 (since SF , SF monotone increasing). The algorithm will always reach the points s∗n , s∗m+1 with SF (s∗n ) = SF (s∗m+1 ) = 1 and stop. Similar algorithms have been presented in the literature, e.g., by Ferson et al. (2002) and Regan et al. (2004). The main difference lies in the fact that algorithm 1 generates half-closed intervals in step 2, while other formulations usually would choose the corresponding closed interval. The latter option is the sensible choice, when the lower and upper step functions enclosing the p-box are not strictly right- and left-continuous, respectively. Closed intervals always give a random set with belE (−∞, s] ≤ F (s) and plE (−∞, s] ≥ F (s), where equality holds almost everywhere. However, if the bounds are left- and rightcontinuous, we can be more precise. Kriegler and Held (2005) have shown that the resulting discretized p-box P[SF , SF ] enclosing P[F , F ] represents a convex set of probabilities, whose lower envelope P : R → [0, 1], with R the Borel algebra on the

32

E. Kriegler et al.

real line, is a belief function. Its associated Möbius inverse is defined by the random set (E, ν) as generated by Algorithm 1. Several methods have been proposed for choosing the lower and upper step functions that enclose a continuous distribution band P[F , F ]. Following Williamson and Downs (1990), a uniform partition of the probability axis is usually employed to generate these step functions. The probability axis is divided into n adjacent intervals (0, 1/k], (1/k, 2/k], . . . , ((k − 1)/k, 1] of equal size. The associated random set resulting from Algorithm 1 assigns probability mass νi = 1/k, 1 ≤ i ≤ k to all focal elements which constitute the inverse images of the lower and upper Cdf at −1 the partition boundaries, i.e., Ei = (F ((i − 1)/k), F −1 (i/k)], 1 ≤ i ≤ k (Ferson et al. 2002). The accuracy of the p-box approximation for a fixed number k of focal elements can be improved by adjusting the step height of the approximating step functions to reflect the shape of the continuous bounding functions. Kriegler and Held (2005) presented a nonlinear program for finding the optimal k-partition of the unit probability interval that minimizes the area between lower and upper step functions enclosing the distribution band. The resulting step functions coincide with the p-box approximation that yields the smallest information loss w.r.t to the original distribution band. In this analysis, however, we will stick to the simpler approach of choosing a uniform partition of the probability axis. It is a reasonable choice if the number of probability intervals is large enough to resolve the continuous lower and upper distribution functions satisfactorily. In this case, the adaptation of the interval width to the shape of the distribution functions would only provide a marginal improvement. Here, we choose a number of k = 100 probability steps {0.01, ..., 0.99, 1}. This resolution has been frequently recommended as a rule of thumb for approximating a continuous distribution band with a p-box (Ferson et al. 2002). Figure 1.4a shows the resulting p-box approximation PT2x [SF , SF ] of the distribution band PT2x [F , F ] for climate sensitivity as well as the associated random set generated with Algorithm 1. The information content of PT2x [SF , SF ] can be compared with the Ipcc estimate of [1.5 ◦ C, 4.5 ◦ C]. The probability for T2x ∈ [1.5 ◦ C, 4.5 ◦ C] lies in the interval [0.36, 0.97], for T2x < 1.5 ◦ C in [0, 0.08], and for T2x > 4.5 ◦ C in [0.03, 0.56]. It is important to note that these probability intervals can be directly calculated from knowledge of the random set (E, ν)T2x = {(E1 , ν1 ), ..., (E100 , ν100 )} by means of Eq. (1.16). The numbers show that PT2x [SF , SF ] does not support the Ipcc estimate, especially for high climate sensitivities T2x > 4.5 ◦ C. This reflects the fact that the upper bound of the Ipcc estimate is not supported by the probability estimates in the literature.

1.7.2

Possibilistic Decomposition

There exists an alternative method for representing a distribution band that builds on possibility theory. Dubois and Prade (1992) and de Cooman and Aeyels (1999)

33


Figure 1.4: Representation of the distribution band PT2x [F , F ] for climate sensitivity in terms of a random set (a) and two possibility distributions (b).

have shown that the natural extension of an upper probability that is defined on a chain of sets is a possibility measure. Since an upper Cdf F : R → [0, 1] specifies an upper probability on the chain (−∞, s], and a lower Cdf F : R → [0, 1] specifies a lower probability on the chain (−∞, s], all probability families which are defined by solely an upper or solely a lower probability distribution have possibilistic envelopes. Lower distr. band:

Upper distr. band:

P[F ] := { P |∀ s ∈ supp(P[F , F ])

P (−∞, s] ≥ F (s)

P[F ] := { P |∀ s ∈ supp(P[F , F ])

P (−∞, s] ≤ F (s)

/ and ∀ A ∩ supp(P[F , F ]) = ◦

P (A) = 0 } ,

and ∀ A ∩ supp(P[F , F ]) = ◦ /

P (A) = 0 } .

Clearly, it is P[F , F ] = P[F ] ∩ P[F ]. The possibility measures ΠF : ℘(S) → [0, 1] and ΠF : ℘(S) → [0, 1] constituting the upper envelope of P[F ] and P[F ] (see Definition 1.20) are characterized by underlying possibility distributions πF and πF , respectively (cf. Eq. 1.19). These possibility distributions can be derived directly from the continuous lower bound F and upper bound F of the distribution band, i.e., F (s) ∀ s ∈ supp(P[F , F ]) , (1.23) πF (s) = 0 ∀ s 6∈ supp(P[F , F ]) 1 − F (s) ∀ s ∈ supp(P[F , F ]) . πF (s) = 0 ∀ s 6∈ supp(P[F , F ]) Thus, we can represent a distribution as the intersection of two probability families P[F ] and P[F ] encompassed by possibility measures ΠF and ΠF , respectively. In the following we will label this method for representing a p-box possibilistic decomposition.

34

E. Kriegler et al.

Figure 1.4b shows the two possibility distributions πS,F and πS,F that are derived from the distribution band PT2x [F , F ] by using Eq. (1.23). Since they are directly obtained from the continuous lower bound F and upper bound F , no information loss due to a step function approximation is entailed as it was the case for the p-box approximation. This initial advantage, however, can be nullified, when projecting the two possibility distributions onto the range of a (dynamically generated) transfer function f : S → X by applying the extension principle (1.21). Let the extended possibility distributions be denoted by πX,F and πX,F , and the corresponding distribution bands by P[πX,F ] and P[πX,F ], respectively. If the transfer function f fulfills a few “good nature” conditions that are not too restrictive, it can be shown that f (P[F ]) = P[πX,F ] and f (P[F ]) = P[πX,F ] (Kriegler and Held 2003). However, even if this is the case, it is f (P[F ] ∩ P[F ]) ⊆ f (P[F ]) ∩ f (P[F ]) = P[πX,F ] ∩ P[πX,F ] ,

(1.24)

where equality holds only in the case of a monotone transfer function. A nonmonotone transfer function can lead to an information loss as will be exemplified in the subsequent section.

1.8

Estimating Global Mean Temperature Change

We use a simple dynamic model to translate a radiative forcing ∆F of the earth system into a change ∆T in global mean temperature (Gmt) since preindustrial times (Dutton 1995). Ce ·

d ∆T (t) dt

∆T (t) T2x effective ocean heat capacity per unit area

= ∆F (t) − F2x · Ce

(1.25)

F2x

radiative forcing for a doubling of atmospheric CO2

T2x

climate sensitivity

Differential equation (1.25) is the simplest type of energy balance model. It equates the net radiative flux at the top of the atmosphere to surface heat uptake Ce d ∆T / dt, which is dominated by the ocean. In an approximation, the net radiative flux into the earth system is separated into a radiative forcing ∆F and a radiative damping proportional to ∆T (t) that includes the effect of temperature feedbacks on the radiation balance. The effective ocean heat capacity Ce is an artificial quantity that arises from the simple form of the energy balance model (1.25). It depends on ocean characteristics like vertical diffusivity, but also on climate sensitivity (Dutton 1995; Cubasch and Meehl 2001). A comparison of model (1.25) with emulations of different atmosphere-ocean general circulation models (Aogcms) suggests a functional γc with 0 < γc < 1. The ensemble dependence of Ce on T2x of the form Ce ∼ T2x of Aogcm emulations considered in Cubasch and Meehl (2001) exhibited a mean


35

Figure 1.5: Transfer functions for the four Wre policies considered in this analysis at two different points in time: (a) year 2100, (b) year 2300.

climate sensitivity of T2x = 2.8 ◦ C. The ensemble mean projections of Gmt change in the 21st century are best reproduced by model (1.25) for a value of approximately C¯e = 43 W yr/m2 ◦ C. We extrapolate this estimate of ocean heat capacity to different climate sensitivities by Ce = C¯e ·

T2x 2.8 ◦ C

γc

with γc = 0.8 .

Effective ocean heat capacity crucially influences the transient temperature path. However, the equilibrium temperature change Teql = T2x ∆Feql /F2x after a stabilization of the forcing at ∆Feql is solely determined by climate sensitivity. In this analysis, the various forcing trajectories F (t) are provided by the Wre policies (see Fig. 1.1b). For each forcing scenario, model (1.25) generates a transfer function ∆T (t) := f (F (·), T0 , t; T2x ) from the set of possible states of the world, i.e., climate sensitivity, onto Gmt change at time t. f (F (·), To , t) : T2x → ∆T (t) constitutes the acts under consideration. Due to the simplicity of Eq. (1.25), the model can be integrated easily over the whole domain of climate sensitivity,. Figure 1.5 shows the resulting transfer functions for the four Wre policies that stabilize atmospheric CO2 concentration at 450, 550, 650 and 750 ppm, respectively, and two different points in time (year 2100 and 2300). Due to the dependence of effective ocean heat capacity on climate sensitivity, the transfer functions are non-monotone for times t ≪ teql in the transient phase of the climate response. Their non-monotonic behavior results from the choice γc = 0.8 which probably exaggerates the dependence of ocean heat capacity on climate sensitivity. However, we have chosen such a high value of γc deliberately here, because we want to study, among others, the effect of a non-monotone transfer function on the extension of random sets and possibility distributions.

36

E. Kriegler et al. (a)

(b)

year 2300 year 2200 year 2100

0

1

pT , F 2x

Possibility

Possibility

pT , F 2x

year 2300 year 2200 year 2100

2

3

4

5

6

7

Global mean temperature change for WRE 550 [oC]

8

0

1

2

3

4

5

6

7

Global mean temperature change for WRE 550 [oC]

8

Figure 1.6: Extension of the two possibility distributions – (a) πT2x ,F (a) and (b) πT2x ,F (b) – resulting from possibilistic decomposition of PT2x [F , F ] onto Gmt change for the Wre 550 policy.

1.8.1

Possibilistic Extension

Once the dynamically generated transfer functions are specified, we can project the possibilistic decomposition and the random set approximation of the p-box PT2x [F , F ] for climate sensitivity onto global mean temperature change ∆T (t) at some time t in the future. We first consider the possibilistic approach for processing the information contained in PT2x [F , F ]. The two possibility distributions πS,F and πS,F that emerge from the possibilistic decomposition (1.23) of PT2x [F , F ] are extended onto ∆T (t) by applying the extension principle (1.21) for fuzzy sets. Since ∗ (t) at which the transfer function reaches there exists exactly one interior point T2x a local extremum in temperature increase ∆T ∗ (t), the possibility distribution πS,F is projected onto two branches over the range ∆T (t). One of the branches origi∗ (t), and increases monotonic in Gmt change, while nates from the domain T2x ≤ T2x ∗ (t), and decreases monotonic in Gmt the other derives from the domain T2x > T2x change (because πS,F increases monotonic in T2x ). The extended possibility distribution πX,F on ∆T (t) is constituted by the maximum possibility on these two branches. Figure 1.6a shows πX,F for the years 2100, 2200, and 2300. Due to the maximum operation on two distinct branches, the extended πX,F is discontinuous. A similar line of reasoning holds for the second possibility distribution πS,F that results from the possibilistic decomposition of PT2x [F , F ]. However, since πS,F is ∗ (t) contains monotone decreasing in T2x , the branch originating from T2x ≤ T2x ∗ larger possibility values than the branch deriving from T2x > T2x (t) throughout the entire range [0, ∆T ∗ (t)] of the transfer function. Thus, the extended possibility distribution πX,F is constituted entirely by one branch, and therefore continuous and monotone decreasing (see Fig. 1.6b). The two extended possibility distributions πX,F and πX,F define possibility measures that are upper envelopes of two probability families P[ΠX,F ] and P[ΠX,F ], respectively, as defined by Eq. (1.20). P[ΠX,F ] is

37

Climate Protection Strategies under Ambiguity equivalent to a distribution band ∗

∗

P[F Π ] := {P | ∀ x ∈ X P (−∞, x] ≤ F Π (x) := sup πX,F (x′ ) } , x′ ≤x

if and only if the possibility distribution πX,F is monotone increasing on the range [0, ∆T ∗ (t)]. Likewise, P[ΠX,F ] is equivalent to a distribution band P[F ∗Π ] := {P | ∀ x ∈ X P (−∞, x] ≥ F ∗Π (x) := 1 − sup πX,F (x′ ) } , x′ >x

if and only if the possibility distribution πX,F is monotone decreasing. This highlights the fact that probability families with possibilistic envelopes cannot be described in terms of p-boxes in general (see Kriegler and Held 2003 for a discussion of such families). In the particular case of our analysis, πX,F and πX,F fulfill the necessary and sufficient conditions cited above in a good approximation. Thus, ∗ we can use the lower Cdf F ∗Π and upper Cdf F Π to represent the information of the two extended probability families P[ΠX,F ] and P[ΠX,F ]. Figure 1.7 shows the lower and upper Cdf for the years 2100 and 2300. The class of probabilities that is enclosed between them represents the intersection P[ΠX,F ] ∩ P[ΠX,F ]. Recalling relation (1.24), this intersection will be a true superset of the projection f (PT2x [F , F ]) of the original distribution band on climate sensitivity onto Gmt increase, since the transfer function f is non-monotone. The information loss that is incurred by the possibilistic decomposition method can be inferred from a comparison with the random set approach6 .

1.8.2

Random Set Extension

The random set approach provides an alternative method, i.e., random set extension, for transferring the distribution band for climate sensitivity onto distribution bands for global mean temperature increase (see Section 1.6, in particular Eq. 1.18). In this subsection, we will employ the method of random set extension to compare the resulting distribution bands for different times in the future and different Wre stabilization policies with the corresponding distribution bands deduced from possibilistic extension. The distribution band PT2x [F , F ] for climate sensitivity has been approximated with a p-box PT2x [SF , SF ] whose information content is described by the random set (E, ν)T2x (see Section 1.7 and Kriegler and Held 2005). (E, ν)T2x can be projected onto an estimate for Gmt increase by means of random set extension (1.18). This requires to calculate the images f (Ei ) = [∆T i , ∆T i ] of 6 In the following, we will frequently compare the information content of distribution bands, random sets, and possibility distributions. This always refers to the convex set of probabilities P that is associated with these uncertainty representations. We say that P1 contains less information than P2 , if it encompasses a larger set of classical probabilities than P2 , i.e., P1 ⊃ P2 . Likewise, P1 contains the same information as P2 , if P1 = P2 , and P1 contains more information than P2 , if P1 ⊂ P2 .

38

E. Kriegler et al.

Figure 1.7: Cdf envelopes in 2100 (a) and 2300 (b) of the intersection of the distribution bands P[ΠX,F ] and P[ΠX,F ].

each of the k = 100 focal elements Ei = (T 2x,i , T 2x,i ] under the dynamically generated transfer functions f (∆F (·), T0 , t) : T2x → ∆T (t), and assigning the probability mass νi = 1/k of the original focal element Ei to them. Since the transfer func∗ (t) in the interior tion is non-monotone with a single local extremum at point T2x of the T2x -domain, the calculation of the images amounts to compute the Gmt change at the boundaries of the interval, i.e., ∆Ti (t) = f (∆F (·), T0 , t; T 2x,i ) and ∗ (t) ∆Ti′ (t) = f (∆F (·), T0 , t; T 2x,i ), and to check whether the climate sensitivity T2x leading to the maximum temperature increase ∆T ∗ (t) is contained in Ei . The resulting random sets (E, ν)∆T can be used to reconstruct the extended pboxes P∆T [SF , SF ] for Gmt increase. The lower and upper cumulative probability envelopes of P∆T [SF , SF ] are determined by X X νi , SF (∆T ) := νi . SF (∆T ) := i | f (Ei )⊆(−∞,∆T ] i | f (Ei )∩(−∞,∆T ]6= ◦ /

Figure 1.8 shows the resulting p-boxes in the years 2100 and 2300 for the Wre 550 policy. It is important to note that P∆T [SF , SF ] might describe a weaker state of information than the extended random set (E, ν)∆T if the transfer function f (∆F (·), T0 , t) is non-monotone. We have identified and discussed this phenomenon in Kriegler and Held 2005. In the present analysis, however, the p-box PT2x [SF , SF ] for climate sensitivity and the various transfer functions f (∆F (·), T0 , t) considered here combine in such a way that P∆T [SF , SF ] and (E, ν)∆T describe nearly the same state of information. This can be seen, e.g., by using Algorithm 1 to generate random sets directly from P∆T [SF , SF ]. They are almost identical with the random sets (E, ν)∆T that were obtained from random set extension. Figure 1.8 also compares the p-boxes P∆T [SF , SF ] for the Wre 550 policy with the corresponding distribution bands P[ΠX,F ] ∩ P[ΠX,F ] derived from possibilistic decomposition and extension. The possibilistic approach leads to an information loss compared to the random set approach due to the decomposition of PT2x [F , F ]


39

Figure 1.8: P-boxes P∆T [SF , SF ] in the years 2100 (a) and 2300 (b) that emerge from the random set extension for the Wre 550 policy. The distribution bands derived from possibilistic decomposition (in red) are also shown for comparison.

into two probability families that are extended separately, and combined later on. The amount of information loss depends on the non-monotonicity of the transfer function. While the information loss is significant for the year 2100, the approaches virtually agree for the year 2300. In the limit of a monotone transfer function, both approaches are completely equivalent except for the step function approximation that is required for generating random sets. In this limit, however, both methods become superfluous, because the lower and upper envelope on the cumulative probability distributions could then be projected directly onto the model estimate. Hence, we find that the random set approach yields more favorable results than possibilistic decomposition when processing the information content of a distribution band. This finding, however, does not translate to other types of probability families. As pointed out in Section 1.6 (see Eq. 1.20 in particular), genuine probability families with possibilistic envelope exhibit a structure that can be very different from distribution bands (Kriegler and Held 2003). Figure 1.9 compares the estimated distribution bands of global mean temperature change in the years 2100 and 2300 for the four different Wre policies of stabilizing atmospheric CO2 at levels of 450, 550, 650 and 750 ppm. The upper panels (a), (b) show the p-box approximations derived from the random set approach, while the two lower panels (c), (d) depict the distribution bands that are computed with the possibilistic approach. As detailed for the case of the Wre 550 policy, the random set approach yields more accurate results for the year 2100, while both methods agree well for the year 2300. In this century, a substantial reduction of Gmt increase relative to the lean Wre 750 policy can only be achieved by Wre 450 (up to 1 ◦ C in 2100). For later times, Wre 550 can also provide a significant reduction of global mean warming compared to Wre 750. However, the maximum possible temperature increase in 2300 lies above 5 ◦ C for all policies. It is important to note that the numbers presented in this prototypical climate policy analysis should not

40

E. Kriegler et al.

Figure 1.9: Distribution bands of global mean temperature change in the year 2100 (a), (c), and 2300 (b), (d) – (a), (b) random set approach; (c), (d) possibilistic decomposition method – for the four policies Wre 450-750 considered in this analysis.

be taken too literally due to the simplicity of model (1.25) (see Kriegler 2005 for a more sophisticated estimation of the imprecise probability of future climate change). The imprecise probability for a temperature change ∆T ≤ 2 ◦ C and ∆T ≤ 4 ◦ C, respectively, in the year 2300 can be directly deduced from Fig. 1.9b, d. The values tabulated in Tab. 1.1 provide one piece of information for determining the imprecise probability to avoid ruin which we want to employ in the policy analysis presented here. The second piece of information has to be derived from an assessment of mitigation costs.

1.9

Estimating Mitigation Costs

As pointed out in Section 1.4, the socio-economic consequences of the Wre mitigation policies will be described by assessing the global economic impact (excluding mitigation benefits) of stabilizing CO2 concentrations in terms of the corresponding percentage losses of gross world product (Gwp), i.e., the relative difference between Gwp in the stabilization case and the business-as-usual Gwp evolution. Thus, Gwp

41


≤ 2 ◦ C in 2300 ≤ 4 ◦ C in 2300

Wre 450

Wre 550

Wre 650

Wre 750

[0, 0.36] [0.41, 0.96]

[0, 0.11] [0.21, 0.84]

[0, 0.05] [0.12, 0.67]

[0, 0.03] [0.03, 0.51]

Table 1.1: Imprecise probabilities for observing the temperature guardrails specified in Section 1.4.

losses heavily depend on the energy-economy model and on the baseline scenarios used to express the reference business-as-usual evolution. Morita and Lee (1998) compiled the most comprehensive database on the global economic impact of stabilizing atmospheric CO2 concentrations to date (see also Morita et al. 2000). The database comprises the results of a multitude of atmospheric CO2 stabilization exercises. They were carried out with different models by taking into account the entire range of plausible baseline scenariosthat were put forward by the Ipcc in the Special Report on Emission Scenarios (Sres) (Nakićenović and Swart 2000). These so-called “Post-Sres” stabilization scenarios do not exactly match the Wre policies presented by Wigley et al. (1996). However, given the generally large uncertainty about mitigation costs, we can make the reasonable assumption that the range of costs of the “Post-Sres” scenarios constitutes a good indicator for the uncertainty about the mitigation costs of the corresponding Wre policy. In this analysis, we consider all “Post-Sres” scenarios contained in the database that provide sufficient information to determine the maximum percentage Gwp loss in any decade of the 21st century as well as the present value percentage Gwp loss calculated on the basis of the time-discounted (at 5% per year) sum of Gwp over the entire 21st century. We will use both indicators for identifying unbearable costs whose burden amounts to a potentially catastrophic loss (see Section 1.4). The considered scenarios comprise (labeled by underlying models, their hosting institutions, the set of investigated Sres baselines and concentration targets): Aim (Nies and Kyoto University, Japan): A1FI, A1B, A2, B1, B2 / 450, 550, 650 ppm (Jiang et al. 2000); Maria (Science University of Tokyo, Japan): A1B, A1T, B1, B2 / 450, 550, 650 ppm (Mori 2000); Message-Macro (Iiasa, Austria): A1FI, A1B, A2, B2 / 550, 750 ppm (Riahi and Roehrl 2000); MiniCam (Pnnl, USA): A1FI, A2, B2 / 450, 550, 650, 750 ppm (Pitcher 2000); and WorldScan (Cpb, Netherlands): A1B, A2, B1, B2 / 450, 550 ppm (Bollen et al. 2000). For every stabilization target, we would like to estimate the uncertainty about the mitigation costs originating from different model assumptions and different baseline scenarios. Unfortunately, none of the models investigated the complete set of plausible combinations of baselines and targets. In order to avoid a bias due to the modelers’ choice of target-baseline combinations, we extrapolated the cost estimates from each model to the minimum and maximum mitigation costs associated with

42

E. Kriegler et al. Maximum loss Present value loss

MARIA

MARIA

Maximum loss Present value loss

(a)

(b)

WorldScan

WorldScan

MiniCAM

0

5

MiniCAM

10

15

MESSAGE

MESSAGE

AIM

AIM

20

25 0


MARIA

1

2

3

4

MARIA (c)

(d) WorldScan

WorldScan

MiniCAM

MiniCAM

MESSAGE

MESSAGE

AIM

AIM 0

1

5


2 3 4 Mitigation cost [% GWP]

5 0

1

2 3 4 Mitigation cost [% GWP]

5

Figure 1.10: Extrapolated estimates of percentage Gwp losses for achieving a stabilization of the atmospheric CO2 concentration from five different model studies: (a) 450 ppm (upper left), (b) 550 ppm (upper right), (c) 650 ppm (lower left), (d) 750 ppm (lower right). The estimated cost intervals are combined to a random set by assigning equal probability weight ν = 0.2 to them. Costs are given either in terms of maximum costs in any decade of the 21st century or in discounted present value Gwp losses.

at least one low (A1T or B1) and high emissions baseline (A1FI, A1G, A1C). The linear extrapolation for each model and each stabilization target was conducted on the basis of cost ratios between either baseline scenarios (for the same target) or different targets (for the same baseline) that were known from the results of the other models. Figure 1.10 shows the extrapolated mitigation cost ranges for each stabilization target and model considered here. The very high maximum percentage Gwp losses of Aim and Message for 450 ppm were obtained by extrapolating the results for 550 ppm using the large 450 ppm/550 ppm loss ratio exhibited by Maria. When assessing the percentage losses depicted in Fig. 1.10, one has to take into account that the relative losses refer to a baseline Gwp evolution that grows in absolute terms by a factor of more than 25 in the best case (Sres scenarios A1F, A1B, A1FI) and still more than 10 times in the worst case (Sres scenario B2) between 1990 and 2100. We combine the model estimates into a random set for each stabilization target by assigning equal probability weights ν = 0.2 to every model estimate. This

43


≤ 2% ≤ 2% ≤ 4% ≤ 4%

maximum ∆Gwp Pv ∆Gwp maximum ∆Gwp Pv ∆Gwp

Wre 450

Wre 550

Wre 650

Wre 750

[0, 1] [0, 1] [0, 1] [0.2, 1]

[0, 1] 1 [0.6, 1] 1

[0.2, 1] 1 1 1

[0.4, 1] 1 1 1

Table 1.2: Imprecise probabilities for observing the economic guardrails on mitigation costs as specified in Section 1.4.

approach is very similar to aggregating subjective probability estimates from different expert sources by assigning equal weights to the source. Such an underlying equiprobability assumption has been frequently criticized, not the least by ourselves in Section 1.2 (for a critique in the context of climate change see, e.g., Keith 1996). Thus, our attempt to construct random sets for the mitigation costs of stabilizing the atmospheric CO2 concentration from literature estimates is clearly unsatisfactory. We wish to remind the reader once more that the concrete numbers used in this analysis should not be taken too literally. Our emphasis is on the conceptual and methodological framework for a policy analysis under ambiguity about catastrophic consequences, not on concrete policy advice. In future work, we intend to use model-based analyses along the lines of Section 1.7 and 1.8 to generate random sets for the mitigation costs. Such analyses need to build on endogenous growth models, which include crucial parameters influencing the mitigation costs like, e.g., the technological learning rate in the renewable energy sector (Edenhofer et al. 2005). If we accept the assumption of equal probability weights over the mitigation cost intervals of the five different models, we can use the resulting random sets to calculate the imprecise probability of avoiding unbearable mitigation costs. As explained in Section 1.4, we have chosen the guardrails ∆Gwp ≤ 2% and ∆Gwp ≤ 4% in terms of either maximum percentage Gwp loss or present value (Pv) percentage Gwp loss to prevent potentially catastrophic economic losses. The resulting imprecise probabilities to avoid such catastrophic economic consequences when aiming at the various stabilization targets entailed by the Wre policies are tabulated in Tab. 1.2. This provides the second piece of information that we need for conducting the envisaged policy analysis on the basis of the imprecise probability to avoid ruin.

1.10

Ranking Stabilization Policies under Ambiguity

The imprecise probability to avert catastrophic climate impacts (see Tab. 1.1) can be combined with the imprecise probability to prevent potentially catastrophic economic losses (see Tab. 1.2) to an overall imprecise probability to avoid ruin (Ipar) for each of the four Wre policies under consideration. If we assume independence between global mean temperature change ∆T and mitigation costs ∆Gwp, such a

44

E. Kriegler et al.

Cost scheme Wre 450 Wre 550 Wre 650 Environmtl. max. ∆Gwp [0, 0.36] [0, 0.11] [0, 0.05] Pv ∆Gwp [0, 0.36] [0, 0.11] [0, 0.05] Capitalist max. ∆Gwp [0, 0.96] [0, 0.84] [0.024, 0.67] Pv ∆Gwp [0, 0.96] [0.21, 0.84] [0.12, 0.67] Easy rider max. ∆Gwp [0, 0.96] [0.126, 0.84] [0.12, 0.67] Pv ∆Gwp [0.082, 0.96] [0.21, 0.84] [0.12, 0.67] Alarmist max. ∆Gwp [0, 0.36] [0, 0.11] [0, 0.05] Pv ∆Gwp [0, 0.36] [0, 0.11] [0, 0.05]

Wre 750 [0, 0.03] [0, 0.03] [0.012, 0.51] [0.03, 0.51] [0.03, 0.51] [0.03, 0.51] [0, 0.03] [0, 0.03]

Table 1.3: Imprecise probabilities to avoid ecological and economic ruin for the four different types of decision makers considered here (see, Section 1.4). Shown are the results for both costing schemes to evaluate percentage Gwp losses.

combination of imprecise probabilities is straightforward. The event of avoiding ruin, i.e., C¯ = { (∆T (2300), ∆Gwp) | ∆T (2300) ≤ ∆T † (2300) ∧ ∆Gwp ≤ ∆Gwp† } (see Section 1.4) is a product set in the consequence space X = ∆T × ∆Gwp. Although there exist various independence concepts for imprecise probabilities (Couso et al. 2000), they lead to the same results on product sets. If two marginal events A and B are independent, the imprecise probability of their joint occurrence, i.e. the event A × B ⊂ X, is given by (Walley 1991, Chapter 9.3.5) [ P (A × B), P (A × B) ] = [ P (A) · P (B), P (A) · P (B) ] . Independence between ∆T and ∆Gwp for a given Wre policy can be justified on the grounds that the uncertainty about these quantities arises from factors that are entirely located in the realm of either the natural system (∆T ) or the socio-economic system (∆Gwp). Table 1.3 shows the resulting Ipar values for the four different Wre policies under different assessments about what “avoiding catastrophe” actually means. As explained in Section 1.4, we consider four “cartoons” of decision makers who differ in their expectation about the resilience of the natural and economic system (see guardrail definitions 1.2-1.5). We also distinguish two schemes for valuing the mitigation costs over the 21st century (see Section 1.9). The Ipar values already reveal a basic characteristic of the decision making problem. The estimates about mitigation costs of all Wre policies exhibit an upper probability P = 1 for avoiding potentially catastrophic economic losses. Due to this characteristic, the upper probability of avoiding ruin is determined solely by the upper probability of averting catastrophic climate impacts, which is very sensitive to the choice of Wre policy. The situation is different for the lower probability of avoiding ruin. It is very small in most cases under consideration since either the lower probability for observing the temperature guardrail is small for CO2 stabilization

45


Undominated interval max. and Pv ∆Gwp Maximal max. and Pv ∆Gwp E-admissible max. and Pv ∆Gwp Bival. undominated max. ∆Gwp Pv ∆Gwp Γ-maximin (λ = 1) max. ∆Gwp Pv ∆Gwp Γ-maximax (λ = 0) max. and Pv ∆Gwp

Environmtl.

Capitalist

Easy rider

Alarmist

all

all

all

all

all

all

all

all

all

all

all

all

450 450

450, 650 450, 550

450, 550 450, 550

450 450

— —

650 550

550 550

— —

450

450

450

450

Table 1.4: Admissible policies for the four different types of decision makers under the various decision making criteria presented in Section 1.5. Shown are the results for both costing schemes to evaluate percentage Gwp losses.

levels of 550 ppm and higher, or the lower probability of avoiding economic ruin is small for the more stringent stabilization policy of 450 ppm. Overall, Ipar is clearly dominated by the imprecise probability of avoiding catastrophic climate impacts. As a result, the Ipar values for the alarmist and environmentalist exhibiting the same choice of guardrail for Gmt increase are identical and independent of the scheme to value mitigation costs. Likewise, capitalist and easy riders exhibit Ipar values that are similar in magnitude. The structure of the Ipar values will be reflected in the results of the policy analysis. Table 1.4 shows the admissible policies for the four different types of decision makers and the various decision criteria under ambiguity discussed in Section 1.5. The admissible policies with regard to interval dominance, bivalent dominance, Γ-maximin / maximax and the Hurwicz criterion can be directly determined from knowledge of the lower and upper probability to avoid ruin as provided in Tab. 1.3. The situation is more involved for the decision criteria of maximality and E-admissibility. In these cases, we need to reconstruct the set of probability measures that are compatible with the Ipar values. Since we consider only four policies with two different outcomes (to ruin or not to ruin society), the reconstruction of the class of compatible probabilities is feasible. In total, we have to consider 24 elementary events with n = 2 the number of outcomes of a policy and m = 4 the number of policies. All probability distributions that distribute the probability

46

E. Kriegler et al.

masses over the 16 elementary events in a way that respects the constraints provided by the Ipar values (lower and upper bound for each policy - 8 constraints) are compatible with the given state of information. To identify a policy as E-admissible, it suffices to construct a compatible probability distribution under which the policy maximizes the probability to avoid ruin. For the Ipar constraints provided in Tab. 1.3, this can be done for all policies independently of the type of decision maker and costing scheme. Hence, all stabilization policies are E-admissible. Since every E-admissible policy is also maximal (see Section 1.5), all stabilization policies need to be maximal as well. It turns out that interval dominance, maximality and E-admissibility are not decisive enough to narrow down the set of available policies. This points to the potential for controversy in the climate policy arena. To resolve the indecision, we need to look for stronger decision criteria under ambiguity. When requiring bivalent dominance, the set of admissible policies is reduced decisively. For environmentalists and alarmists, the dominance criterion already suffices to agree on the Wre 450 policy constituting the lowest stabilization target under consideration. Even easy riders and capitalists agree on a low stabilization target (Wre 450 or 550) except for the case of capitalists who value the mitigation costs according to the maximum percentage loss experienced in any decade. In the perspective of environmentalists and alarmists, the lower probability to avoid ruin vanishes for all Wre policies. Thus, both types of decision makers are struck by indifference under a Γ-maximin criterion symbolizing a precautionary attitude. Adopting a precautionary attitude in a situation, where no confidence whatsoever in avoiding catastrophe can be gained, leads to an indifference that resembles the agony of the desperate. The situation is different for capitalists and easy riders. If the mitigation costs are assessed in terms of their present value, both groups would agree on the Wre 550 policy that avoids economic ruin with certainty, but provides only half of the lower probability of observing the climate guardrail of the Wre 450 policy. If maximum percentage Gwp losses in any decade are considered capitalists become more conservative and favor the Wre 650 policy. While the different types of decision makers exhibit different choice sets under Γ-maximin, they all agree on the Wre 450 policy under Γ-maximax. This is so because the upper probability to avoid ruin is determined solely by the plausibility of catastrophic climate impacts. In this situation, opportunity minded decision makers, no matter of what type, will tend to low stabilization targets. The overall tendency towards the Wre 450 policy can be seen clearly, when considering the Hurwicz criterion over the range of possible values of the security index λ ∈ [0, 1]. Figure 1.11 shows the change of optimal policies, when reducing λ from a precautionary (λ = 1, Γ-maximin) to an opportunity-seeking attitude (λ = 0, Γ-maximax). For λ < 0.36 all types of decision makers would choose Wre 450 independently of the scheme to value mitigation costs. In the overwhelming majority of the remaining cases, they would choose either the Wre 450 or Wre 550 policy (for any value

47


Best policy

WRE 750

Environmentalist maximum and present value D GWP loss

Alarmist maximum and present value D GWP loss

(a)

(b)

WRE 650

WRE 550

WRE 450

0

Best policy

WRE 750

0.2

0.4 0.6 Security index l

0.8

Capitalist maximum DGWP loss present value D GWP loss

1

0

0.2


0.8

Easy Rider maximum DGWP loss present value D GWP loss

(c)

1

(d)

WRE 650

WRE 550

WRE 450

0

0.2


0.8

1

0

0.2


0.8

1

Figure 1.11: Best policies for the four different types of decision makers – (a) environmentalist, (b) alarmist, (c) capitalist, (d) easy rider – in dependence of the security index λ, when the Hurwicz criterion is used to rank the policies (see, Section 1.5). Shown are the results for both costing schemes to evaluate percentage Gwp losses. The dots in panels (a) and (b) indicate the indecision of environmentalist and alarmist in the extreme case λ = 1 (agony of the desperate).

of λ ∈ [0, 1] when considering Pv Gwp loss, and for λ < 0.96, when considering maximum percentage Gwp loss). Thus, we find a rather robust result. In the majority of decision maker perspectives considered here, the Wre 450 or Wre 550 policies are the preferred choice under bivalent dominance, Γ-maximin, Γ-maximax and the Hurwicz criterion. This reflects the structure of the Ipar values tabulated in Tab. 1.3, which are mainly dominated by the uncertainty about climate impacts.

1.11

Summary and Conclusion

We have presented a conceptual and methodological framework for conducting policy analyses under ambiguity about catastrophic consequences. The Ellsberg experiment has shown that the presence of ambiguity can give rise to a qualitatively new

48

E. Kriegler et al.

type of uncertainty aversion, which is very different from the risk aversion related to the shape of the utility function. Likewise, the possibility of potentially catastrophic consequences can fundamentally alter the decision behavior by, e.g., giving rise to lexicographic preferences. Environmental policy problems constitute a prime example where both ambiguity and potentially catastrophic consequences are part and parcel of the decision situation. In this analysis, the challenge posed by global climate change has served as an example to conduct a policy analysis under ambiguity about catastrophic consequences. We have employed the imprecise probability of ruin as a decision indicator that accounts for both ambiguity and potentially catastrophic losses. Four exemplary climate policies aiming at stabilizing the atmospheric CO2 concentrations at levels from 450 to 750 ppm were considered. For each policy, we calculated the imprecise probability on global mean temperature (Gmt) change in the year 2300 by constructing an imprecise probability for climate sensitivity and feeding it into a simple energy balance model of the temperature response to a radiative forcing of the earth system. A literature survey was employed to generate imprecise probabilities for the mitigation costs associated with the Wre policies. The climatic and socio-economic uncertainty was combined into a joint imprecise probability to avoid ruin for four different specifications of the catastrophic domain. We considered seven different decision criteria under ambiguity to identify the set of admissible policies. Results show that low stabilization targets (450 and 550 ppm) are favored over less ambitious targets (650 and 750 ppm) in the large majority of decision maker perspectives for all decision criteria that focus on the lower and upper extreme of the plausible values for the probability to avoid ruin. On the basis of this prototypical analysis, we conclude that • the ambiguity in the estimates for future climate change as well as mitigation costs is very large. The conceptual and methodological framework that we employed is capable to process such poor states of information, and to account for the particular nature of ambiguity. • the imprecise probability to avoid ruin is a good indicator to explore the structure of the uncertainty that is relevant for the decision making problem. Therefore, it provides a meaningful basis for conducting a decision analysis over acts under ambiguity about catastrophic consequences. • even in cases where the imprecision in the probability estimates for avoiding ruin is large, a small set of optimal policies might be identified. Thus, large ambiguity does not necessarily lead to a dilution of the policy recommendation. Although these findings are promising, a host of open questions and challenges remains. It does not suffice that imprecise probability concepts carry the potential to consistently capture the different degrees of information that are encountered in assessments of global environmental change. They also need to be applicable to


49

dynamic models of the environmental problem in question which usually exhibit a high-dimensional continuous uncertainty space. In this analysis, we have employed a simple dynamic model that could be solved quasi-analytically for extending random sets and possibility distributions onto prognostic model variables. Methods to solve the extension problems (1.18) and (1.21) for more complex models, such as branch and bound techniques, need to be further investigated with respect to their applicability to global environmental policy analyses. Another challenge is the construction of imprecise probabilities from a direct comparison of historical data with model prognoses. Our approach to combine probabilistic estimates from the literature into a distribution band might be satisfactory to establish an imprecise prior for the model parameters, but this prior should be updated with observational data by using some type of Bayesian procedure. In Kriegler (2005), we have presented a methodology to update a prior belief function with a likelihood of reproducing the observations. Our analysis shows, however, that much more work has to be done in this area to avoid the emergence of overly imprecise posterior information. A detailed investigation reveals that the particular nature of distribution bands can give rise to excessively imprecise estimates for the posterior parameter uncertainty as well as for prognostic model variables. Distribution bands contain all degenerate probabilities whose Cdfs are enclosed by the lower and upper bound of P[F , F ]. While it can be argued whether such probabilities should be included in the analysis (assumption (A) in Section 1.7), they are having a large impact on the resulting uncertainty estimates for prognostic variables (Kozine and Krymsky 2003). An alternative might be to consider more general classes of probability families, e.g., those that are bounded by lower and upper probability density functions. However, such probability families cannot be decomposed easily into random sets or even possibility distributions. An algorithm to approximate a probability density family with a random set was presented in Hall and Lawry (2004). Another challenge will arise when it becomes necessary to combine random sets or possibilistic estimates for several uncertain parameters into a joint imprecise probability. In general, the classes of possibilistic probability families as well as belief functions (associated with random sets) are not closed under such an operation. Sentz and Ferson (2002) and Hall and Lawry (2004) discuss a variety of combination rules for random sets. Kriegler and Held (2005) give some guidance for combining marginal random sets under the assumption of independence or unknown dependency. However, more application-oriented research on the construction of joint imprecise probabilities from marginal estimates is needed. Hence, there is no doubt that much more research is needed to enhance the generality of the framework presented here, and its applicability to more complex models of environmental change. However, our prototypical climate policy analysis offers a starting point. It shows how distribution bands, random sets and possibility distributions can be utilized to assess the imprecise probability to avoid ruin

50

E. Kriegler et al.

for a given set of policies. It also shows how this information can be fed into a policy analysis based on a variety of decision criteria that can deal with ambiguous information. This underlines our assertion that ambiguity and the possibility of catastrophic consequences have to be included in an environmental policy analysis, and that there exists an operational framework to do so in a meaningful manner.

References Alley, R. B., J. Marotzke, W. D. Nordhaus, J. T. Overpeck, D. M. Peteet, R. A. Pielke, R. T. Pierrehumbert, P. B. Rhines, T. F. Stocker, L. D. Talley, and J. M. Wallace (2003). Abrupt climate change. Science 299, 2005–2010. Andronova, N. G. and M. E. Schlesinger (2001). Objective estimation of the probability density function for climate sensitivity. J. of Geophysical Research 106 (D19), 22605– 22611. Anscombe, F. J. and R. J. Aumann (1963). A definition of subjective probability. The Annals of Mathematics and Statistics 34, 199–205. Arrow, K. J. and L. Hurwicz (1972). An optimality criterion for decision making under ignorance. In C. F. Carter and J. L. Ford (Eds.), Uncertainty and Expectations in Economics. Oxford: Basil Blackwell. Basu, S. and A. DasGupta (1990). Bayesian analysis under distribution bands. Technical Report 90–48, Department of Statistics, Purdue University. Billingsley, P. (1995). Probability and Measure. New York: Wiley Interscience. Bollen, J., T. Manders, and H. Timmer (2000). The benefits and costs of waiting - early action versus delayed response in post-sres stabilization scenarios. Environmental Economics and Policy Studies 3, 143–158. Casadesus-Masanell, R., P. Klibanoff, and E. Ozdenoren (2000). Maxmin expected utility over savage acts with a set of priors. Journal of Economic Theory 92, 35–66. Chateauneuf, A. and J. Y. Jaffray (1989). Some characterization of lower probabilities and other monotone capacities through the use of M¨ obius inversion. Mathematical Social Sciences 17, 263–283. Chevé, M. and R. Congar (2003). Managing environmental risks under scientific uncertainty and controversy. Paper presented at the EARE 2003 conference, Bilbao, Spain. Chichilnisky, G. (2000). An axiomatic approach to choice under uncertainty with catastrophic risks. Resource and Energy Economics 22, 221–231. Couso, I., S. Moral, and P. Walley (2000). A survey of concepts of independence for imprecise probabilities. Risk, Decision and Policy 5, 165–181. Cubasch, U. and G. Meehl (2001). Projections of future climate change. In J. Houghton and Y. Ding (Eds.), Climate Change 2001: The Scientific Basis, Contribution of Working Group I to the Third Assessment Report of the IPCC, pp. 525–582. Cambridge: Cambridge University Press. de Cooman, G. and D. Aeyels (1999). Supremum preserving upper probabilities. Information Sciences 118, 173–212.


51

Debreu, G. (1954). Representation of a preference ordering by a numerical function. In R. M. Thrall, C. H. Combs, and R. L. Davids (Eds.), Decision Processes, New York, pp. 159–165. Wiley & Sons. Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38, 325–339. Dessai, S. and M. Hulme (2003). Does climate policy need probabilities? Tyndall Centre Working Paper 34, Tyndall Centre. Dubois, D. and H. Prade (1988). Possibility Theory: An Approach to Computerized Processessing of Uncertainty. New York: Plenum Press. Dubois, D. and H. Prade (1990). Consonant approximations of belief functions. International Journal of Approximate Reasoning 4, 419–449. Dubois, D. and H. Prade (1991). Random sets and fuzzy interval analysis. Fuzzy Sets and Systems 42, 87–101. Dubois, D. and H. Prade (1992). When upper probabilities are possibility measures. Fuzzy Sets and Systems 49, 203–244. Dutton, J. A. (1995). An analytical model of atmospheric feedback and global temperature change. Journal of Climate 8, 1122–1139. Edenhofer, O., N. Bauer, and E. Kriegler (2005). The impact of technological change on climate protection and welfare: Insights from the model mind. Ecological Economics 54, 277–292. Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics 75, 643–669. European Commission (2000). Communication of the Commission on the precautionary principle. COM(2000) 1. Ferson, S., L. R. G. V. Kreinovich, D. S. Meyers, and K. Sentz (2002). Constructing probability boxes and Dempster-Shafer structures. Technical Report SAND2002-4015, Sandia National Laboratories, Albuquerque. Forest, C. E., P. H. Stone, A. P. Sokolow, M. R. Allen, and M. D. Webster (2002). Quantifying uncertainties in climate system properties with the use of recent climate observations. Science 295, 113–117. Gilboa, I. and D. Schmeidler (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18, 141–153. Gr¨ ubler, A. and N. Nakicenovic (2001). Identifying dangers in an uncertain climate. Nature 412, 15. Hall, J. and J. Lawry (2004). Generation, combination and extension of random set approximations to coherent lower and upper probabilities. Reliability Engineering and System Safety 85, 89–101. Hurwicz, L. (1951). Optimality criteria for decision making under ignorance. Cowles Commision Discussion Paper. Jaffray, J. Y. (1989). Linear utility theory for belief functions. Operations Research Letters 8, 107–112.

52

E. Kriegler et al. Jaffray, J. Y. (1994). Dynamic decision making with belief functions. In R. R. Yager, J. Kacprzyk, and M. Fedrizzi (Eds.), Advances in the Dempster-Shafer theory of evidence, pp. 331–352. Wiley, New York. Jiang, K., T. Morita, T. Matsui, and Y. Matsuoka (2000). Global long-term greenhouse gas mitigation emission scenarios based on AIM. Environmental Economics and Policy Studies 3, 239–254. Kaysen, C., S. E. Miller, M. B. Malin, W. D. Nordhaus, and J. D. Steinbruner (2002). War with Iraq: Costs, Consequences, and Alternatives. Report, American Academy of Arts and Sciences, Cambridge. Keith, D. W. (1996). When is it appropriate to combine expert judgements? Climatic Change 33, 139–143. Knutti, R., T. F. Stocker, F. Joos, and G. K. Plattner (2002). Constraints on radiative forcing and future climate change from observations and climate model ensembles. Nature 416, 719–723. Kozine, I. O. and V. Krymsky (2003). Reducing uncertainty by imprecise judgements on probability distributions: Application to system reliability. In Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, pp. 335–344. Kriegler, E. (2005). Imprecise probability analysis for integrated assessment of climate change. Ph. D. thesis, University of Potsdam. http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:kobv:517opus-5611. Kriegler, E. and H. Held (2003). Transformation of probability families using possibility theory. Working Paper, Potsdam Institute for Climate Impact Research. Kriegler, E. and H. Held (2005). Utilizing random sets for the estimation of future climate change. International Journal of Approximate Reasoning 39, 185–209. Kuznetsov, V. P. (1991). Interval Statistical Methods (in Russian). Radio i Svyaz Publ., Moscow. Levi, I. (1980). The Enterprise of Knowledge. MIT Press, Cambridge, Masachusetts. Morgan, M. G. and D. W. Keith (1995). Subjective judgments by climate experts. Environmental Policy Analysis 29, 468–476. Mori, S. (2000). Effects of carbon emission mitigation options under carbon concentration stabilization scenarios. Environmental Economics and Policy Studies 3, 125–142. Morita, T. and H.-C. Lee (1998). IPCC SRES database, Version 1.0. Emissions scenario database prepared for IPCC Special Report on Emission Scenarios. Morita, T., N. Nakicenovic, and J. Robinson (2000). Overview of mitigation scenarios for global climate stabilization based on new IPCC emission scenarios (SRES). Environmental Economics and Policy Studies 3, 65–88. Murphy, J. M., D. M. H. Sexton, D. N. Barnett, G. S. Jones, M. J. Webb, M. Collins, and D. A. Stainforth (2004). Quantification of modelling uncertainties in a large ensemble of climate change simulations. Nature 430, 768–772.


53

Nakićenović, N. and R. Swart (2000). Emissions Scenarios. Special Report of the IPCC. Cambridge: Cambridge University Press. OECD (2001). OECD in Figures. Report, Organisation for Economic Co-operation and Development, Paris. Parry, M., N. Arnell, T. McMichael, R. Nicholls, P. Martens, S. Kovats, M. Livermore, C. Rosenzweig, A. Iglesias, and G. Fischer (2001). Millions at risk: defining critical climate change threats and targets. Global Environmental Change 11, 181–183. Pitcher, H. M. (2000). Mitigation options in a sustainable development world. Environmental Economics and Policy Studies 3, 173–194. Regan, H. M., S. Ferson, and D. Berleant (2004). Equivalence of methods for uncertainty propagation of real-valued random variables. International Journal of Approximate Reasoning 36, 1–30. Riahi, K. and R. A. Roehrl (2000). Energy technology strategies for carbon dioxide mitigation and sustainable development. Environmental Economics and Policy Studies 3, 89–124. Savage, L. J. (1954). The Foundations of Statistics. New York: Wiley & Sons. Schmidt, U. (1998). Axiomatic utility theory under risk: non-archimedian representations and application to insurance economics. Heidelberg: Springer-Verlag. Schneider, S. H. (2001). What is dangerous climate change? Nature 411, 17–19. Seidenfeld, T., M. J. Schervish, and J. B. Kadane (1995). A representation of partially ordered preferences. Annals of Statistics 23, 2168–2217. Sentz, K. and S. Ferson (2002). Combination of evidence in Dempster-Shafer theory. Technical Report SAND2002-0835, Sandia National Laboratories, Albuquerque, NM. Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton: Princeton University Press. Smets, P. and R. Kennes (1994). The transferable belief model. Artificial Intelligence 66, 191–234. Smith, J., H.-J. Schellnhuber, and M. Q. Mirza (2001). Lines of evidence for vulnerability to climate change: A synthesis. In O. Canziani and J. McCarthy (Eds.), Climate Change 2001: Impacts, Adaptation and Vulnerability, pp. 914–967. Cambridge: Cambridge University Press. Troffaes, M. C. M. (2004). Decision making with imprecise probabilities: A short review. In F. G. Cozman (Ed.), SIPTA Newsletter, Volume 2(1), Manno, Switzerland, pp. 4–7. Society for Imprecise Probability Theory and Applications. van Asselt, M. and J. Rotmans (1996). Uncertainty in perspective: a cultural perspective based approach. Global Environmental Change 6, 121–158. Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London. WBGU (1995). Scenario for the Derivation of Global CO2 Reduction Targets and Implementation Strategies. Special Report for the First Conference of the Parties to the FCCC. Bremerhaven: German Advisory Council on Global Change (WBGU).

54

E. Kriegler et al. Weichselberger, K. (2000). The theory of interval probability as a unifying concept for uncertainty. International Journal of Approximate Reasoning 24, 149–170. Wigley, T. M. L., R. Richels, and J. A. Edmonds (1996). Alternative emissions pathways for stabilizing concentrations. Nature 379, 240–243. Williamson, R. C. and T. Downs (1990). Probabilistic arithmetic I: numerical methods for calculating convolutions and dependency bounds. International Journal of Approximate Reasoning 4, 89–158. Yager, R. R. (1986). Arithmetic and other operations on Dempster-Shafer structures. International Journal of Man-Machine Studies 25, 357–366. Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1, 3–29. Zaffalon, M., K. Wesnes, and O. Petrini (2003). Reliable diagnosis of dementia by the naive credal classifier inferred from incomplete cognitive data. Artificial Intelligence in Medicine 29, 61–79.

CHAPTER 2

An Introduction to Viability Theory and Management of Renewable Resources Jean-Pierre Aubin and Patrick Saint-Pierre

2.1

Introduction

The main purpose of viability theory is to explain the evolution of the state of a control system, governed by non-deterministic dynamics and subjected to viability constraints, to reveal the concealed feedbacks which allow the system to be regulated and provide selection mechanisms for implementing them. It assumes implicitly an “opportunistic” and “conservative” behavior of the system: a behavior which enables the system to keep viable solutions as long as its potential for exploration (or its lack of determinism) — described by the availability of several evolutions — makes possible its regulation. We illustrate the main concepts and results of viability theory by revisiting the Verhulst type models in population dynamics, by providing the class of all Malthusian feedbacks (mapping states to growth rates) that guarantee the viability of the evolutions, and adapting these models to the management of renewable resources. Other examples of viability constraints are provided by architectures of networks imposing constraints described by connectionist tensors operating on coalitions of actors linked by the network. The question raises how to modify a given dynamical system governing the evolution of the signals, the connectionist tensors and the coalitions in such a way that the architecture remains viable.

2.1.1

From Malthus to Verhulst and Beyond

A population for which there is a constant supply of resources, no predators, but limited space provides a simple one-dimensional example: at each instant t ≥ 0, the population x(t) must remain confined in an interval K := [a, b] where 0 < a < b. The maximal population size b is called the carrying capacity. The dynamics are unknown, really, and several models have been proposed. They are all particular cases of a general dynamical systems of the form x′ (t) = u e x(t) x(t) ,

where u e : [a, b] 7→ R is a model of the growth rate of the population feeding back on the size of the population. Malthus advocated in 1798 to choose a constant positive

56

J.-P. Aubin and P. Saint-Pierre

growth rate u e(x) = u > 0, leading to an exponential evolution x(t) = xeut starting at x, which cannot be viable in any bounded interval. This the is price to pay for linearity of the dynamic of the population: “population, when unchecked, increases in a geometrical ratio”. Population dynamical models providing evolutions growing fast when the population is small and declining when it becomes large where looked for to compensate for the never ending expansion of the Malthusian model. The purely logistic Verhulst model with feedback of the form u e(x) := r(b − x) ,

proposed in 1838 by the Belgian mathematician Pierre-Fran¸cois Verhulst (rediscovered in the 1930’s by Raymond Pearl) does the job. The solution to the logistic ′ differential equation x (t) = rx(t) b − x(t) starting from x ∈ [a, b] are respectively equal to bx . x(t) = x + (b − x)e−rt They remain confined in the interval [a, b] and converge to the equilibrium b when t 7→ +∞. The logistic model and the S-shape graph of its solution became popular in the 1920’s and stood as the evolutionary model of a large manifold of growths, from the tail of rats to the size of men. Instead of finding one feedback u e satisfying the above requirements by trial and error, we proceed systematically to design feedbacks by leaving the choice of the growth rates open, regarding them as regulons of the system x′ (t) = u(t) x(t) .

(2.1)

These regulons are chosen to govern evolutions confined in the interval [a, b]. Imposing a bound on the velocities of the growth rates of the population further restrict the selection of such regulons. We suggest to characterize all of the feedbacks governing evolutions viable on the interval [a, b] under inertia bound c. Among them, we will find the Verhulst feedback, to which we add two explicit other ones, providing inert and heavy evolutions. Inert evolutions are obtained by taking the maximal velocities allowed: u′ (t) = ±c. Heavy evolutions combine Malthusian and inert growth. They are obtained starting with constant growth rate (Malthusian evolution) of the state (by taking u′ (t) = 0) until the time when the inertia bound is met. This provides warning signals telling when, where and how the regulons must evolve: the feedback becomes a specific one, the inert feedback, providing constant (negative) velocities u′ (t) = −c of growth rates driving the state at its carrying capacity b. Before detailing these facts and describing other results, we need for this purpose the concept of viability kernel, a central concept of viability theory.

Viability Theory and Management of Renewable Resources

2.1.2

57

Purpose of this Chapter

The purpose of this survey paper is thus to present a brief introduction of viability theory that studies adaptive type evolution of complex systems under uncertain environment and viability constraints that are found in many domains involving living beings, from biology to cognitive sciences, from ecology to sociology and economics. Instead of applying only known mathematical and algorithmic techniques, most of them motivated by physics and not necessarily adapted to such problems, viability theory designs and develops mathematical and algorithmic methods for studying the evolution of such systems, organizations and networks of systems, 1. constrained to adapt to a (possibly co-evolving) environment, 2. evolving under contingent, stochastic or tychastic1 uncertainty, 3. using for this purpose regulation controls, and in the case of networks, connectionist matrices or tensors, 4. the evolution of which is governed by regulation laws that are then “computed” according to given principles such as the inertia principle, 5. the evolution being either continuous, discrete, or an “hybrid” of the two when impulses are involved, 6. the evolution concerning both the variables and the environmental constraints mutational viability), 7. the non-viable dynamics being corrected by introducing adequate controls when necessary (viability multipliers), 8. or by introducing the “viability kernel” of a constrained set under a nonlinear controlled system (either continuous or hybrid), that is the set of initial states from which starts at least one evolution reaching a target in finite time while obeying state (viability) constraints. 1

Uncertainty is translated mathematically by parameters on which actors, agents, decision makers, etc. These parameters are often perturbations, disturbances (as in “robust control” or “differential games against nature”) or more generally, tyches (meaning “chance” in classical Greek, from the Goddess Tyche) ranging over a state-dependent tychastic map. They could be called “random variables” if this vocabulary were not already confiscated by probabilists. This is why we borrow the term of tychastic evolution to Charles Peirce who introduced it in a paper published in 1893 under the title evolutionary love: ”Three modes of evolution have thus been brought before us: evolution by fortuitous variation, evolution by mechanical necessity, and evolution by creative love. We may term them tychastic evolution, or tychasm, anancastic evolution, or anancasm, and agapastic evolution, or agapasm. One can prove that stochastic viability is a (very) particular case of tychastic viability (see Aubin and Da Prato 1998 and Aubin and Doss 2003 for instance).

58


It is by now a consensus that the evolution of many variables describing systems, organizations, networks arising in biology and human and social sciences do not evolve in a deterministic way, and may be, not even in a stochastic way as it is usually understood, but with a Darwinian flavor, where inter-temporal optimality selection mechanisms are replaced by several forms of “viability”, a word encompassing polysemous concepts as stability, confinement, homeostasis, tolerable windows approach (see, e.g., Petschel-Held et al. 1999; Bruckner et al. 1999; Tóth 2003) etc., expressing the idea that some variables must obey some constraints. Inter-temporal optimization is replaced by myopic selection mechanisms that involve present knowledge, sometimes the knowledge of the history (or the path) of the evolution, instead of anticipations or knowledge of the future (whenever the evolution of these systems cannot be reproduced experimentally). Uncertainty does not necessarily obey statistical laws, but only unpredictable rare events (tyches, or perturbations, disturbances) that obey no statistical law, that must be avoided at all costs (precautionary principle or robust control). These systems can be regulated by using regulation (or cybernetical) controls that have to be chosen as feedbacks for guaranteeing the viability of a system and/or the capturability of targets and objectives, possibly against tyches (perturbations played by nature). The sets of controls as well as the constrained set can be “toll sets” (a variant of fuzzy sets) as in Aubin and Dordan 1996, for instance. The purpose of viability theory is to attempt to answer directly the question that some economists or biologists ask: Complex organizations, systems and networks, yes, but for what purpose? One can propose the following answer: to adapt to the environment. This is the case in biology, since the Claude Bernard’s “constance du milieu intérieur” and the “homeostasis” of Walter Cannon. This is naturally the case in ecology and environmental studies. This is also the case in economics when we have to adapt to scarcity constraints, balances between supply and demand, and many other ones. The environment is described by constraints of various kinds (representing objectives, physical and economic constraints, “stability” constraints, etc.) that can never be violated. At the same time, the actions, the messages, the coalitions of actors and connectionist operators do evolve, and their evolution must be consistent with the constraints, with objectives reached at (successive) finite times (and/or must be selected through inter-temporal criteria). There is no reason why collective constraints are satisfied at each instant by evolutions under uncertainty governed by stochastic or tychastic control dynamical systems. This leads to the study of how to correct either the dynamics, and/or the constraints in order to re-establish this consistency. This may allow us to provide an explanation of the formation and the evolution of the architecture of the system and of their variables. Presented in such an evolutionary perspective, this approach of (complex) evolution departs from the main stream of modeling studying static networks with graph


59

theory and dynamical complex systems by ordinary or partial differential equations, a task difficult outside the physical sciences. For dealing with these issues, one needs concepts and formal tools, algorithms and mathematical techniques motivated by complex systems evolving under uncertainty. For instance, and without entering into the details, we can mention systems sharing such common features arising in economics, where the viability constraints are the scarcity constraints. We can replace the fundamental Walrasian model of resource allocations by a decentralized dynamical model in which the role of the controls is played by the prices or other economic decentralizing messages (as well as coalitions of consumers, interest rates, and so forth). The regulation law can be interpreted as the behavior of Adam Smith’s invisible hand choosing the prices as a function of the allocations, dynamical connectionist networks and/or dynamical cooperative games, where coalitions of players may play the role of controls: each coalition acts on the environment by changing it through dynamical systems. The viability constraints are given by the architecture of the network allowed to evolve, genetics and population genetics, where the viability constraints are the ecological constraints, the state describes the phenotype and the controls are genotypes or fitness matrices. sociological sciences, where a society can be interpreted as a set of individuals subject to viability constraints. They correspond to what is necessary to the survival of the social organization. Laws and other cultural codes are then devised to provide each individual with psychological and economical means of survival as well as guidelines for avoiding conflicts. These cultural codes play the role of regulation controls. cognitive sciences, where, at least at one level of investigation, the variables describe the sensory-motor activities of the cognitive system, while the controls translate into what could be called a conceptual control (which is the synaptic matrix in neural networks.) control theory and differential games, conveniently revisited, can provide many metaphors and tools for grasping the above problems. Many problems in control design, stability, reachability, inter-temporal optimality, viability and capturability, observability and set-valued estimation can be formulated in terms of viability kernels. The viability kernel algorithm computes this set. Outline: We devote the three first sections to the description of the main concepts and basic results of viability theory: evolutions, viability kernels and capture basins under evolutionary systems in the first section, characterization of viability and of

60


the adaptation law in the second one, construction of static and dynamic feedbacks, including the slow and heavy ones, in the third. We shall illustrate the main concepts and results of viability theory by revisiting the Verhulst type models in population dynamics, by providing the class of all Malthusian feedbacks (mapping states to growth rates) that guarantee the viability of the evolutions, and adapting these models to the management of renewable resources. More complex applications go beyond the format of this introduction to viability theory. We finally address the issues related to the restoration of the viability when the constraints are not viable under an evolutionary system: correcting the dynamics by viability multipliers, re-initializing the state whenever the viability is at stake, providing hybrids of continuous and discrete time evolutions, or changing the constraints by governing their evolutions by mutational equations.

2.2

The Mathematical Framework

For more details on viability theory, we refer to Aubin (2000) in the case of differential equations, to Aubin (1991) and Aubin (1997) for the state of the art and economic applications up until 1995 and to the book Aubin et al. (2006) for recent advances .

2.2.1

Viability and Capturability

Let X denote the state space of the system. Evolutions describe the behavior of the state of the system as a function of time t ∈ R+ := [0, . . . , +∞[ ranging over the set of nonnegative real numbers or scalars t ∈ R+ . We shall assume all along that 1. the state space is a finite dimensional vector space X := Rn , 2. evolutions are continuous functions x(·) : t ∈ R+ 7→ x(t) ∈ X describing the evolution of the state x(t). We denote the space of continuous evolutions x(·) by C(0, ∞; X) or, in short, C(X). Some evolutions, mainly motivated by physics, are classical: equilibria and periodic evolutions. But these properties are not necessarily adequate for problems arising in economics, biology, cognitive sciences and other domains involving living beings. Hence we add the concept of evolutions viable in a constrained set K ⊂ X (the environment) or capturing a target C ⊂ K in finite time to the list of properties satisfied by evolutions. Therefore, we consider mainly evolutions x(·) viable in a subset K ⊂ X representing a constrained set (an environment) in which the trajectory of the evolution must remain forever: ∀ t ≥ 0, x(t) ∈ K .

(2.2)


61

Alternatively, a “target” C ⊂ K being given, we distinguish evolutions x(·) capturing the target C in the sense that they are viable in K until they reach the target C in finite time: x(T ) ∈ C . (2.3) ∃ T ≥ 0 such that ∀ t ∈ [0, T ], x(t) ∈ K We devote our paper to the study of the set of evolutions viable in K outside C, i.e. that are viable in K forever or until they reach the target C in finite time.

2.2.2

The Evolutionary System

Next, we provide the mathematical description of one of the “engines” governing the evolution of the state. We assume that there exists a control parameter, or, better, a regulatory parameter, called a regulon, that influences the evolution of the state of the system. This dynamical system takes the form of a control system with (multi-valued) feedbacks :

(i) x′ (t) = f x(t), u(t) (ii) u(t) ∈ U x(t)

(action) (contingent retro-action)

(2.4)

taking into accountthe a priori availability of several regulons u(t) ∈ U x(t) chosen in a subset U x(t) ⊂ Y of another finite dimensional vector-space Y subjected to state-dependent constraints. Once the initial state is fixed, the first equation describes how the regulon acts on the velocities of the system whereas the second inclusion shows how the state (or an observation on the state) can retroact through (several) regulons in a vicariant way. We observe that there are many evolutions starting from a given initial state x0 , one for each time-dependent regulon t 7→ u(t). The set-valued map U : X Y also describes the state-dependent constraints on the regulons. In this case, system (2.4) can no longer be regarded as a parameterized family of differential equations, as in the case when U (x) ≡ U does not depend upon the state, but as a differential inclusion (see, Aubin and Cellina 1984, for example). Fortunately, differential inclusions enjoy most of the properties of differential equations. A solution to system (2.4) is an evolution t → x(t) satisfying this system for some (measurable) open-loop control t → u(t) (almost everywhere). We associate with the control system the evolutionary system x S(x) associating with any initial state x ∈ K the subset S(x) ⊂ C(0, ∞; X) of solutions starting at x. Most of the results on viability kernels and capture basins depend upon few properties of this evolutionary system, that are shared by other “engines of evolution”, such as diffusion-reaction systems, path (or history) dependent systems, mutational equations governing the evolution of compact sets.

62


2.2.3

Viability Kernels and Capture Basins

The problems we shall study are all related to the viability of a constrained subset K and/or the capturability of a target C ⊂ K under the dynamical system modeling the dynamic behavior of the system. 1. The subset Viab(K) of initial states x0 ∈ K such that one solution x(·) to system (2.4ii) starting at x0 is viable in K for all t ≥ 0 is called the viability kernel of K under the control system. A subset K is a repeller if its viability kernel is empty. 2. The subset Capt(K, C) of initial states x0 ∈ K such that the target C ⊂ K is reached in finite time before possibly leaving K by one solution x(·) to system (2.4ii) starting at x0 is called the viable-capture basin of C in K. A subset C ⊂ K such that Capt(K, C) = C is said to be isolated in K. We say that 1. a subset K is viable under S if K = Viab(K), 2. that K is a repeller if Viab(K) = ◦ /.

In other words, the viability of a subset K under a control system is a consistency property of the dynamics of the system confronted to the constraints it must obey during some length of time. To say that a singleton {c} is viable amounts to saying that the state c is an equilibrium (equilibria, equal balance) — sometimes called a fixed point. The trajectory of a periodic solution is also viable. Contrary to the century-old tradition going back to Lyapunov, we require the system to capture the target C in finite time, and not in an asymptotic way, as in mathematical models of physical systems. However, there are close mathematical links between the various concepts of stability and viability. For instance, Lyapunov functions can be constructed using tools of viability theory. Or one can prove that the attractor is contained in the viability kernel of an absorbing set under the backward (negative) system. This needs much more space to be described: we refer to Chapter 8 of Aubin (1991) and Chapter 8 of Aubin (1997) for more details on this topic. One can prove that the viability kernel Viab(K) of the subset K is the “largest” subset of K viable under the control system. Hence, all interesting features such as equilibria, trajectories of periodic solutions, limit sets and attractors, if any, are all contained in the viability kernel. One can prove that the viability kernel is the unique subset D ⊂ K viable and isolated in K such that K\D is a repeller. If K\C is a repeller, the capture basin Capt(K, C) of C ⊂ K is the unique subset D between C and D such that D is isolated in K and D\C is locally viable.


63

The viability kernels of a subset and the capture basins of a target can thus be characterized in diverse ways through tangential conditions thanks to the viability theorems. They play a crucial role in viability theory, since many interesting concepts are often viability kernels or capture basins. See Aubin (2001b) and Aubin (2002) for more properties of viability kernels and capture basins. Furthermore, algorithms designed in Saint-Pierre (1994) allow us to compute viability kernels and capture basins (see, e.g., Cardaliaguet et al. 1999, Quincampoix and Saint-Pierre 1998). In general, there are no explicit formulas providing the viability kernel and capture basins.

2.3

Characterization of Viability and/or Capturability

The main task is to characterize the subsets having this viability/capturability property. To be of value, this task must be done without solving the system for checking the existence of viable solutions for each initial state.

2.3.1

Tangent Directions

An immediate intuitive idea jumps to the mind: at each point on the boundary of the constrained set outside the target, where the viability of the system is at stake, there should exist a velocity which is in some sense tangent to the viability domain and serves to allow the solution to bounce back and remain inside it. This is, in essence, what the viability theorem below states. Before stating it, the mathematical implementation of the concept of tangency must be made. We cannot be content with viability sets that are smooth manifolds (such as spheres, which have no interior), because inequality constraints would thereby be ruled out (as for balls, that possess distinct boundaries). So, we need to implement the concept of a direction v tangent to K at x ∈ K, which should mean that starting from x in the direction v, we do not go too far from K: The adequate definition due to G. Bouligand and F. Severi proposed in 1930 states that a direction v is tangent to K at x ∈ K if it is a limit of a sequence of directions vn such that x + hn vn belongs to K for some sequence hn → 0+. The collection of such directions, which are in some sense inward, constitutes a closed cone TK (x), called the tangent cone2 to K at x. Naturally, except if K is a smooth manifold, we lose the fact that the set of tangent vectors is a vector-space, but this discomfort is not unbearable, since advances in set-valued analysis built a calculus of these cones allowing us to compute them (see, e.g., Aubin and Frankowska 1990, Rockafellar and Wets 1997). 2 Replacing the linear structure underlying the use of tangent spaces by the tangent cone is at the root of set-valued analysis.

64

2.3.2


The Adaptive Map

We then associate with the dynamical system (described by (f, U )) and with the viability constraints (described by K) the (set-valued) adaptive or regulation map RK . It maps any state x ∈ K\C to the subset RK (x) (possibly empty) consisting of regulons u ∈ U (x) which are viable in the sense that f (x, u) is tangent to K at x: RK (x) := {u ∈ U (x) | f (x, u) ∈ TK (x)} . We can for instance compute the adaptive map in many instances.

2.3.3

The Viability Theorem

The viability theorem states that the target C can be reached in finite time from each initial condition x ∈ K\C by at least one evolution of the control system viable in K, if and only if, for every x ∈ K\C, there exists at least one viable control u ∈ RK (x). This viability theorem holds true when both C and K are closed and for a rather large class of systems, called Marchaud systems: beyond imposing some weak technical conditions, the only severe restriction is that, for each state x, the set of velocities f (x, u) when u ranges over U (x) is convex (This happens for the class of control systems of the form x′ (t) = f x(t) + G x(t) u(t) ,

where G(x) are linear operators from the control space to the state space, when the maps f : X 7→ X and G : X 7→ L(Y, X) are continuous and when the control set U (or the images U (x)) are convex. Curiously enough, viability implies stationarity, i.e., the existence of an equilibrium. Equilibria being specific evolutions, their existence requires stronger assumptions. The equilibrium theorem states that when the constrained set is assumed to be viable, convex and compact, then there exists a (viable) equilibrium. Without convexity, we deduce only the existence of minimal viable closed subsets. The proofs of the above viability theorem and the equilibrium theorem are difficult: The equilibrium theorem is derived from the 1910 Brouwer fixed point theorem, and the proof of the viability theorem uses all the theorems of functional analysis except the closed graph theorem and the Lebesgue convergence theorem. However, their consequences are much easier to obtain and can be handled with moderate mathematical competence.

2.3.4

The Adaptation Law

Once this is done, and whenever a constrained subset is viable for a control system, the second task is to show how to govern the evolution of viable evolutions. We thus


65

prove that viable evolution of system (2.4) are governed by

(i) x′ (t) = f (x(t), u t) (ii) u(t) ∈ RK x(t)

(adaptation law)

(2.5)

until the state reaches the target C. We observe that the initial set-valued map U involved in (2.4ii) is replaced by the adaptive map RK in (2.5ii). The inclusion u(t) ∈ RK (x(t)) can be regarded as an adaptation law (rather than a learning law, since there is no storage of information at this stage of modeling).

2.3.5

Planning Tasks: Qualitative Dynamics

Reaching a target is not enough for studying the behavior of control systems, that have to plan tasks in a given order. This issue has been recently revisited in Aubin and Dordan (2002), in the framework of qualitative physics (see, e.g., Kuipers 1994; Dordan 1995; Aubin 1996; Eisenack and Petschel-Held 2002; Eisenack et al. 2006, and Chapter 3 in this book, for more details on this topic). We describe the sequence of tasks or objectives by a family of subsets regarded as qualitative cells. Giving an order of visit of these cells, the problem is to find an evolution visiting these cells in the prescribed order.

2.3.6

The Meta-System

In order to bound the chattering (rapid oscillations or discontinuities) of the regulons, we set a priori constraints on the velocities of the form ∀ t ≥ 0, ku′ (t)k ≤ c . Let B(0, c) ⊂ Y denote the ball of radius c centered at the origin. The bound on the velocity of the regulons is taken into account by the meta-system, associating it with the initial viability problem is the system

(i) x′ (t) = f x(t), u(t) (ii) u′ (t) ∈ B(0, c)

(2.6)

subjected to the meta-constraints ∀ t ≥ 0, x(t) ∈ K and u(t) ∈ U x(t) .

(2.7)

Unfortunately, the above meta-constraints may no longer be viable under the metasystem.

66

2.4 2.4.1


Selecting Viable Feedbacks Static Feedbacks

A (static) feedback r is a map x ∈ K 7→ r(x) ∈ X which isused to pilot evolutions governed by the differential equation x′ (t) = f x(t), r(x(t)) . A feedback r is said to ′ be viable if the solutions to the differential equation x = f x, r(x) are viable in K. The most celebrated examples of linear feedbacks in linear control theory designed to control a system have no reason to be viable for an arbitrary constrained set K, and, according to the constrained set K, the viable feedbacks are not necessarily linear. However, the viability theorem implies that a feedback r is viable if and only if r is a selection of the adaptive map RK in the sense that ∀ x ∈ K\C, r(x) ∈ RK (x) .

(2.8)

Hence, the method for designing feedbacks for control systems to evolve in a constrained subset amounts to find selections r(x). One can design a factory for designing selections (see, Chapter 6 of Aubin (1991), for instance). Ideally, a feedback should be continuous to guarantee the existence of a solution to the differential ′ equation x = f x, r(x) . But this is not always possible. This is the case of slow selection r ◦ of RK of minimal norm, governing the evolution of slow viable evolutions (despite its lack of continuity).

2.4.2

Dynamic Feedbacks

One can also look for dynamic feedbacks gK : K × Y 7→ Y that governs the evolution of both the states and the regulons through the meta-system of differential equations

(i) x′ (t) = f x(t), u(t) (ii) u′ (t) = gK x(t), u(t) .

(2.9)

A dynamic feedback gk is viable if the meta-constraints (2.7) are viable under the meta-system (2.9). As for the (static) feedbacks, one can prove that all the viable dynamic feedbacks are selections of a dynamical adaptive map GK : K × Y Y obtained by differentiating the adaptation law (2.4ii); due to the differential calculus of set-valued maps (see, Aubin and Frankowska 1990).

2.4.3

Heavy Evolutions and the Inertia Principle

Among the viable dynamic feedbacks, one can choose the heavy viable dynamic ◦ ∈ G feedback gK K with minimal norm that governs the evolution of heavy viable


67

solutions, i.e., viable evolutions with minimal velocity. They are called heavy viable evolutions3 in the sense of heavy trends in economics. Heavy viable evolutions offer convincing metaphors of the evolution of biological, economic, social and control systems that obey the inertia principle. It states in essence that the regulons are kept constant as long as viability of the system is not at stake. Heavy viable evolutions can be viewed as providing mathematical metaphors for the concept of punctuated equilibrium introduced in paleontology by Eldredge and Gould (1972). In our opinion, this is a mode of regulation of control systems (see, Chapter 8 of Aubin (1996) for further justifications). Indeed, as long as the state of the system lies in the interior of the constrained set (i.e., away of its boundary), any regulon will do. Therefore, the system can maintain the regulon inherited from the past. This happens if the system obeys the inertia principle. Since the state of the system evolves while the regulon remains constant, it may reach the viability boundary with an outward velocity. This event corresponds to a period of viability crisis: To survive, the system must find other regulons such that the new associated velocity forces the solution back inside the viability set until the time when a regulon can remain constant for some time.

2.5

Management of Renewable Resources

Let us consider the regulons of the system (2.1) x′ (t) = u(t) x(t) chosen to govern evolutions viable in the interval [a, b]. The equilibrium map U∝ is defined by U∝ (x) = {0} and the monotonic maps U+ and U− by U+ (x) := R+ and U− (x) := R− . The regulation map is equal to   R+ if x = a RK (x) := R if x ∈]a, b[  R− if x = b .

It is set-valued, has non-empty values, but has too poor continuity property, a source of mathematical difficulties. The interval [a, b] is obviously viable under such a control system. As mentioned in Section 2.1.1, we regard the growth rates as regulons of the system (2.1): x′ (t) = u(t)x(t). The affine feedback map r(b−x) providing the Pearl-Verhulst logistic equation is always positive on the interval [a, b], so that the velocity of the population is always 3 When the regulons are the velocities, heavy solutions are the ones with minimal acceleration, i.e., maximal inertia.

68


non-negative, even though the population slows down. In order to have negative velocities (still with positive growth rates), we should require that the feedback satisfies the following phenomenological properties: for some ξ ∈ [a, b[,  ∀ x ∈ [a, ξ[, u e′ (x) > 0  (i) (ii) ∀ x ∈]ξ, b], u e′ (x) < 0  (iii) u(b) = 0 .

The increasing behavior of u e(x) on the interval [a, ξ[ is called the allee effect, stating that at a low population size, an increase of the population size is desirable and has positive effects on population growth, whereas the decreasing behavior of u e(x) on the interval ]ξ, b] is called the logistic effect, stating that at high population, an increase of the size has a negative effect on the growth of the population. Inert evolutions are obtained by taking the maximal velocities allowed: u′ (t) = ±c. Heavy evolutions combine Malthusian and inert growth. They are obtained starting with constant growth rate (Malthusian evolution) of the state (by taking u′ (t) = 0 until the time when the inertia bound is met (and thus, implying a (weak) allee effect during this phase of the evolution). This provides warning signals telling when, where and how the regulons must evolve: The feedback becomes a specific one, the inert feedback, providing constant (negative) velocities u′ (t) = −c of growth rates driving the state at its carrying capacity b.

2.5.1

Discrete Versus Continuous Time

Discrete evolutions are maps associating with each discrete time j ∈ N := {0, . . . , +∞{ ranging over the set of non-negative integers a state x(j) =: xj ∈ X. Unfortunately, for discrete time evolutions, tradition imposes upon us to regard discrete → evolutions as sequences and to use the notation − x : j ∈ N 7→ xj := x(j) ∈ X. The choice between these two representations of time, the discrete and the continuous, is not easy. The natural one, that appears the simplest for the nonmathematicians, is the choice of the set N of discrete times. It has drawbacks, though. On the one hand, it may be difficult to find a common time scale for the different components of the state variables of the state space of a given type of models. On the other, by doing so, we deprive ourselves of the concepts of velocity, acceleration and other dynamical concepts that are not well taken into account by discrete time systems as well as of the many results of the differential and integral calculus gathered for more than four centuries. However, for computational purposes, we shall approximate continuous-time systems by discrete time ones where the time scale becomes infinitesimal. However, viability properties of the discrete analogues of continuous-time systems can be drastically different: we shall see in the simple example of the Verhulst logistic equation that the interval [0, 1] is invariant under the continuous system x′ (t) = rx(t) 1 − x(t) ,


69

whereas the viability kernel of [0, 1] under its discrete analog xn+1 = rxn+1 (1 − xn+1 ) is a Cantor subset of [0, 1] when r > 4. See Fig. 2.1.

Figure 2.1: Viability kernel of [0, 1] under the quadratic map: The interval [0, 1] has been magnified several times for easier visibility and for highlighting the Cantor structure of the viability kernel.

2.5.2

Introducing Inertia Bounds

The constraint on the velocities of the growth rates imposed by the inertia property means that |u′ (t)| ≤ c . This suggests to take the velocity of the regulons as “meta-regulons”, required to range over the interval [−c, +c]. The state-regulon pairs (x, u) are the “meta-states” of the “meta-system”: (i) x′ (t) = u(t) x(t) (2.10) (ii) |u′ (t)| ≤ c , where the meta-regulon is chosen to be the velocity or the growth rate of the system. Unfortunately, the meta-constrained set [a, b] × R+ is obviously not viable under the above meta-system: Every solution starting from (a, u) with u < 0 leaves the set [a, b] × R+ immediately, as do evolutions starting from (b, u) with u > 0. We thus define the graph of the set-valued map Uc by Graph(Uc ) := Viab(2.10) [a, b] × R+ .

70


Figure 2.2: Viability kernel and inert evolution. Courtesy of Patrick Saint-Pierre.

The regulation map Uc for the one-dimensional model in Eq. (2.1) can be explicitly described by two feedbacks maps: we associate the inert feedbacks s r x h i b ♯ ♭ r (x) := 2 log , and R(x) := −r ♭ (x), +r ♯ (x) . , r (x) := 2 log x a

We can prove that for system x′ (t) = u(t)x(t), the graph of the regulation map Uc √ √ is limited by the graphs of − c r ♭ below and c r ♯ above: The regulation map is equal to i √ h √ Uc (x) := c −r ♭ (x), +r ♯ (x) = c R(x) .

The graph of Uc can also be computed by the Saint-Pierre viability kernel algorithm, as it is shown in Fig. 2.2. We shall now construct several feedbacks as selections of the regulation map Uc . 2.5.2.1

Affine Feedbacks and the Verhulst Logistic Equation

Affine feedbacks defined by u e(x) := r(b − x) ,

√ are selections of the regulation map Uc when r ≤ c 2b , so that the viability of the interval [a, b] under the models using such affine feedbacks is guaranteed. The Verhulst logistic differential equation x′ (t) = rx(t)(b − x(t)) corresponds to such an affine feedback. The solutions starting from x ∈ [a, b] are equal to x(t) =

bx . x + (b − x)e−rt


71

They remain confined in the interval [a, b] and converge to the equilibrium b when t 7→ +∞. We observe that the velocity and the growth rate u(t) are respectively given by u(t) = re−rt (b − x) They start derivatives

r(b−x) , b

and

u(t) =

r(b − x)e−rt . x + (b − x)e−rt

converge to 0 when t 7→ +∞ and are always decreasing since their u′ (t) =

are negative.

−r 2 (b − x)e−rt (x + (b − x)e−rt )2

Remark: Other Example of Feedback — We mention another example of an explicit feedback used in population dynamics, defined by 1 1 u e(x) := r √ − √ x b

which also produces only a logistic effect, the derivative u e′ (x) being always nonpositive. 2.5.2.2

Inert Evolutions

For a given inertia bound c, inert evolutions are governed by the feedbacks u(t) = √ ♯ √ c r (x(t)) if u > 0 and u(t) = − c r ♭ (x(t)) if u < 0. See Fig. 2.2. Let us consider the case when u > 0 (the case when u < 0 being symmetrical). √ Then the derivative of the feedback c r ♯ being negative on [a, b], it is a purely logistic feedback with no allee effect. In both cases, the velocity governing the inert evolution is constant and equal to u′ (t) = −c, so that u(t) = u − ct. Therefore the evolution of the inert regulon is given by ! ut u(t) = u 1 − 2 log xb and the evolution of the inert state by

ut−

x(t) = xe

u2 t2 b 4 log x

( ).

The state reaches the equilibrium (b, 0) at time log xb τ (x, u) = 2 . u

72


After, taking x(t) ≡ b and u(t) ≡ 0, the solution remains at equilibrium for t ≥ τ (x, u). √ The derivative of the feedback u(x) := c r ♯ (x) governing the evolution of the inert evolution being negative whenever a ≤ x < b, the inert evolution does not show any allee effect, but only a logistic one. This is the same situation as with the Verhulst equation. For obtaining feedbacks conveying the allee effect, we introduce heavy evolutions associated with a bound c > 0 on the velocities of the regulons. 2.5.2.3

Heavy Evolutions

Heavy solutions xc are obtained when the regulon is kept constant as long as possible. Starting from (x, u), the state xc (·) of the heavy solutions evolves according to xc (t) = xeut log( b ) and reaches b at time u x (half the time needed for the inert evolution to reach equilibrium) with velocity equal to u > 0 and ub > 0, so that xc (·) leaves the interval [a, b] in finite time. This evolution reaches the boundary of the graph of Uc at  u2  warning state ξc (x, u) = be− 2c b  warning time σ (x, u) := log( x ) − u . c

u

2c

Hence, once a velocity limit c is fixed, the heavy solution evolves with constant regulon u until the last instant σc(x, u) when the state reaches ξc (x, u) and the velocity of the regulon α ξc (x, u), u = c. This is the last time when the regulon has to change by taking ! log xb u + uc (t) = u − c t − u 2c so that the evolution follows the inert solution starting at ξc (x, u), u . It reaches u : equilibrium (b, 0) at time t⋆ := σc (x, u) + 2c log xb u t := + u 2c ⋆

with an advance equal to σc (x, u) over the inert solution. The heavy evolution xc (·), uc (·) is associated with the heavy feedback uec defined by  u if x ≤ y ≤ ξc (x, u) √ c r ♯ (y) if ξc (x, u) ≤ y < b uec (y) :=  0 if y = b .

73


Figure 2.3: Graph of the heavy evolution. Both the graphs of the heavy evolution (in blue) and of its control (in red) are plotted. They are not computed from the analytical formulas given below, but extracted from the viability kernel algorithm. The control remains constant until the trajectory of the exponential solution hits the boundary of the viability kernel and then slows down when it is controlled with a decreasing linear time dependent controls with velocity equal to −c. It reaches in finite time the boundary of the constrained interval with a velocity equal to 0 and may remain at this equilibrium. Courtesy of Patrick Saint-Pierre.

See Fig. 2.2. The heavy feedback has an allee effect on the interval [x, ξc (x, u)] and a logistic effect on [ξc (x, u), b]. Bounding the inertia by c, the feedback governing the heavy evolution maximizes the allee effect. Indeed, we observe that the part of the graph of any feedback u e passing through (x, u) (u = u e(x)) has an allee effect only when it lies above the horizontal line √ passing through (x, u). It necessarily intersects the graph of c r ♯ before reaching (ξc (x, u), u). The allee effect of the heavy evolution is weak in the sense that the velocity of the regulon is equal to 0 instead of being strictly positive. But it lasts longer (cf. Fig. 2.3). In summary, the heavy evolution under bound c > α(x, u) is described by the following formulas: the regulons are equal to

   u      uc (t) = u−c t−       

log( xb ) u

+

u 2c

if t ∈ 0,

if t ∈

log( xb ) u

log( xb ) u

−

−

u 2c

b u log( x ) 2c , u

+

u 2c

,

74


and the states to xc (t) =    xeut     12 0   log b  ! u2 @t− ( x ) + u A u 2c log( b ) 2 u − − u2c +u t− u x + 2c 2    be      

if t ∈ 0,

if t ∈

log( xb ) u

log( xb ) u

−

−

u 2c

b u log( x ) 2c , u

+

u 2c

.

Remark: — Let f : [a, b] 7→ [0, ∞[ be a function such that the primitives Fa (x) := R x dx R b dx a f (x) and Fb (x) := x f (x) exist. In the above examples, we took f (x) = x, so that Fa (x) = log xa and Fb (x) = log xb . We can adapt the previous formulas to the case of control systems of the form x′ (t) = u(t)f x(t) defined on the interval [a, b]. The corresponding feedbacks r ♯ and r ♭ are given by p p and r ♭ (x) := 2Fa (x) , r ♯ (x) := 2Fb (x)

the derivatives of which are given by ′

r ♯ (x) :=

−1

r ♯ (x)f (x)

and

′

r ♭ (x) :=

1 r ♭ (x)f (x)

.

The regulation map Uc is equal to Uc (x) :=

i √ h ♭ c −r (x), +r ♯ (x) ,

so that the meta-viability constrained set [a, b] ∈ R is viable under the control system x′ (t) = u(t)f x(t) where u′ (t) ∈ Uc x(t)

under bounded inflation. We may choose among the evolutions governed by this system the inert evolution, or the heavy evolution maximizing the allee effect on the interval for a given inertia bound c as we did for x′ (t) = u(t)x(t). 2.5.2.4

The Heavy Hysteresis Cycle

The heavy evolution (xc (·), uc (·)) starting at (x, u) where u > 0 stops when reaching u the equilibrium (b, u) at time t⋆ := σc (x, u) + 2c . But as (b, u) lies on the boundary of [a, b]×R, there are (many) other possibilities to find evolutions starting at (b, 0) remaining viable while respecting the velocity

75


limit on the regulons. For instance, for t ≥ t⋆ , we keep the evolutions xh (t) defined by ut−

⋆

∀ t ≥ t , xh (t) = xe

u2 t2 b 4 log x

( )

associated with the regulons ut ∀ t ≥ t , uh (t) = u 1 − 2 log xb ⋆

!

.

√ The meta-state (xh (·), uh (·)) ranges over the graph of the map − c r ♯ . Therefore the regulon uh (t) and the state velocity x′h (t) become negative, so that the population size xh (t) starts decreasing. The velocity of the negative regulon is constant. But it is no longer viable on the interval [a, b], because with such a strictly negative velocity, xh (·) leaves [a, b] in finite time. Hence regulons have to be switched before √ the evolution leaves the graph of Uc by crossing through the graph of − c r ♭ . Therefore, letting the heavy solution bypass the equilibrium by keeping its velocity equal to −c instead of switching it to 0, allows us to build a periodic evolution by taking velocities of regulons equal successively to 0, −c, 0, +c, and so on. We obtain in this way a periodic evolution showing an hysteresis property: The evolution oscillates between a and b back and forth by ranging alternatively two different trajectories on the meta-state space X × U. More precisely, we introduce the following notations: Denote by ac (u) and bc (u) the roots u2

ac (u) = ae 2c

and

u2

bc (u) = be− 2c

of the equations r ♭ (x) = u and r ♯ (x) = u. Then x⋆ := ac (u) = bc (u), if and only if √ u := c u⋆ where s √ b x⋆ := ab and u⋆ := log . a √ Therefore ac (u) ≤ bc (u) if and only if u ≤ cu⋆ . We also set log ab ⋆ . τ (u) = 2 u The periodic heavy hysteresis cycle xh (·) (of period 2τ ⋆ (u) + 3u c ) is described in the following way: 1. The meta-state xh (·), uh (·) starts from ac (u), u by taking the velocity of u the regulon equal to 0. It remains viable on the time interval [0, τ ⋆ (u) − 2c ] until it reaches the meta-state (bc (u), u). u by taking 2. The meta-state xh (·), uh (·) starts from bc (u), u at time τ ⋆ (u)− 2c √ ♯ the velocity of the regulon equal to −c. It ranges over the graph of c r on the u time interval [τ ⋆ (u)− 2c , τ ⋆ (u)+ 3u 2c ] until it reaches the meta-state bc (u), −u .

76


Figure 2.4: Graph of the inert evolution. Both the graphs of the inert evolution (in blue) and of its control (in red) are plotted. They are not computed from the analytical formulas given below, but extracted from the viability kernel algorithm. The velocity of the control remains constant until the trajectory of the solution hits the boundary of the viability kernel and then switches to the other extremal control with opposite sign and so on. The evolution is then periodic, alternatively increasing and decreasing from the lower bound of the constrained interval to its upper bound. Source: Patrick Saint-Pierre.

3. The meta-state xh (·), uh (·) starts from bc (u), −u at time τ ⋆ (u) + 3u 2c by taking the velocity of the regulon equal to 0. It remains viable on the time u ⋆ interval [τ ⋆ (u) + 3u 2c , 2τ (u) + c ] until it reaches the meta-state (ac (u), −u). 4. The meta-state xh (·), uh (·) starts from (ac (u), −u) at time 2τ ⋆ (u) + uc by taking the velocity of the regulon equal to +c. It ranges over the graph of √ ♭ c r on the time interval [2τ ⋆ (u) + uc , 2τ ⋆ (u) + 3u c ] until it reaches the metastate (ac (u), u). √ For the limiting case when u := c u⋆ , it becomes the inert hysteresis cycle xh (·) (of ⋆ √ ) described in the following way (cf. Fig. 2.4): period 4u c √ 1. The meta-state xh (·), uh (·) starts from (x⋆ , c u⋆ ) at time 0 with the velocity √ of the regulon equal to −c. It ranges over the graph of c r ♯ on the time interval √ ⋆ √ ] until it reaches the meta-state (x⋆ , − c u⋆ ), [0, 2u c

√ 2. The meta-state xh (·), uh (·) starts from (x⋆ , − c u⋆ ) at time

⋆ 2u √ c

by taking √ the velocity of the regulon equal to +c. It ranges over the graph of c r ♭ on √ ⋆ ⋆ √ , 4u √ ] until it reaches the meta-state (x⋆ , c u⋆ ). the time interval [ 2u c c

2.5.3

Verhulst and Graham

Biologists and ecologists are rather interested to long horizons and survival — viability — problems whereas economists most of the time are preoccupied with short horizons, concentrating on efficiency and substitutability of commodities (see, Gabay 1994 for this basic point).


77

The simplest ecological models are based on the logistic dynamics proposed by Verhulst. This evolution is slowed down by economic activity which depletes it. The basic model was originated by Graham (1935) and taken up by Schaeffer (1954), where it is assumed that the exploitation rate is proportional to the biomass and the economic activity. The equilibria of such dynamics are called in the literature the sustainable yields. The purpose of many investigations is to find an equilibrium yield maximizing some profit function. Beyond this static approach, optimal inter-temporal optimization was used in economics for proposing “bang-bang” solutions, which may be not viable economically, and which bypass the inertia principle, translating many rigidities in the way of operating the capture of the renewable resource. We present here a qualitative study by Luc Doyen and Daniel Gabay (see, Doyen and Gabay 1996 for more details) dealing with the viability of both economic and ecological constraints (see, also Scheffran 2000; Eisenack et al. 2006 and Chapter 3 in this book for further discussions on these issues). We denote by x ∈ R+ the biomass of the renewable resource and by v ∈ R+ the economic effort for exploiting it, playing the role of the regulon. The state and the regulon is subjected to viability constraints of the form 1. Ecological constraints ∀ t ≥ 0, 0 ≤ x(t) ≤ b , where b is the carrying capacity of the resource. 2. Economic constraints ∀ t ≥ 0, cv(t) + C ≤ γv(t) x(t) , where C ≥ 0 is a fixed cost, c ≥ 0 the unit cost of economic activity and γ ≥ 0 the price of the resource. 3. Production constraints ∀ t ≥ 0, 0 ≤ v(t) ≤ v , where v is maximal exploitation effort. We assume that γb > c Therefore, setting a :=

and

C ≤ v. γb − c

C + cv , the economic constraint implies that γv ∀ t ≥ 0, x(t) ∈ [a, b] .

78


The above constraints are summarized under the set-valued map V : [a, b] defined by C ,v . ∀ x ∈ [a, b], V (x) := γx − c

R+

The dynamics involve the Verhulst logistic dynamics and the Schaeffer proposal:  x(t)  ′  − v(t)x(t)  (i) x (t) = rx(t) 1 − b (2.11) C   ,v .  (ii) v(t) ∈ V (x(t)) := γx(t) − c

The “equilibrium curve” is here the of the singleton {0, 0} and the graph of union x . They are called “sustainable yields” in the Verhulst feedback ve(x) = r 1 − b C the literature. They are stable and viable if the graph of ve(x) := intersects γx − c the equilibrium line: viable equilibrium belong to the interval [x− , x+ ] where x± are the real roots of the equation γrx2 − rx(γb − c) + (C + rc)b = 0 . x± Setting v± := r 1 − , we can see that [a, b] is viable whenever v− ≥ v, i.e., if b and only if the growth rate r is large enough: r ≥

bγv 2 + rC bγ − c

.

Otherwise, it easy to check that the viability kernel of the interval [a, b] under system (2.11) is equal to the interval [x− , x+ ]. We set a bound to the velocity of the economic effort, which translates the rigidity of the economic behavior: ∀ t ≥ 0, −d ≤ v ′ (t) ≤ +d . Therefore the meta-system governing the evolution of the state and the regulon are described by  x(t)  ′ (i) x (t) = rx(t) 1 − − v(t)x(t) (2.12) b  (ii) |v ′ (t)| ≤ d and the meta-constrained set is the graph of V . The viability kernel is equal to

Viab(Graph(V )) = {(x, v) ∈ Graph(V ) | x ≥ ρ♯ (v)} , where ρ♯ is the solution to the differential equation ρ♯ (v) dρ♯ = r 1− − uρ♯ (v) −d dv b


79

satisfying the initial condition ρ♯ (v− ) = x− . This viability kernel can also be computed by the Saint-Pierre viability kernel algorithm, as it is shown in Fig. 2.5a. In this case, danger happens when the economic effort v is larger than v− . One cannot maintain the economic effort constant. The heavy evolution consists in keeping the economic effort constant as long as the biomass is larger than ρ♯ (v). At this level, the economic effort has to be drastically reduced with the velocity d, i.e., using v(t) = −d, while the biomass decreases until it reaches the level v− .

Figure 2.5: Regulation Maps and heavy solutions under (left) and Verhulst-Schaeffer x(t) ′ Verhulst-Inert Meta–systems (right) x (t) = r x(t) 1 − b − v(t) x(t) and r √ b − v(t) respectively. The equilibrium lines are the graphs x′ (t) = x(t) α 2 log x(t) q √ r 1 − xb and α 2 log xb . Heavy evolutions stop when their trajectories hit the equilibrium line. (a) The heavy evolutions starting from A or B stop at equilibrium position C. (b) The heavy evolution starting from position A stops at equilibrium position D.

2.5.4

The Inert-Schaeffer Meta–System

We have chosen the Verhulst feedback x 7→ r(1 − xb ) to represent the growth of the resource to be exploited for anchoringq our study in history. But we could have √ chosen instead the inert feedback x 7→ α 2 log xb and study the viability of the interval [a, b] under the system  s !  √ b    (i) x′ (t) = x(t) α 2 log − v(t) x(t) (2.13)  C    (ii) v(t) ∈ V x(t) := ,v . γx(t) − c

80


The “equilibrium curve” is here s again the union of the singleton {0, 0} and the graph √ b . They are stable and viable if the graph of the inert feedback ve(x) = α 2 log x C of e v (x) := intersects the equilibrium line: viable equilibrium belong to the γx − c interval [x− , x+ ] where x± are the real roots of the equation 1 − 2α

x = be s

“

C γx−c

”2

b , we can see that [a, b] is viable whenever v− ≥ v, Setting ve(x) = α 2 log x± i.e., if and only if the inertia bound α is large enough. We set a bound to the velocity of the economic effort, which translates the rigidity of the economic behavior: √

∀ t ≥ 0, −d ≤ v ′ (t) ≤ +d Therefore the Inert-Schaeffer meta-system governing the evolution of the state and the regulon are described by s  !  √ b  ′ α 2 log − v(t) i) x (t) = x(t) (2.14) x(t)   ′ ii) |v (t)| ≤ d and the meta-constrained set is the graph of V . The viability kernel can be computed by the viability kernel algorithm, see Fig. 2.5b.

2.5.5

The Crisis Function

We regard K ⊂ L as a “soft” constrained set embedded in a “hard” constrained set L. If K is viable, then the evolution may stay in K forever, whereas it has to leave K in finite time if it is a repeller. The questions arises whether an evolution reaches the viability kernel of K in finite time, so that it will remain forever in K and, otherwise, if the evolution x(·) reaches K outside its viability kernel, so that the evolution will leave K in finite time and enters a new era of crisis. This crisis may be endless if the evolution enters the complement of the capture basin of K viable in L. Otherwise, same scenario plays again. Hence the complement in L of the viability kernel of K can be partitioned in two sub–sets, one from which the evolutions will never return to the target (before leaving L), the other one from which at least one evolution returns and remains in the viability kernel of the target after a crisis lasting for a finite time of crisis.


81

′ Figure 2.6: Crisis function under the Verhulst-Schaeffer meta–system x (t) = x(t) − v(t)x(t) modeling the evolution of renewable resources depleted by an rx(t) 1 − b economic activity v(t). The meta-controls are the velocities |v ′ (t)| ≤ d of economic activity C bounded by a constant d. The constrained set {(x, v) ∈ [a, b] × [0, v] | v ≥ γx−c } translates economic constraints. The figure represents the graph of the crisis function, measuring the time spent by an evolution outside of the constrained set equal to zero on the viability kernel (in green), taking infinite values at states from which it is impossible to reach the constrained set (projection of pink area, which is the brown area). It is strictly positive and finite on states defined on the union of the purple and yellow areas. The curve inlayed in the graph of the crisis time function indicates the evolution of this crisis time along the optimal trajectory starting from position A in the space (x, v, θC ) where θC (xA , vA ) is the minimal crisis time one can expect when starting from (xA , vA ).

For any x ∈ L, the crisis function, introduced in Doyen and Saint-Pierre (1997), measures the minimal time spent outside the subset K by an evolutions starting at x. In other words, it measures the duration of crisis of not remaining in K. This takes into account the fact the zero damage within, infinite damage outside the emission corridor, as it is said in (Tóth 2003), is not the only pre-occupation of viability theory, as well as the tolerable window approach. It happens that the epigraph of the crisis function is a viability kernel, and can be computed by the viability kernel algorithm, as it is is shown in Fig. 2.6 (for the Verhulst-Schaeffer meta–system) and Fig. 2.8 (for the Inert-Schaeffer meta–system). Figure 2.7 provides the description of the domain of the crisis function and of the projections of the trajectories shown in Fig. 2.6.

82


Figure 2.7: Domain of the crisis function x′ (t) = rx(t) 1 −

x(t) b

− v(t)x(t) modeling the

evolution of renewable resources depleted by an economic activity v(t). The constrained set C } is the epigraph of the hyperbola in these two diagrams. {(x, v) ∈ [a, b] × [0, v] | v ≥ γx−c Its viability kernel under Verhulst-Schaeffer meta-system is the green area and the graph of the equilibrium curve is the red line: the viable equilibria range over the intersection of the equilibrium line and of the viability kernel of the constrained set. The projections of the trajectories of heavy and inert evolutions of the crisis function starting from A are shown. As long as they are above the constrained set (purple area), they use the constant meta–control equal to u = −1 (slowing down the fishing effort with the maximal (negative) allowed velocity). Being viable, the crisis function remains constant. It starts to decrease when the evolution leaves the constrained set (yellow area) until the time when they reach the state B when the meta-control has to be changed to the constant meta–control equal to u = +1 (increasing the fishing effort with the maximal (positive) allowed velocity). Then the evolution reaches the viability kernel at state E. Either we follow the heavy evolution, taking for meta–control the one with minimal velocity. Then the evolution leaves E to reach the equilibrium line in red at state F, where it then remains forever. Or, we keep the meta– control equal to +1, which is no longer a viable meta–control: the evolution remains viable until it reaches the boundary of the viability kernel at state C. In order to survive, we have to choose again the meta–control equal to u = −1 which brings the evolution to another equilibrium D = x− . The evolution remains there forever taking for new meta–control 0. Source: Patrick Saint-Pierre.

2.5.6

Towards Dynamical Games

Actually, since we do not really know what are the dynamical equations governing the evolution of the resource, we could take Malthusian feedbacks u e in a given class Ue of continuous feedbacks as parameters and study the viability kernel Viabue ([a, b]) of the interval [a, b] under the system


83

Figure 2.8: Crisis function under the Verhulst-Inert meta–system x′ (t) = r √ b x(t) α 2 log x(t) − v(t) modeling the evolution of renewable resources depleted by

an economic activity v(t). The meta–controls are the velocities |v ′ (t)| ≤ d of economic activity bounded by a constant d. The constrained set {(x, v) ∈ [a, b] × [0, v] } translates economic constraints. (a) represents the graph of the crisis function, equal to zero on the viability kernel (in green), taking infinite values at states from which it is impossible to reach the constrained set. (b) is a projection of (a): the union of the brown and light blue areas is the complement in the constrained set of the domain of the crisis function. It is strictly positive and finite on states defined on the union of the purple and yellow areas. The yellow curve is the trajectory of the inert evolution. (cf. also the comments of Figs. 2.6 and 2.7, A-F corresponds to those in Fig. 2.7) Courtesy of Patrick Saint-Pierre.

  (i)

x′ (t) = (e u(x(t)) − v(t))x(t) C ,v  (ii) v(t) ∈ V (x(t)) := γx(t) − c

(2.15)

This suggests to introduce the Guaranteed Viability Kernel [

e v e∈V

Invev ([a, b])

This is the very first question with which one can study viability issues of dynamical games, that are dynamical systems parameterized by two parameters under the control of two different players. See the paper of Cardaliaguet et al. (1999) for a summary on a viability approach to differential games. We can also study “meta–games” by setting bounds c and d on the velocities of the growth rate u(t) and the exploitation effort v(t), regarded as meta–controls, whereas the meta–states of the meta–game are the triples (x, u, v):  x′ (t) = (u(t) − v(t))x(t)  (i) (ii) u′ (t) ∈ B(0, c) (2.16)  ′ (iii) v (t) ∈ B(0, d)

84


Figure 2.9: Guaranteed viability kernel of the dynamical game (2.16) modeling the evolution of renewable resources x(t) with a growth rate ku(t)k depleted by an economic activity v(t). The meta–controls are the velocities |v ′ (t)| ≤ d of economic activity bounded by a constant d and the “meta-tyches” the velocities of the growth rates of the renewable reC } translates sources. The constrained set {(x, v, u) ∈ [a, b] × [0, v] × [−c, +c] | v ≥ γx−c economic constraints. The guaranteed viability kernel is represented in the axes (x, v, u). Courtesy of Patrick Saint-Pierre.

subjected to the viability constraints u(t) ∈ R and

C ≤ v(t) ≤ v γx(t) − c

The guaranteed viability kernel of dynamical game (2.16) is computed by the adequate version of the viability kernel algorithm, as it is shown in Fig. 2.9.

2.6

Viability and Optimality

Interestingly enough, viability theory implies the dynamical programming approach for optimal control. Denote by S(x) the set of pairs (x(·), u(·)) solutions to the control problem given in Eq. (2.4) starting from x at time 0. We consider the minimization problem  inf t∈[0,T ]  V (T, x) = inf (x(·),u(·))∈S(x) Z t  c(T − t, x(t)) + l(x(τ ), u(τ ))dτ , 0

where c and l are cost functions. We can prove that the graph of the value function (T, x) 7→ V (T, x) of this optimal control problem is the capture basin of the graph of the cost function c under an auxiliary system involving (f, U ) and the cost function l. The regulation map of this auxiliary system provides the optimal solutions and the tangential conditions furnish Hamilton-Jacobi-Bellman equations of which the


85

value function is the solution (see, for instance, Frankowska 1989b; Frankowska 1989a; Frankoswka 1991; Frankowska 1993, and more recently, Aubin 2001a). This is a very general method covering numerous other dynamic optimization problems. However, contrary to optimal control theory, viability theory does not require any single decision-maker (or actor, or player) to guide the system by optimizing an inter-temporal optimality criterion4 . Furthermore, the choice (even conditional) of the controls is not made once and for all at some initial time, but they can be changed at each instant so as to take into account possible modifications of the environment of the system, allowing therefore for adaptation to viability constraints. Finally, by not appealing to inter-temporal criteria, viability theory does not require any knowledge of the future5 (even of a stochastic nature.) This is of particular importance when experimentation6 is not possible or when the phenomenon under study is not periodic. For example, in biological evolution as well as in economics and in the other systems we shall investigate, the dynamics of the system disappear and cannot be recreated. Hence, forecasting or prediction of the future are not the issues which we shall address in viability theory. However, the conclusions of the theorems allow us to reduce the choice of possible evolutions, or to single out impossible future events, or to provide explanation of some behaviors which do not fit any reasonable optimality criterion. Therefore, instead of using inter-temporal optimization7 that involves the future, viability theory provides selection procedures of viable evolutions obeying, at each instant, state constraints which depend upon the present or the past. (This does not exclude anticipations, which are extrapolations of past evolutions, constraining in the last analysis the evolution of the system to be a function of its history.)

2.7

Restoring Viability

The above example shows that there are no reasons why an arbitrary subset K should be viable under a control system. Therefore, the problem of reestablishing viability arises. One can imagine several methods for this purpose: 1. Keep the constraints and change initial dynamics by introducing regulons that are viability multipliers, 4

The choice of which is open to question even in static models, even when multi-criteria or several decision makers are involved in the model. 5 Most systems we investigate do involve myopic behavior; while they cannot take into account the future, they are certainly constrained by the past. 6 Experimentation, by assuming that the evolution of the state of the system starting from a given initial state for a same period of time will be the same whatever the initial time, allows one to translate the time interval back and forth, and, thus, to know the future evolution of the system. 7 Which can be traced back to Sumerian mythology which is at the origin of Genesis: one decision-maker, deciding what is good and bad and choosing the best (fortunately, on an intertemporal basis, thus wisely postponing to eternity the verification of optimality), knowing the future, and having taken the optimal decisions, well, during one week.

86

J.-P. Aubin and P. Saint-Pierre 2. or change the initial conditions by introducing a reset map Φ mapping any state of K to a (possibly empty) set Φ(x) ⊂ X of new initialized states, (impulse control), 3. Keep the same dynamics and looking for viable constrained subsets by letting the set of constraints evolve according to mutational equations, as in Aubin (1999b).

We shall describe succinctly these methods.

2.7.1

Designing Regulons

When an arbitrary subset is not viable under an intrinsic system x′ (t) = f x(t) , the question arises to modify the dynamics by introducing regulons and designing feedbacks so that the constrained subset K becomes viable under the new system. Using the above results and characterizations, one can design several mechanisms. We just highlight three of them, that are described in more details in Aubin (1997). 2.7.1.1

Viability Multipliers

If the constrained set K is of the form K := {x ∈ X such that h(x) ∈ M } , where h : X 7→ Z := Rm and M ⊂ Z, we regard elements u ∈ Z as viability multipliers, since they play a role analogues to Lagrange multipliers in optimization under constraints. They are candidates to the role of regulons regulating such constraints. Indeed, we can prove that K is viable under the control system x′j (t)

= fj (x(t)) +

m X ∂hk (x(t)) k=1

∂xj

uk (t)

in the same way than the minimization of a function x 7→ J(x) over a constrained set K is equivalent to the minimization without constraints of the function x 7→ J(x) +

m X ∂hk (x) k=1

∂xj

uk

for an adequate Lagrange multiplier u ∈ Z (see for instance Aubin 1998c). 2.7.1.2

Connection Matrices

Instead of introducing viability multipliers, we can use a connection matrix W ∈ L(X, X) as in neural networks (see, Aubin 1996, for instance). We replace the intrinsic system x′ = If (x) (where I denotes the identity) by the system x′ (t) = W (t) f x(t)


87

and choose the connection matrices W (t) in such a way that the solutions of the above system are viable in K. The evolution of the state no longer derives from intrinsic dynamical laws valid in the absence of constraints, but requires some self-organization — described by connection matrices — that evolves together with the state of the system in order to adapt to the viability constraints, the velocity of connection matrices describing the concept of emergence. The evolution law of both the state and the connection matrix results from the confrontation of the intrinsic dynamics to the viability constraints. One can prove that the regulation by viability multipliers u is a particular case of the regulation by connection matrices W : we associate with x and u the matrix W the matrix the entries of which are equal to m fi (x) X ∂hk x(t) uk (t) if i 6= j wi,j := − kf (x)k2 ∂xj k=1

and, of j = i, wi,i

m fi (x) X ∂hk x(t) uk (t) . := 1 − kf (x)k2 ∂xi k=1

The converse is false in general. However, if we introduce the connectionist complexity index kI − W k, one can prove that the viable evolutions governed with connection matrices minimizing at each instant the connectionist complexity index are actually governed by the viability multipliers with minimal norms. See Aubin (1998b), Aubin (1998a), and Aubin (2003). The concept of heavy evolution when the regulon is a connection matrix amounts to minimize the norm of the velocity W ′ (t) of the connection matrix W (t) starting from the identity matrix, that can be used as measure of dynamical connectionist complexity. Such a velocity could encapsulate the concept of emergence in the systems theory literature. The connection matrix remains constant — without emergence — as long as the viability of the system is not at stakes, and evolves as slowly as possible otherwise. 2.7.1.3

Hierarchical Organization

One can also design dynamic feedbacks for obeying constraints of the form ∀ t ≥ 0, W m−1 (t) · · · W j (t) . . . W 0 (t)x(t) ∈ M ⊂ Y is satisfied at each instant. Such constraints can be regarded as describing a sequence of m planning procedures. Introducing at each level of such a hierarchical organization xi (t) := W i (t)xi−1 (t), on can design dynamical systems modifying the evolution of the intermediate states xi (t) governed x′j (t) = gj xj (t)

88


and the entries of the matrices W i (t) by j′ (t) = ejk,l W j (t) . wk,l

Using viability multipliers, one can prove that dynamical systems of the form  ⋆ (1)0 x′0 (t) = g0 x0 (t) − W 0 (t)p1 (t)(j = 0)   ⋆    (1)j x′j (t) = gj xj (t) + pj (t) − W j (t)pj+1 (t)    (j = 1, . . . , m − 1) m (1) x′m (t) = gm xm (t) + pm (t) (j = m)    j   (2)(k,l) wj ′ k,l (t) = ejk,l e W j (t) − xjk (t)p(j+1)l (t)    (j = 0, . . . , m − 1, k = 1, . . . , n, l = 1, . . . m)

govern viable solutions. Here, the viability multipliers pj are used as messages to both modify the dynamics of the jth level state xj (t) and to link to “consecutive levels” j + 1 and j. Furthermore, the connection matrices evolve in a Hebbian way, since the correction of the velocity W j ′ k,l of the entry is the product of the kth component of the j-level intermediate state xj and the lth component of the (j +1)-level viability multiplier pj+1 . 2.7.1.4

Evolution of the Architecture of a Network

This hierarchical organization is a particular case of a network. Indeed, the simplest general form of coordination is to require that a relation Q between actions of the form g(A(x1 , . . . , xn )) ∈ M must be satisfied. Here A : ni=1 Xi 7→ Y is a connectionist operator relating the individual actions in a collective way. Here M ⊂ Y is the subset of the resource space Y and g is a map, regarded as a propagation map. We shall study this coordination problem in a dynamic environment, by allowing actions x(t) and connectionist operators A(t) to evolve according to dynamical systems we shall construct later. In this case, the coordination problem takes the form ∀ t ≥ 0, g A(t) x1 (t), . . . , xn (t) ∈ M .

However, in the fields of motivation under investigation, the number n of variables may be very large. Even though the connectionist operators A(t) defining the architecture of the network are allowed to operate a priori on all variables xi (t), they actually operate at each instant t on a coalition S(t) ⊂ N := {1, . . . , n} of such variables, varying naturally with time according to the nature of the coordination problem, as in dynamic cooperative games (see for instance Aubin 2005, Scheffran 2001; Scheffran 2001). Therefore, our coordination problem in a dynamic environment involves the evolution Q 1. of actions x(t) := x1 (t), . . . , xn (t) ∈ ni=1 Xi , Q 2. of connectionist operators AS(t) (t) : ni=1 Xi 7→ Y ,


89

3. acting on coalitions S(t) ⊂ N := {1, . . . , n} of the n actors and requires that ∀ t ≥ 0, g

AS (t) x(t) S⊂N ∈ M ,

Q where g : S⊂N YS 7→ Y . The question we raise is the following. Assume that we may know the intrinsic laws of evolution of the variables xi (independently of the constraints), of the connectionist operator AS (t) and of the coalitions S(t), there is no reason why collective constraints defining the above architecture are viable under these dynamics, i.e, satisfied at each instant. One may be able, with a lot of ingeniousness and the intimate knowledge of a given problem, and for simple constraints, to derive dynamics under which the constraints are viable. However, we can investigate whether there is a kind of mathematical factory providing classes of dynamics correcting the initial (intrinsic) ones in such a way that the viability of the constraints is guaranteed. One way to achieve this aim is to use the concept of viability multipliers q(t) ranging over the dual Y ∗ of the resource space Y that can be used as controls involved for modifying the initial dynamics. This may allow us to provide an explanation of the formation and the evolution of the architecture of the network and of the active coalitions as well as the evolution of the actions themselves. In order to tackle mathematically this problem, we shall 1. restrict the connectionist operators to be multi-affine, and thus, involve tensor products, 2. next, allow coalitions S to become fuzzy coalitions so that they can evolve continuously. Fuzzy coalitions χ = (χ1 , . . . , χn ) are defined by memberships χi ∈ [0, 1] between 0 and 1, instead of being equal to either 0 or 1 as in the case of usual coalitions. Q The membership γS (χ) := i∈S χi is by definition the product of the memberships of the members i ∈ S of the coalitions. Using fuzzy coalitions allows us to defined their velocities and study their evolution. The viability multipliers q(t) ∈ Y ∗ can be regarded as regulons, i.e., regulation controls or parameters, or virtual prices in the language of economists. They are chosen adequately at each instant in order to show that the viability constraints describing the network can be satisfied at each instant, and the main theorem of this paper guarantees that it is possible. Another one tells us how to choose at each instant such regulons (the regulation law). For each actor i, the velocities x′i (t) of the state and the velocities χ′i (t) of its membership in the fuzzy coalition χ(t) are corrected by subtracting 1. the sum over all coalitions S to which he belongs of adequate functions weighted by the membership γS (χ(t)):

90

J.-P. Aubin and P. Saint-Pierre 2. the sum over all coalitions S to which he belongs of the costs of the constraints associated with connectionist tensor AS of the coalition S weighted by the membership γS\i (χ(t)). This type of dynamics describes a panurgean effect. The (algebraic) increase of actor i’s membership in the fuzzy coalition aggregates over all coalitions to which he belongs the cost of their constraints weighted by the products of memberships of the actors of the coalition other than him.

As for the correction of the velocities of the connectionist tensors AS , their correction is a weighted multi-Hebbian” rule: for each component of the connectionist tensor, the correction term is the product of the membership γS(χ(t)) of the coalition S, of the components xik (t) and of the component q j (t) of the regulon. In other words, the viability multipliers appear in the regulation of the multi-affine connectionist operators under the form of tensor products, implementing the Hebbian rule for affine constraints (see, Aubin 1996), and multi-Hebbian rules for the multi-affine ones (Aubin and Burnod 1998). Even though viability multipliers do not provide all the dynamics under which a constrained set is viable, they provide classes of them exhibiting interesting structures that deserve to be investigated and tested in concrete situations. Remark: Learning Laws and Supply and Demand Law — It is curious that both the standard supply and demand law, known as the “Walrasian tâtonnement process”, in economics and the Hebbian learning law in cognitive sciences were the starting points of the Walras general equilibrium theory and neural networks. In both theories, this choice of putting such adaptation laws as a prerequisite led to the same “cul de sacs”. Starting instead from dynamic laws of agents, viability theory provides dedicated adaptation laws, so to speak, as the conclusion of the theory instead as the primitive feature. In both cases, the point is to maintain the viability of the system, that allocation of scarce commodities satisfy the scarcity constraints in economics, that the viability of the neural network is maintained in the cognitive sciences. For neural networks, this approach provides learning rules that possess the features meeting the Hebbian criterion. For the general networks studied here, these features are still satisfied in spirit. We refer to Aubin (2003) for more details on this topic.

2.7.2

Impulse Systems

There are many other dynamics that obey the inertia principle, among which heavy viable evolutions are the smoothest ones. At the other extreme, on can study also the (discontinuous) impulsive variations of the regulon. Instead of waiting the system to find a regulon that remains constant for some length of time, as in the case of heavy solutions, one can introduce another (static) system that resets a new constant regulon whenever the viability is at stakes, in such a way that the system


91

evolves until the next time when the viability is again at stakes. This regulation mode is a particular case of what are called impulse control in control theory (see, for instance, Aubin 1999a; Aubin et al. 2002), hybrid systems in computer sciences and integrate and fire models in neurobiology, etc. Impulse systems are described by a control system governing the continuous evolution of the state between two impulsions, and a reset map resetting new initial conditions whenever the state enters the domain of the reset map. An evolution governed by an impulse dynamical system, called a run in the control literature, is defined by a sequence of cadences (periods between two consecutive impulse times), of re-initialized states and of motives describing the continuous evolution along a given cadence, the value of a motive at the end of a cadence being reset as the next re-initialized state of the next cadence. Given an impulse system, one can characterize the map providing both the next cadence and the next re-initialized state without computing the impulse system, as a set-valued solution of a system of partial differential inclusions. It provides a summary of the behavior of the impulse system from which one can then reconstitute the evolutions of the continuous part of the run by solving the motives of the run that are the solutions to the dynamical system starting at a given re-initialized state. A cadenced run is defined by constant cadence, initial state and motive, where the value at the end of the cadence is reset at the same re-initialized state. It plays the role of discontinuous periodic solutions of a control system. We prove in Aubin and Haddad (2001) that if the sequence of re-initialized states of a run converges to some state, then the run converges to a cadenced run starting from this state, and that, under convexity assumptions, that a cadenced run does exist.

2.7.3

Mutational Equations Governing the Evolution of the Constrained Sets

Alternatively, if the viability constraints can evolve, another way to resolve a viability crisis is to relax the constraints so that the state of the system remains inside the new viability set. For that purpose, kind of differential equation governing the evolution of subsets, called mutational equations, have been designed. This requires ◦

an adequate definition of the velocity K (t) of a tube t K(t), called mutation, that makes sense and allows us to prove results analogous to the ones obtained in the domain of differential equations. This can be done, but cannot be described in few lines. Hence the viability problem amounts to find evolutions of both the state x(t) and the subset K(t) to the system (

i)

x′ (t) = f x(t), K(t)

ii) K (t) ∋ m x(t), K(t) ◦

(differential equation) (mutational equation)

(2.17)

viable in the sense that for every t, x(t) ∈ K(t). For more details, see Aubin (1999b).

92

2.8


Conclusion

Viability theory, dealing with the confrontation of uncertain dynamical systems mathematically translated by differential inclusions with constraints and targets has been motivated since the end of the 1970’s by an attempt to provide mathematical metaphors of Darwinian evolution and by what appeared as shortcomings of general equilibrium theory in mathematical economics, centered on static considerations dealing with optimal or stationary evolutions. At that time, differential inclusions were mainly studied in Eastern Europe and former Soviet Union (around Filippov, Krasovski, Olech, to quote a few) and viability was restricted to differential equations after the Nagumo Theorem proved in 1943 – and rediscovered at least 14 times since. Since the beginning of the 1980’s when the main viability theorems were proved by Haddad (in the framework of differential inclusions with memory, see Haddad 1981), many advances have been accomplished when the concepts of viability kernels and capture basins have been characterized after 1985. It happened that not only they are appealing and natural concepts, but also that they appear as mathematical tools to solve many other problems : We mentioned in this paper the concepts of metasystems and crisis functions, but more and more mathematical objects happen to be viability kernels or capture basins : Among them, the value functions of many diverse inter-temporal finite or infinite horizon optimal control problems, attractors and fluctuation basins, solutions to first-order systems of partial differential equations or inclusions, issues in qualitative physics, etc., can be characterized through viability kernels and capture basins of auxiliary subsets under auxiliary dynamical systems. Recent advances are being gathered in the Aubin et al. (2006). If mathematical investigations allow us to derive properties of viability kernels and capture basins, it is quite impossible to characterize by explicit analytical formulas except for very simple examples as the ones we presented here. However, for computational purposes, such formulas are not necessarily needed, and the SaintPierre viability kernel algorithm provides not only computations of viability kernels and capture basins, but also viable evolutions such as heavy and inert ones. For the time, as for dynamical programming in optimal control theory, the implementation of this algorithm in computer softwares faces the sadly celebrated dimensional curse. For the time, the general algorithm can be implemented to systems. Much remains to be done as the theoretical level, the numerical and computing level, and at the modeling level. It is time to cross the interdisciplinary gap and to confront and hopefully to merge the points of view rooted in different disciplines. Mathematics, thanks to its abstraction power by isolating only few key features of a class of problems, can help to bridge these barriers as long as it proposes new methods motivated by these new problems instead of applying the classical ones only motivated until now by physical sciences. If we accept that physics studies much simpler phenomena than the ones investigated by social and biological sciences, and that for this very purpose, they motivated and used a more and more complex


93

mathematical apparatus, we have to accept also that social sciences require a new and dedicated mathematical arsenal which goes beyond what is presently available. Paradoxically, the very fact that the mathematical tools useful for social and biological sciences are and have to be quite sophisticated impairs their acceptance by many social scientists, economists and biologists, and the gap menaces to widen. Acknowledgments. The author thanks warmly J¨ urgen Kropp and J¨ urgen Scheffran for inviting this contribution, and Noël Bonneuil for their hidden collaboration.

References Aubin, J.-P. (1991). Viability Theory. Boston: Birkhäuser. Aubin, J.-P. (1996). Neural Networks and Qualitative Physics: A Viability Approach. Cambridge: Cambridge University Press. Aubin, J.-P. (1997). Dynamic Economic Theory: A Viability Approach. Berlin: SpringerVerlag. Aubin, J.-P. (1998a). Connectionist Complexity and its Evolution, pp. 50–79. Amsterdam: Elsevier. Aubin, J.-P. (1998b). Knowledge and Information in a Dynamic Economy, Chapter Minimal Complexity and Maximal Decentralization, pp. 83–104. Berlin: Springer-Verlag. Aubin, J.-P. (1998c). Optima and Equilibria. Berlin: Springer-Verlag. Aubin, J.-P. (1999a). Impulse differential inclusions and hybrid systems: A viability approach. Lecture notes, University of California at Berkeley, Berkeley. Aubin, J.-P. (1999b). Mutational and morphological analysis: tools for shape regulation and morphogenesis. Boston: Birkhäuser. Aubin, J.-P. (2000). Applied functional analysis. New York: Wiley-Interscience. Aubin, J.-P. (2001a). Concise introduction to viability theory, optimal control and robotics. Technical Report Notes, Ecole Normale Supérieure de Cachan. Aubin, J.-P. (2001b). Viability kernels and capture basins of sets under differential inclusions. SIAM J. Control 40, 853–881. Aubin, J.-P. (2002). Boundary-value problems for systems of Hamilton-Jacobi-Bellman inclusions with constraints. SIAM Journal on Control and Optimization 41 (2), 425– 456. Aubin, J.-P. (2003). Regulation of the evolution of the architecture of a network by tensors operating on coalitions of actors. Journal of Evolutionary Economics 13, 95–124. Aubin, J.-P. (2005). Dynamic core of fuzzy dynamical cooperative games. In A. S. Nowak and K. Szajowski (Eds.), Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control, Boston, pp. 129–162. 9th Int. Symp. on Dynamical Games and Applications: Birkhäuser.

94

J.-P. Aubin and P. Saint-Pierre Aubin, J.-P., A. Bayen, N. Bonneuil, and P. Saint-Pierre (2006). Viability, Control and Game Theories: Regulation of Complex Evolutionary Systems Under Uncertainty. Berlin: Springer-Verlag. to appear. Aubin, J. P. and Y. Burnod (1998). Hebbian learning in neural networks with gates. Technical Report #981, Université de Paris Dauphine, Cahiers du Centre de Recherche Viabilité, Jeux, Contrˆ ole, Paris. Aubin, J.-P. and A. Cellina (1984). Differential Inclusions. Berlin: Springer-Verlag. Aubin, J.-P. and G. Da Prato (1998). The viability theorem for stochastic differential inclusions. Stochastic Analysis and Applications 16, 1–15. Aubin, J.-P. and O. Dordan (1996). Hung Nguyen, Chapter Fuzzy Systems, Viability Theory and Toll Sets, pp. 461–488. Dordrecht: Kluwer Academic Publ. Aubin, J.-P. and O. Dordan (2002). Hybrid systems: Computation and control. In C. J. Tomlin and M. R. Greenstreet (Eds.), Dynamical Qualitative Analysis of Evolutionary Systems, Lecture Notes in Computer Science, pp. 62ff. Berlin: Springer-Verlag. Aubin, J.-P. and H. Doss (2003). Characterization of stochastic viability of any nonsmooth set involving its generalized contingent curvature. Stochastic Analysis and Applications 21 (5), 955 – 981. Aubin, J.-P. and H. Frankowska (1990). Set-Valued Analysis. Boston: Birkhäuser. Aubin, J.-P. and G. Haddad (2001). Cadenced runs of impulse and hybrid control systems. International Journal of Robust and Nonlinear Control 11 (5), 401–415. Aubin, J.-P., J. Lygeros, M. Quincampoix, S. Sastry, and N. Seube (2002). Impulse differential inclusions: A viability approach to hybrid systems. IEEE Transactions on Automatic Control 47, 2–20. Bruckner, T., G. Petschel-Held, F. Tóth, H. M. F¨ ussel, C. Helm, and M. Leimbach (1999). Climate change decision-support and the tolerable windows approach. Enviromental Modelling and Assessment 4, 217–234. Cardaliaguet, P., M. Quincampoix, and P. Saint-Pierre (1999). Set-valued numerical analysis for optimal control and differential games. In M. Bardi, T. Parthasarathy, and T. E. S. Raghavan (Eds.), Stochastic and Differential Games: Theory and Numerical Methods (Annals of International Society of Dynamic Games ed.), Vol. 4, pp. 177–247. Boston: Birkh¨ auser. Dordan, O. (1995). Analyse qualitative. Paris: Masson. Doyen, L. and D. Gabay (1996). Economie des ressources renouvelables et viabilité. Actes des journèes Vie, Environnement évolution Sociétés. Doyen, L. and P. Saint-Pierre (1997). Scale of viability and minimal time of crisis. SetValued Analysis 5, 227–246. Eisenack, K. and G. Petschel-Held (2002). Graph theoretical analysis of qualitative models in sustainability science. In Working Papers of 16th Workshop on Qualitative Reasoning, Sitges, pp. 53–60. available from http://www.upc.es/web/QR2002/Papers/QR2002%20-%20Eisenack.pdf. Eisenack, K., J. Scheffran, and J. P. Kropp (2006). Viability analysis of management frameworks for fisheries. Environmental Modelling and Assessment 11 (1), 69–79.


95

Eisenack, K., H. Welsch, and J. P. Kropp (2006). A qualitative dynamical modelling approach to capital accumulation in unregulated fishery. Journal for Economic Dynamics and Control . in press, online: DOI 10.1016/j.jedc.2005.08.004. Eldredge, N. and S. J. Gould (1972). Punctuated equiibria: an alternative to phyletic gradualism. In S. T. J. M. (Ed.), Models in Paleobiology, pp. 82–115. San: Francisco: Freeman, Cooper & Company. Frankoswka, H. (1991). Lower semicontinuous solutions to Hamilton-Jacobi-Bellman equations. In Proc. 30th Conference on Decision and Control, Brighton, pp. 265–270. Frankowska, H. (1989a). Hamilton-Jacobi-equation: viscosity solutions and generalized gradients. J. of Math. Analysis and Appl. 141, 21–26. Frankowska, H. (1989b). Optimal trajectories associated to a solution of tangent Hamilton-Jacobi-equations. Applied Mathematics and Optimization 19, 291–311. Frankowska, H. (1993). Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation. SIAM J. on Control and Optimization 31 (1), 257–272. Gabay, D. (1994). Modeling the articulation between the economy and the environment. In J.-L. Diaz and J.-L. Lions (Eds.), Environment, economics, and their mathematical models, pp. 67–86. Paris: Masson. Graham, M. (1935). Modern theory of exploiting a fishery and applications to North-Sea trawling. J. Com. Perm. Intern. Exploitation des Mers 10, 264–274. Haddad, G. (1981). Monotone viable trajectories for functional differential inclusions. J. Diff. Eq. 42, 1–24. Kuipers, B. J. (1994). Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. Cambridge: MIT Press. Petschel-Held, G., H. J. Schellnhuber, T. Bruckner, F. Tóth, and K. Hasselmann (1999). The tolerable windows approach: Theoretical and methodological foundations. Climatic Change 41, 303–331. Quincampoix, M. and P. Saint-Pierre (1998). An algorithm for viability kernels in H¨ olderian case: Approximation by discrete dynamical systems. Journal of Mathematical Systems, Estimation, and Control 8 (1), 17–29. Rockafellar, R. T. and R. Wets (1997). Variational Analysis. Berlin: Springer-Verlag. Saint-Pierre, P. (1994). Approximation of the viability kernel. Applied Mathematics & Optimisation 29, 187–209. Schaeffer, M. B. (1954). Some aspects of the dynamics of populations. Bull. Inter. Amer. Trop Tuna Comm. 1, 26–56. Scheffran, J. (2000). The dynamic interaction between economy and ecology - cooperation, stability and sustainability for a dynamic-game model of resource conflicts. Mathematics and Computers in Simulation, 371–380. Scheffran, J. (2001). Stability and control of value-cost dynamic games. European Journal of Operations Research 9, 197–225. Tóth, F. L. (2003). Climate policy in light of climate science: The Iclips project. Climatic Change 56(1-2), 7–36.

Part II

Qualitative Methods

CHAPTER 3

Qualitative Modeling Techniques to Assess Patterns of Global Change ¨deke, Gerhard Klaus Eisenack, Matthias K.B. Lu 1 ¨rgen Scheffran, and Ju ¨rgen P. Kropp Petschel-Held , Ju

3.1

Introduction

Global society is facing a new type of environmental problems. Anthropogenic influence on the earth system approaches a dimension where it has a fundamental ascertainable impact on the system (Turner II et al. 1990). Today civilization is a significant factor of interference in the global ecosphere and the variety of involved mechanisms are characterized by complex trans-sectoral interdependencies and interrelationships (Schellnhuber and Kropp 1998). Therefore context-dependence of socio-ecological dynamics makes it extremely difficult to draw general conclusions about determinants for their robustness and for success or failure of management and/or steering strategies. There are considerable efforts to develop theories of socio-ecological systems or of management success factors, e.g. based on large samples of case studies. Nevertheless these approaches are often criticized for being too abstract for an application to concrete environmental or institutional problems. The frequently stated reason for this circumstance is that every case has its distinct features which makes it problematic to compare it with other cases. This argument is based on idiographic traditions in science, which aims to identify the particularities of single case studies. Such traditions deserve attention in their own right, in particular, if we consider that formal modeling disregards contextual differences, at least for technical issues of tractability. On the other hand, formal methods play an important role in nomothetic traditions which aim at identifying results on a general level (e.g. fundamental laws of physics), and therefore abstract from particularities in a more rigorous way. Having this in mind, it appears that although every case in a comparative case study is different, there can also be strong resemblance. This knowledge may be very valuable for the design of strategies to deal with, e.g. management or steering targets for CO2 emission management or sustainable resource utilization. 1 Gerhard Petschel-Held was one of the most inspiring contributors to the syndrome concept during the recent years. In scientific discussions he always aimed to generalize his ideas beyond imagination at first and then pursuing a serious effort to gather a solid support for his theories and assumptions. On September 9th , 2005 he suddenly passed away in his office at the age of 42. In remembrance we wish to dedicate this chapter to him.

100

K. Eisenack et al.

Nevertheless in sustainability science we observe the phenomenon of complexity, i.e. we often have to consider an exceptional dynamics involving innumerable system parts and a multitude of (non-linear) interrelations between the socio-economic and the natural sphere. Thus, it is more than ever essential to understand the most relevant mechanisms, driving forces, and/or feedback loops of systems, at least on an intermediate scale of complexity (cf. Schellnhuber et al. 2002). At this point any systematic analysis enters the “bottleneck of knowledge representation and derivation”, since up to now no formal strategy exists how to generalize from single observations. One reasonable strategy is to develop a quantitative and/or qualitative symbolic coding system (cf. e.g. Boardman 1995), but any representation may be imperfect and can be a source of errors (Davis et al. 1993). We present archetypes as a pattern approach to this task. The general idea is that each problem class is structured by core interactions allowing to describe it as a typical pattern. The pattern approach is an essential technique to cope with complex situations and is also part of the learning process in human-environment interaction. It is the closest concept to the reality of brain representation and is based on both phylogeny and ontogeny. The brain of mammals is extremely efficient in reconstructing fragmentary patterns and in providing solutions for unknown situations by referring to analogous cases. This advantage is, in particular, used by artificial neural networks (cf. Kropp and Schellnhuber 2006). In this Chapter we will operationalize the pattern idea which has a long lasting history in different fields of science (see, for example, Polya 1954; Hayek 1973; Vámos 1995; Kelso 1997; Kropp et al. 2006a) for the assessment and simulation of complex global change pattern. A first attempt to do this was the so-called syndrome approach, originated by the German Advisory Council on Global Change to the Federal Government (Wbgu 1994), which was often criticized due to its less formal foundation and mainly heuristic characteristics. In a more general context of global change research the syndrome approach was suggested as an instrument to analyze complex transsectoral phenomena. It provides a semi-quantitative and transsectoral overview of the “dynamical degradation patterns” that characterize contemporary human-environment interactions across the planet (cf. e.g. Schellnhuber et al. 1997; Schellnhuber et al. 2002). It decomposes the mega-process of “global change” into archetypal patterns, named syndromes, under the hypothesis that the web of relationships governing the planetary development is made up by a finite set of transsectoral sub-webs of distinct causal typology. It is a formidable task to elicit more knowledge on the complex interrelationships governing global change, but the syndrome approach can help to identify hot spots, but also to identify key mechanisms. The latter is - at least one precondition for the design of successful management regimes. We will reconsider this example by assuming archetypes as simple qualitative models which can (if necessary) be refined to the particularities of each case. We will show that by the utilization of adequate methodological concepts a formal basis of the syndrome analysis is possible and will provide further insights.

Modeling Techniques to Assess Patterns of Global Change

101

This Chapter is organized as follows. In Section 3.2 we introduce archetypes of social-ecological systems as a formal framework based on model ensembles and demonstrate in an abstract way how it can be used to pose well-defined and relevant questions within the domain of sustainability science. The framework is underpinned with an abstract semantic specification which relates it to established methods in case study research. In the subsequent Section 3.3 we introduce the syndrome concept. First we discuss the conceptualization, advantages, and shortcomings of the traditional syndrome approach (Section 3.3.1). This will be shown by the example of the Overexploitation syndrome. Subsequently the semantic and formal aspects of three syndromes are illustrated in more detail. We start the formalization with the Overexploitation syndrome (Section 3.3.2) and present the same strategy for the Dust-Bowl syndrome indicating the non-sustainable use of soils and water bodies (Section 3.3.3) and for the Sahel syndrome dealing with the overuse of marginal land (Section 3.3.4). Then we grasp again the thread of the Overexploitation syndrome (Section 3.4) and introduce a multi-actor approach in order to derive further knowledge about successful management regimes. Finally, we summarize the strengths and limitations of this methodology and indicate directions for further research (Section 3.5).

3.2 3.2.1

Describing and Analyzing Archetypes as Model Ensembles Model Ensembles

We understand archetypes as building blocks of society-nature interaction which appear in multiple case studies. The underlying hypothesis is that insights can be transferred from one case to another if the cases share archetypes. For this goal an adequate notion of similarity is required which respects particularities where needed, but generalizes features from other case studies, if they are of limited relevance for the knowledge transfer from one case to the other. Such similarities introduce equivalence classes of models. This is necessary to examine real-world systems, which cannot be formally described in a unique way, also due to uncertainty. Although not concisely described as a general strategy, this style of reasoning is common – not only – in sustainability science, e.g. for parameter variation (e.g. Stainforth et al. 2005), model comparison (cf. Gregory et al. 2005), or scenario development (e.g. Nakićenović et al. 2000; Swart et al. 2004; Mea 2005). We formalize and generalize these ideas by introducing the notion of a model ensemble, which is a structured set of ordinary differential equations (Ode). These are considered simultaneously instead of investigating only single models. A model ensemble M is defined as a set of functions f : X × R+ → Rn on a state space X ⊆ Rn . These functions are called models, each describing a possible configuration of a real-world system under investigation or one example of the pat-

102

K. Eisenack et al.

tern to be analyzed. The set E contains functions x(·) : R+ → X, being the space of admissible trajectories of the systems, e.g. E = C 1 (R+ , X). Each model f ∈ M defines a family of initial value problems x˙ = f (x, t), x(0) = x0 , with x0 ∈ X. We call the set of all initial value problems given by a model ensemble the systems of the model ensemble M. It is also possible to consider model ensembles which only contain autonomous models. Of course, the systems of the model ensemble have (in general) different solutions. Thus, a set of trajectories is assigned to each initial value x0 . The set-valued solution operator SM (·) : X → P(E) (of a model ensemble M with respect to a state space X and admissible trajectories E), assigning to an initial state a subset of E, is defined by SM (x0 ) := {x(·) ∈ E | x(0) = x0 , ∃f ∈ M ∀t ∈ R+ : x(t) ˙ = f (x(t), t)}. Depending on E it may be sufficient that the Ode only holds almost everywhere. We call the elements of SM (X) the solutions of the model ensemble M (with respect to a state space X and admissible trajectories E). If an application requires a very general model, M is the collection of all cases which have to be analyzed. Similarly, in the case of uncertainties, M is defined to subsume all systems which must be considered. These are given by the part of the knowledge base which is certain to a high degree, while variation is admitted for uncertain parameters, functions or processes. It is assumed that all these cases and/or all considered variations can be described as dynamical systems on the same state space X. The solution operator is closely related to the concept of an evolutionary system as defined by Aubin (1991) (cf. also Chapter 2 in this book for details). The main challenge in reasoning with model ensembles is to find relevant structures in SM (X). This includes 1. representing a model ensemble in a way that is adequate to the modeler and allows for a formal treatment, 2. efficient algorithms to determine SM (X) from a (possibly infinite) model ensemble M, 3. detecting structural features of the solutions of the model ensemble. An example for the latter is to introduce normative settings in a formalized way by a partition X = A ∪ B of the state space into a preferable region A and a problematic region B. If for all x(·) ∈ SM (x0 ) and for all t ≥ 0 the relation x(t) ∈ B holds such that the system is “locked in” B due to its intrinsic dynamic interactions, it may be said that “a catastrophic outcome is unavoidable”. Or if ∃x(·) ∈ SM (x0 ), t ≥ 0 : x(t) ∈ A, then “it is possible to sustain preferable conditions”. Such features are


103

very robust in that they hold for a whole model ensemble and not just one model. They are introduced as invariant or viable sets in Chapter 2. A discrete analogon in described in Section 3.2.3. We now provide some examples for model ensembles (cf. also Schellnhuber 1998). Example: Let M contain only one function f : X × R+ → Rn which is Lipschitz on X, and let the admissible trajectories be E = C 1 (R+ , X). Then, SM (x0 ) contains the usual solutions of the initial value problem with x(0) = x0 which exist on R+ . Example: Given a function f : X × R+ × Rn → Rn , (x, u) 7→ f (x, u, t), depending on a control vector u(t) ∈ U (x(t)), and a finite set P of possible parameterizations, define the finite model ensemble M := {f ′ ∈ C(X × R+ , Rn ) | f (x, t) = f ′ (x, t; p), p ∈ P }. Then, the solution operator with respect to a set of admissible trajectories provides all “scenario runs” for the different parameterizations. Example: In analogy to evolutionary systems as discussed in Chapter 2 for a given autonomous measurable function f : X ×U → Rn , (x, u) 7→ f (x, u), depending on a control vector u(t) ∈ U (x(t)) where U (·) assigns a set of admitted control vectors to each state, the infinite model ensemble, we define M := {f ′ : X × R+ → Rn measurable | ∃u ∈ U (x(t)) : f ′ (x, t) = f (x, u)}. Taking absolutely continuous functions as admissible trajectories, the solution operator SM (x0 ) describes all trajectories starting from x0 which result from any measurable open-loop control u(·) : R+ → U . Example: A further example (which will be introduced in detail below) are qualitative differential equations (Qdes). Basically, such a model ensemble is defined by a prescribed matrix of signs Σ via M = {f ∈ C 1 (X, Rn ) | ∀x ∈ X : sgn J (f )(x) = Σ},

(3.1)

where J denotes the Jacobian and the sign operator is applied component wise. The signs of the Jacobian may, for example, result from a formalized causal-loop diagram (cf. Richardson 1986). We will see below how it can be used to define archetypes which describe the profit-driven or poverty-driven overexploitation of natural resources. If we take the set of continously differentiable functions which

104

K. Eisenack et al.

have only a finite number of critical points on any compact interval, SM (·) can be computed by using the Qsim algorithm developed by Kuipers (1994) and his group at the University of Texas at Austin. Within the framework of model ensembles, further questions can be posed which are highly relevant for sustainability science. If it is not possible to find relevant features common to all solutions of a model ensemble M we can try to identify subsets M′ ⊆ M for which such robust properties can be determined. The characterization of M′ is associated with the discovery of structural features which bring about problematic or desirable system behavior. In other terms, conditions are found under which certain (sub)pattern evolve. If M is partially determined by certain control measures imposed on the system, and M′ by alternative control measures, the differences between the solution operators SM (X) and SM′ (X) are of interest.

3.2.2

Qualitative Differential Equations

Qualitative differential equations (Qdes) are a prominent methodology in qualitative modeling (Kuipers 1994). The basic idea of Qdes is not to determine all quantitative solutions, but all possible sequences of sign vectors sgn(x) ˙ which can be brought about by at least one solution x(·) ∈ S(X) (and some further information, see below). These can be determined from the definition of M using the Qsim algorithm. The set of the possible sign vectors is finite, so that the result of the algorithm can be displayed as a directed state-transition graph, where each node represents a sign vector, and there is an edge between nodes v1 , v2 , if the sign vector v2 occur as a direct successor of v1 in at least one solution. The nodes in such a graph are also called qualitative states, and the edges transitions. The input for a modeling task is a Qde comprising the following parts: 1. a set of state variables; 2. a quantity space for each variable, specified in terms of an ordered set of symbolic landmarks; 3. a set of constraints expressing the algebraic, differential or monotonic relationships between the variables. This defines the model ensemble in a twofold manner: (i) variables take values from the set of symbolic landmarks or intervals between landmarks. Each landmark represents a real number, e.g. maximum sustainable yield, of which the exact quantitative value may be unknown or uncertain. Nevertheless, it is analytically distinguished whether a grain yield or catch is above or below this threshold. The landmark or the interval between landmarks where the value of a variable is at a given time, is called its qualitative magnitude. (ii) Monotonic relationships specified between variables, e.g. that the yield is monotonically decreasing with a decreasing stock, are expressed by constraints. They introduce a (complex) relation between the qualitative magnitudes and the direction of change of the state variables in time (as seen in the last example above).


105

Qualitative simulation achieves its result by performing a constraint satisfaction scheme (for a general introduction see Tsang 1993; Dechter 2003, for a more detailed description of constraint satisfaction problems (Csp) we refer to Chapter 5), where all combinations of qualitative magnitudes inconsistent with the constraints are filtered out. The outcome is a set of trajectories organized as a graph. This so-called state-transition graph describes all possible solutions of a model ensemble M defined by the Qde in the following sense. The sign vectors of the velocity vector of a solution x(t) with increasing t can be written as a well-defined sequence: Definition 3.2.1 For a given solution x(·) ∈ SM (x0 ) on [0, T ] we have an ordered sequence of sign jump points (tj ) with t0 = 0 which subsequently contains all boundary points of the closures of all sets {t ∈ [0, T ]|[x(t)] ˙ = v} with sign vectors v ∈ {−, +}n . We construct a sequence of sign vectors x ˜ = (˜ xj ) := x(τ ˙ j ) , where we arbitrarily choose τj ∈ (tj , tj+1 ). If the sequence (tj ) is finite with m elements, we choose τm ∈ (tm , T ). The sequence x ˜ is called abstraction of x(·). Then, the state-transition graph is defined as Definition 3.2.2 Denote the set of the abstractions of the solutions by S˜M := {˜ x | ∃x0 ∈ X, x(·) ∈ SM (x0 ) : x ˜ is the abstraction of x(·)}. Then, the directed state-transition graph of the monotonic ensemble is defined by the nodes V (G) := {v ∈ {−, +}n | ∃ x ˜ ∈ S˜M , j ∈ N : x ˜j = v}, called qualitative states, and the edges E(G) := {(v, w) | ∃ x ˜ ∈ S˜M , j ∈ N : x ˜j = v and x ˜j+1 = w}, called qualitative transitions. The guaranteed coverage theorem (an in-depth discussion of this theorem is beyond the scope of this paper, but cf. Kuipers 1994, p. 118) ensures that the algorithm computes a graph which contains the abstraction of all solutions of the model ensemble as a path. Obviously, due to the generality of the model ensemble, the state-transition graph usually has more than one unique path, and every path represents a set of quantitative trajectories development paths.

3.2.3

Viability Concepts

For larger model ensembles, the resulting state-transition graph can grow tremendously, such that various techniques are used for their analysis. One method is a graph theoretical analogon to concepts from viability theory (cf. Chapter 2), developed by Aubin (1991), which is increasingly used in sustainability science (see, for

106

K. Eisenack et al.

example, Kropp et al. 2004; Mullon et al. 2004; Cury et al. 2004; Eisenack et al. 2006a). We briefly recall the necessary concepts from viability theory. Let K ⊆ X be a subset of the the state space called a constrained set. A trajectory x(·) which remains in K, i.e. ∀t : x(t) ∈ K, is called viable in K. In the case of a model ensemble, multiple trajectories can start from a given initial value x0 ∈ K. The set of all initial values such that at least one trajectory is viable is called the viability kernel of K. The set of initial values such that all trajectories are viable is called the invariance kernel of K. In the context of a state-transition graph G, these definitions are modified as follows. The sets V (G) and E(G) are the nodes and edges of G, respectively, and the set-valued map Γ : V (G) → P(V (G)) assigns to every node the set of its successors. Definition 3.2.3 A set D ⊆ V (G) is 1. Viable, if for all v0 ∈ D ∃ path v0 , . . . , vi , . . . in G

∀i ≥ 0 : vi ∈ D

or ∃ path v0 , . . . , vm in G : Γ(vm ) = ∅ and ∀i = 0, . . . , m : vi ∈ D. 2. Invariant, if for all v0 ∈ D ∀ paths v0 , . . . , vi , . . . in G, i ≥ 0 : vi ∈ D. 3. No-return, if for all v0 ∈ D ∀ paths v0 , . . . , vm , . . . , vi in G with vm ∈ D, i ≥ m : vi ∈ D. In a viable set a path starts from every vertex which remains in the set. Invariant sets correspond to regions in the phase space which cannot be left once they are entered. The no-return set is new concepts here, describing sets which cannot be reentered once they have been left. In the context of sustainability science, invariant sets correspond to robust facts under uncertainty or generality. Since there is no edge leaving an invariant set, no model of an ensemble has a solution leaving the associated region. In contrast, no-return sets correspond to a fragile configuration of states and velocities: Since there is no re-entering path, no solution of the model ensemble re-enters the region. A negative consequence holds for viable sets. If D is not viable, there are vertices in D where all successors are outside D, i.e. there is a region in the state and velocity space where any solution of the model ensemble necessarily leaves this region – a problematic situation if such a region is valued as positive. No-return sets can be computed easily with standard algorithms from graph theory since it can be shown that every strongly connected component and every


107

node which belongs to no strongly connected component is a no-return set, called a no-return basis. All other no-return sets are unions of such basis sets. Moreover, all invariant sets – which have the structure of a set lattice – can be generated from the no-return basis (see Eisenack 2006 for details).

3.3 3.3.1

Syndromes of Global Change The Traditional Approach

The syndrome concept is a pattern approach which was developed by the German Advisory Council on Global Change to the Federal Government (Wbgu 1994) and successively extended during the recent decade. The early syndrome concept was mainly based on expert elicitations and case study evaluations. With the help of this knowledge those symptoms and interrelations are identified which are at least necessary, but - possibly - not sufficient for the occurrence of the syndrome mechanism. This procedure ends up in a first systematization represented by a syndrome specific network of interrelations (Fig. 3.2, the black ellipses and bold arrows are the core of a syndrome). In it original form it comprises 16 syndromes (Tab. 3.3.1). The names (Tab. 3.3.1, left column) are chosen either to sketch the main processes or to represent a paradigmatic area, where the respective processes (Tab. 3.3.1, right column) can be observed. The syndromes are classified into three groups, reflecting more general properties of the underlying processes. These syndromes are non-exclusive, i.e. distinct syndromes can occur simultaneously at the same location and they can be coupled. The approach seeks for typical functional patterns of human-environment interactions by defining their essential mechanisms. In any case, this is a formidable task, since it implies an extensive evaluation of case studies, expert elicitations, and field work. Before we explain this in more detail, the following definitions have to be introduced which are essential for a general understanding of the syndrome concept. • Symptoms are the basic entities for the description of the earth system with respect to problematic developments. A symptom is a functional aggregate of detailed variables describing a single sub-process of global change closely related to the human-environment interface. Examples are the loss of species diversity, urbanization, or freshwater scarcity. The concept works with approximately 80 symptoms associated to different spheres (atmosphere, social organization, science and technology, biosphere, etc.). They also include a temporal characteristics of the...specific trends; thus, a symptom X is charac˙ X, ¨ X , . . .). terized by the tuple (X, X, • Interrelations are the connecting elements for the symptoms and specify the causal relations. They are defined as monotonic relations, i.e. with increasing (enforcing) or decreasing (mitigating) effect.

108

Syndrome Name Sahel Overexploitation Rural Exodus Dust Bowl Katanga Mass Tourism Scorched Earth

K. Eisenack et al.

Utilization Syndromes Overuse of marginal land Overexploitation of natural ecosystems Degradation through abandonment of traditional agricultural practices Non-sustainable agro-industrial use of soils and bodies of water Degradation through depletion of non-renewable resources Development and destruction of nature for recreational ends Environmental destruction through war and military action

Development Syndromes Aral Sea Green Revolution Asian Tiger Favela Urban Sprawl Disaster

Damage of landscapes as a result of large-scale projects Degradation through the transfer and introduction of inappropriate farming methods Disregard for environmental standards in the course of rapid economic growth Socio-ecological degradation through uncontrolled urban growth Destruction of landscapes through planned expansion of urban infrastructures Singular anthropogenic environmental disasters with long-term impacts

Sink Syndromes Environmental degradation through large-scale diffusion of long-lived substances Environmental degradation through controlled and Waste Dumping uncontrolled disposal of waste Local contamination of environmental assets at Contaminated Land industrial locations Smokestack

Table 3.1: List of 16 Syndromes as proposed by the Wbgu (1994). These patterns of non-sustainable development can be grouped according to basic human usage of nature: as a source for production, as a medium for socio-economic development, as a sink for civilizational outputs.


109

Figure 3.1: Overview of the global distribution of seven syndromes of global change. This image represents the achievements of the early syndrome approach and shows the local distribution of syndromatic patterns. This used some kind of heuristics and was only less formalized with respect to a homogenized methodology (after L¨ udeke et al. 2004).

This approach is nothing more than a qualitative and intuitive typifying approach, but it helps already to structure the single facets of global change in the global manifestation (see Fig. 3.1)2 . The underlying working hypothesis was that the overall phenomenon of global change should not divided into regions, sectors, or processes but be understood but as a co-evolution of dynamical partial patterns of unmistakable character. Nevertheless, the description of these pattern is not sufficient, in particular, if it the aim to provide adequate policy advice. Therefore, some kind on formalization is needed, which we will apply for the subsequently discussed examples.

3.3.2

The Overexploitation Syndrome: Terrestrial and Marine Overexploitation

Before we start with the formal description of syndromes, we will show by the example of overexploitation of terrestrial and marine natural resources how we can proceed from traditional to a more formal syndrome approach. As mentioned syndromes are patterns of interactions, frequently with clear feedback character. It is a holistic and transsectoral approach, based on expert knowledge allowing to structure our knowledge on the mega-process of global change. These experts and our 2 Discussions of specific syndromes can be found in Schellnhuber et al. 1997; Petschel-Held et al. 1999; Cassel-Gintz and Petschel-Held 2000; Kropp et al. 2001; Kropp et al. 2006b.

110

K. Eisenack et al.

intuition allows to define relevant variables (symptoms) and interactions between them. Finally this ended up in syndrome specific network of interrelations representing the essential mechanisms (see Fig. 3.2), e.g. “increasing exploitation of natural resources” leads to an “degradation of ecosystem’s structure and function”. Marine overexploitation is a typical pattern of global environmental change posing threats to mankind’s food security and marine biodiversity (Mea 2005). Fish contributes to, or exceeds, 50% of total animal proteins in a number of countries, such as Bangladesh, Cambodia, Congo, Indonesia, and Japan. Overall, fish provides more than 2.6 billion people with at least 20% of their average per capita intake of animal protein. The share of fish in total world animal protein supply amounted to 16% in 2001 (Fao 2004). Today the impacts of overexploitation and the subsequent consequences are no longer locally nested, since 52% of marine stocks are exploited at their maximum sustainable level and 24% are overexploited or depleted (Fao 2004). Global deforestation must be also regarded as an important threat to ecosystem services and forest degradation is currently more extensive in the tropics than in the rest of the world (Mea 2005). There is multiple relevance of forest ecosystems, reaching from the cultural services for indigenous people and recreation in modern life, via regulating services for soils, biodiversity and climate, to provisioning services like water and wood. Overexploitation of these ecosystems may result in ecological regime shifts, soil degradation and destruction of livelihood. It is estimated that approximately half of global forests has been lost due to human activity since the end of last ice age (Kapos et al. 2000). The non-sustainable path is characterized by strongly increasing timber extraction, fuelled by strong development of extraction and processing infrastructure, often in combination with lacking enforcement of natural protection laws and corruption (Cassel-Gintz and Petschel-Held 2000; L¨ udeke et al. 2004). For example, in Amazonia 80% of the deforested areas are located within 20 km of legally permitted roads (Barreto et al. 2006). Common features of overexploited ecosystems are that they are often beyond their capacity to regenerate which results in further severe damages. One main driving force behind this dynamic are good profit opportunities for actors to utilize the ecosystem, but also poverty alleviation plays an important role. In terms of numbers forest conversation have mainly contributed to poverty mitigation throughout history (Fao 2003). However, any kind of overexploitation is characterized by a temporal discrepancy between of socio-economic use and renewal of the resource (Cassel-Gintz and Petschel-Held 2000; Eisenack et al. 2006b), but education concerning forests and marine resources - on the single actor’s side as well as on the institutional side - has been identified as crucial for sustainable management (cf. Barreto et al. 2006). Marine and terrestrial Overexploitation syndrome share - amongst others - the symptoms “degradation of ecosystem structure and functioning”, “expansion of infrastructure”, and “policy failure”, but differ in a set of accompanying problems (cf.


111

Figure 3.2: Syndrome specific networks of interrelations for the marine: (a) after Kropp et al. (2006b) and the terrestrial overexploitation syndrome: (b) courtesy of Cassel-Gintz and Petschel-Held (2000). The black ellipses (symptoms) considered as the necessary elements for an occurrence of a syndromatic pattern. Arrows indicate forcing and bullets mitigating mechanisms. Obviously there exist similarities, but also differences between the two expressions of the syndrome.

112

K. Eisenack et al.

Fig. 3.2). In the marine case these are problems of the exact estimation of biomass, the migration of fish stocks and higher surveillance costs for management policies. Terrestrial overexploitation is associated with a larger proportion of non-mobile infrastructure which cannot be shifted to other regions, but to other uses (e.g. roads). Further forcing factors like certain institutional settings, industry lobbyism and corruption strongly differ between ecosystems and countries and they are therefore rarely or unsuitably considered in analytical approaches (cf. e.g. Smith et al. 2003; Anderies et al. 2004; Jentoft 2004). For advanced discussions on problems in fishery and forestry, see, for example, Hutchings et al. 1997; Nepstad et al. 1999; Munro 1999; Charles 2001; Eisenack and Kropp 2001; Potter 2002; Petersen 2002; Freire and Garcia-Allut 2000; Pauly et al. 2002; Wickham 2003; Smith et al. 2003; Fao 2003; Barreto et al. 2006 and also Chapter 8. Whilst the the early syndrome approach mainly aims to diagnose potential hotspots of hazardous developments, mitigation is closely connected with the need to enter into the fundamental dynamics of certain syndromes, in particular, if we are interested in an assessment of management strategies. We have shown in the previous Section that a variety of descriptions exists for resource overexploitation, which are not identical, but typically they share a common kernel of symptoms. Thus, a formal dynamic description of marine and terrestrial overexploitation on an integrated and intermediate functional scale of complexity is provided to assess the general development paths in forestry and fisheries. It is shown that in spite of uncertain process knowledge, a variety of conclusions regarding sustainable resource use can be drawn, such that common patterns of forestry and fishery can be identified. 3.3.2.1

Basic Interactions

In the following Section we present a formalized and updated version of the Overexploitation syndrome (cf. Tab. 3.3.1), which generalizes previous work of Kropp et al. (2006b) and Cassel-Gintz and Petschel-Held (2000). We introduce a common core pattern which combines both types of overexploitation. For that we use model ensembles which are solved with the Qsim algorithm and analyzed with viability criteria. In order to reveal more insights in the dynamics of this pattern, the archetypes will be examined in an abstract and systematic manner, i.e. by identification of those processes/features that are characteristic. In the subsequent Sections we will do this in an equal way for the Sahel and the Dust-Bowl syndrome in order to make clear that our approach will work in general, before we discuss the influence of single actors in syndrome analysis again for the example of overexploitation. Broad profit opportunities are the starting point for the overexploitation dynamic to unfold. This can be related to different reasons. Of course, there has to be an ecosystem which provides a rich but subtractable resource, which can be extracted at low costs or sold for high prices. In many cases, one cause for low costs is an de jure or de facto open access nature of the resource and available capital. Prices


113

can be high due to (international) demand. Although profitable, the extraction requires to build-up the necessary infrastructure (e.g. boats, lorries sawmills, harbors), which can partially not be converted to other uses. These mechanism imply that multi-national corporations play an important role. With high investment in infrastructure, extraction also increases, potentially beyond the regeneration capacity of the resource. This capacity itself is under pressure due to degradation and conversion of the whole (terrestrial or marine) ecosystems structure and functions, e.g. losses of biodiversity, changed water balance (in the terrestrial case), anthropogenically induced climate change, and shifts in the food-web. A socio-economic consequence of these processes is the displacement of traditional use, economic marginalization of indigenous people, a subsequent decline in the traditional structures, and finally migration. All these effects have the potential to reduce profit opportunities from resource extraction. Yet, a vicious cycle closes if these effects are over-compensated by the efficiency gains due to investment in more profitable infrastructure (Kropp et al. 2002; Eisenack et al. 2006b). Further effects can be observed in the policy domain. A positive one relates to the increasing environmental awareness which may lead to the establishment of management regulations and international agreements (e.g. Un 1995). However, in many cases efficient management strategies are rejected or fail (cf. Smith et al. 2003; Daw and Gray 2005). This can be related to various reasons, in particular high levels of corruption or lobby pressure on local and regional administrations. Usually, profits resulting from resource exploitation are made by transnational companies, which may be supported by national policies to develop export opportunities. If there are already high levels of infrastructure, there is a stronger pressure to keep money flowing to sustain returns on former investments. Thus, there are incentives for the public to reduce environmental standards or even to introduce subsidies for further infrastructure, leading to overcapitalization and resource overuse, thus closing another vicious cycle where private losses are compensated by public. The overexploitation dynamics can be attenuated by an increasing proportion of infrastructure which is not directly related to resource extraction but to marketing and transport of the resource. This is characterized by increasing overall costs which do not contribute to profits from extraction. Such infrastructure is spatially fixed to a region of the ecosystem, but can sometimes be converted to other uses, enforcing the dynamics of other syndromes. This is different for mobile extraction infrastructure, where investments need not be accounted for as sunk costs, since extraction units can be moved to other resources if the adverse effects of overexploitation make the industry less profitable. For the marine case, this is known as serial overfishing, indicating that fishing firms change target species or fishing grounds (Go˜ ni 1998). It is also widely described in the literature for the terrestrial case (e.g. Lambin and Mertens 1997; Power 1996).

114

K. Eisenack et al.

3.3.2.2

Model Ensemble

Based on a detailed literature survey and on previous work, we propose the following variables and relations for the syndrome core (see Tab. 3.2, Fig. 3.3). They can basically be grouped along three feedback cycles related to the processes outlined in the previous Section, one related to investment in mobile extraction infrastructure (“infra”), to lobby pressure (“lobby”) and to sunk costs of immobile infrastructure (“sunk”). Also some driving forces can be identified. For clarity, we omit consequences which are not part of a feedback cycle. In the following we use the above DEMAND PROF−OPP

sunk OTHR−INFRA

lobby POL−PROF REC

EXTR−INFRA

SUPPLY

RES infra

GEC EXPLOIT

TECH

Figure 3.3: Core processes of the Overexploitation syndrome. Arrows denote enforcing, bullets attenuating and boxes ambiguous influences. The marks “infra”, “lobby” and “sunk” indicate feedback loops. Regarding the abbreviations cf. Tab. 3.2.

qualitative description of the Overexploitation syndrome to set up a formal model using the model ensemble method and Qdes. This should provide additional insights into the syndrome dynamics without the need for quantitative data retrieval. We focus on the “infra” and “lobby” feedback loop, where Prof-Opp, PolProf, Extr-Infra, Exploit and Res occur. Here we assume the driving forces to be constant. To derive a formal model of this part of the influence diagram, more specifications need to be made. Since there is a considerable degree of freedom in this choice, the syndrome core as described by the causal-loop diagram (Fig. 3.3) ˜ We narrow it represents a very general archetype, i.e. a general model ensemble M. ˜ down to a concisely defined, still infinite model ensemble M ⊆ M by the following specification. As state variables K, R, L we chose Extr-Infra, Res and Pol-Prof, which represent stocks, while we assume Prof-Opp (P ) and Exploit (E) to be interme-

Modeling Techniques to Assess Patterns of Global Change Variable

115

Definition

For feedback cycles: Prof-Opp (P ) Extr-Infra (K)

Exploit (E) Res (R) Othr-Infra

Pol-Prof (L) Rec

Increasing profit opportunities. Expected profit rates for the extraction industry. Expansion of extraction infrastructure. Extraction units which enhance efficiency of exploitation, cannot be converted to other uses, but are mobile to be shifted to other ecosystems. Increasing resource exploitation. Extraction rate of resource under consideration. Change of resource stock. The difference between natural regeneration and extraction rate. Expansion of non-extraction infrastructure. Those investments which do not contribute to extraction efficiency and cannot be shifted to other regions. Policies for profit opportunities. Political efforts to subsidize exploitation or decrease environmental regulations. Regional environmental change. Indirect effects of exploitation (as ecosystem conversion), and other regional environmental drivers which influence regeneration capacity.

For driving forces: Gec

Demand

Supply

Tech

Global environmental change. Changes affecting the regional regeneration capacity of the ecosystem, but which can only marginally be influenced by decisions on the regional level. Increasing demand for resource consumption. Regional or international increase in demand for products from the resource, which may be indicated by high prices on the world market. Increasing supply of capital for resource extraction. Financial and physical capital available for investment in infrastructure. Development of new extraction technology. Technological changes which alter efficiency or sustainability of exploitation.

Table 3.2: Core variables of the Overexploitation syndrome.

116

K. Eisenack et al.

diate variables which determine flows: K˙ = f1 (P, L), R˙ = f2 (E), L˙ = f3 (P ),

(3.2)

P = f4 (K, R), E = f5 (K, R). It is assumed that DP f1 , DL f1 , DP f3 > 0, DE f2 < 0 and DK f4 , DR f4 , DK f5 , DR f5 > 0.3 Moreover, we make the premise that there exists a regeneration capacity m > 0 so that f1 (m) = 0 and a political power threshold z > 0 with f3 (z) = 0. Beyond these assumptions, the exact derivatives, the regeneration capacity and investment threshold are not known quantitatively (due to uncertainties and due to differences between the social-ecological systems which should be covered by the archetype). In a systems dynamics context, the next step would be to decide on functional and parametric specifications to run a quantitative simulation of the syndrome dynamics. However, due to the generality of the syndrome archetype, we want to consider all systems which comply with the above requirements, yielding an infinite model ensemble M = {f = (f1 f2 f3 f4 f5 )t ∈ C 1 (R5+ , R5+ ) | ∃m, z ∈ R+ : f1 (m) = f3 (z) = 0 



(3.3)

0 0 + + 0  0 0 0 0 −    and ∀x ∈ R5+ : sgn(J (f )(x)) =   0 0 0 + 0 }, + + 0 0 0  + + 0 0 0

where J denotes the Jacobian. By x(·) = (K(·) R(·) L(·) P (·) E(·))t we denote a trajectory of the state variables, so that the solution operator reads S(x) = {x(·) ∈ C 1 (R5+ , R5+ ) |

x(0) = x, ∃f ∈ M ∀t ∈ R+ : ˙ K(t) = f1 (P (t), L(t)), ˙ R(t) = f2 (E(t)), ˙ L(t) = f3 (P (t)), P (t) = f4 (K(t), R(t)),

E(t) = f5 (K(t), R(t))}. 3

For sake of readability DX Y is equivalent to ∂Y /∂X.

(3.4)


117

It is defined for all initial values x = (K R L P E)T satisfying P = f4 (K, R) and E = f5 (K, R). The task to determine S(R5+ ) can be solved using qualitative differential equations (see Section 3.2.2). 3.3.2.3

Results

The resulting state-transition graph has 52 nodes and 102 edges. There are many final states with R = 0 or K = 0 which differ only slightly. For a readable output of the graph such final states and equilibrium states are omitted which unlikely occur in a changing environment. The graph can be further simplified by automatically eliminating nodes which represent state variables being constant over a time interval (called non-analytical states), and eliminating edges which are not likely to occur (called marginal edges, see Eisenack 2006 for a justification, formalization, and algorithmic treatment). The outcome is a graph consisting of 8 nodes and 16 edges (see Fig. 3.4). It can be easily seen that the whole graph is strongly connected, meaning that it is – in principle – possible to reach every state from every state. This implies that it is possible that the system perpetuates infinitely through different stages. The are also bi-directional edges, which allows to draw the conclusion that the system may shift back and forth between two states forever. However, the definition of the state-transition graph only guarantees that for every edge between two nodes there exists at least on right-hand side in M, such that the associated quantitative solution visits these qualitative states in the prescribed order (see Eisenack 2006 for details). Note that it is not claimed that every solution visits these states. For longer paths such a quantitative solution does not need to exist at all (making them so-called spurious behaviors, cf. Kuipers 1994). However, we may ask which subsets M′ ⊂ M can guarantee that every solution evolves along a given path, or, alternatively, eliminates a given edge from the graph. We will provide a detailed example for the latter below. But, for a first analysis, suppose that a system covered by the Overexploitation syndrome avoids a “catastrophic” outcome (as R = 0 or K = 0) for a long time: what can already be learned about the dynamics represented by this strongly connected graph? To make one striking result more obvious, we apply a projection as further simplification technique to the graph. Here, only differences in some selected variables are considered, while nodes which only differ in other variables are “joined”. If there exists a path in the projected graph, there also exists a path of corresponding nodes with the same qualitative values for the selected variables in the original graph (see Clancy 1997; Eisenack 2006 for a technical introduction). By selecting K and E the existence of boom-and-bust cycles as a central feature of the Overexploitation syndrome becomes clear (see Fig. 3.5 and also Eisenack et al. 2006b). In the green state the resource is relatively unexploited and little capital is assigned to the extraction sector. However, the profitability of the forests or fish stocks attracts investment, leading to increasing extraction (growth phase). This

118

K. Eisenack et al.

Figure 3.4: Simplified state-transition graph of the Overexploitation syndrome. Every ˙ sgn (R), ˙ sgn (L) ˙ are displayed, box represents a qualitative state, where the signs sgn (K), indicated by an upward or downward pointing triangle. Arrows between nodes indicate possible transitions between qualitative states. Catastrophic transitions are not depicted here, but R = 0 is a possible successor for all states with R˙ < 0, and K = 0 for all states with K˙ < 0.

state is left when exploitation is above the sustainable level, where net investment still takes place. This is due to the fact that the absolute resource stock is still high enough to sustain profits and, due to the lobby pressure, potentially decreasing profit rates are compensated (overexploitation phase). If the system is in this state for a longer time, extraction decreases because less of the resource is available. However, investment is still positive due to the lobby loop and the positive effect of accumulating extraction infrastructure on the profit opportunities, for example caused by efficiency gains (orange red state, Fig. 3.5). It is thus interesting that expansion of infrastructure and contraction of resource use occur at the same time. A situation which can be characterized as built-up of overcapacities: Obviously more capital is used to obtain less from the resource. Depending on the rates of lobby pressure and extraction efficiency, the system may shift back to a state with increasing resource use. But it is also possible to move to the situation where the degrading resource shifts extraction costs to a level where investment becomes negative (collapse phase). If this results in extraction rates becoming sustainable again (E < m), the system cannot directly shift back to a level with decreasing capital and high exploitation. The resource has to regenerate first (recovery phase), and then the system may go through the whole cycle again with extraction beginning to increase and investment becoming positive thereafter. It should be noted that this boom-and-bust cycle can only develop in one direction (if we disregard some episodes of shifting back and forth). This is well-known for many case studies in fishery and forestry (e.g. Hilborn and Walters 1992; Power 1996). It is also interesting to observe that every such cycle has to visit the state where overcapacities are built-up, making it a valid statement that every system described by the overexploitation syndrome inevitably undergoes such a period of increasing inefficiency – except for the case of a collapse (see Eisenack et al. 2006b, for a detailed economic analysis of this property for the case of unregulated fisheries).


119

Figure 3.5: Projection of state-transition graph, considering only changes in K˙ and E˙ (for the symbols cf. Fig. 3.4). Triangles for E˙ are in the upper part if E > m, and in the lower part for E < m. In the red state, extraction overcapacities are built-up. The green state is a typical initial state of a boom-and-bust cycle, beginning with an increasing but low exploitation rate and expanding investment in extraction infrastructure.

3.3.2.4

A Refined Model Ensemble

As indicated above it is important to know whether new structures appear if the model ensemble is narrowed down. This can provide important information for the management of a system governed by the archetype. We demonstrate this idea by introducing so called ordinal assumptions on the Jacobian and employing the viability concepts from Section 3.2.3 (cf. also Chapter 2). Define M′ := {f ∈ M | ∀x ∈ R5+ : DR f4 (x) < DK f4 (x) and DK f5 (x) < DR f5 (x)}, (3.5) which implies to consider only those cases of the Overexploitation syndrome where profit expectations react more sensible on changes in capital than on the state of the resource, while the marginal productivity of the extraction sector is stronger with respect to R than with capital K. We call such specification ordinal assumptions. These ordinal assumptions may hold because a given special case of the syndrome is considered, or they may be brought about by management interventions. In the following it is indicated how a state-transition graph for M′ can be determined (see ˙ sgn (R), ˙ sgn (L), ˙ sgn (P˙ ), Fig. 3.6 for the result). At first, not only the signs sgn (K), ˙ sgn (E) are considered, but also whether ˙ − |R| ˙ δ := |K|

(3.6)

is positive or negative, distinguishing situations where extraction capacities change faster than the resource or vice versa. In the first step, every state in the original state-transition graph is splitted in two states with sgn (δ) = (+) or sgn (δ) = (−). These states inherit the edges of the original state and have a bi-directional edge between them. In the second step every edge which contradicts the ordinal assumptions is eliminated. For some edges there is a second elimination criterion, since due to the introduction of δ. If, e.g. δ > 0, it is impossible that K˙ vanishes, so that every edge starting at such a state and where K˙ changes its sign has to be refuted.

120

K. Eisenack et al.

˙ 0 < L, ˙ and 0 < δ. From the Consider, for example, a state with K˙ < 0 < R, original state-transition graph (leftmost box in Fig. 3.4), the following edges need to be assessed: (i) K begins to increase, (ii) L begins to decrease. Furthermore, it has to be checked if (iii) δ can become negative. The first edge has to be eliminated due to the reason given above. For edge (ii), note that ¨ = DP f3 P˙ = DP f3 (DK f4 K˙ + DR f4 R) ˙ L ˙ sgn (K) ˙ + DR f4 |R| ˙ sgn (R) ˙ = DP f3 (DK f4 |K|

(3.7)

˙ ˙ ˙ ˙ = DP f3 (|R|(D K f4 sgn (K) + DR f4 sgn (R)) + DK f4 sgn (K)δ).

˙ R˙ (which characterize the state), the signs of DP f3 , DK f4 , Due to the signs of δ, K, ˙ DK f4 (see definition of M), and due to the ordinal assumption DK f4 K˙ > DR f4 R, ¨ ˙ it follows that L > 0. Consequently, L cannot vanish and edge (ii) has to be ˙ and that δ can only eliminated. Regarding edge (iii), it is obvious that δ = K˙ + R, ˙ ˙ vanish if K = −R. Thus, by differentiating δ at the locus where δ = 0, we obtain ¨ +R ¨ δ˙ = K ˙ + DL f1 L˙ + DE f5 (DK f5 K˙ + DR f5 R) ˙ = DP f1 (DK f4 K˙ + DR f4 R) ˙ P f1 (DK f4 − DR f4 ) + DE f5 (DK f5 − DR f5 )) + DL f1 L. ˙ = K(D

(3.8)

Again, this expression is positive due to the ordinal assumptions, the properties of the model ensemble and the characteristics of the state under consideration. Therefore, also edge (iii) has to be eliminated. Since all three edges can be refuted, ˙ 0 < L, ˙ and 0 < δ has no successor, making it an the state with K˙ < 0 < R, invariant set as introduced in Section 3.2.3: no system included in M′ can leave this situation where capital decreases – although the resource recovers and increasing lobby pressure compensates losses. These types of arguments can be applied to all other states, bringing about a structure where a boom-and-bust cycle still exists, but where several invariant sets and no-return sets can be entered (see Fig. 3.6). These irreversibly bring the system either to an economic (light blue boxes) or resource decline (red boxes). Interestingly, the states where over-capacities are builtup (orange boxes) lead to economic and not to resource decline, a situation which complements observations in many case studies (e.g. Go˜ ni 1998). It should be stressed again that this property holds for every system contained in M′ , so that every system has a corresponding set of repellers and invariance kernels (cf. also Chapter 2). 3.3.2.5

Results

To come up with management conclusions for the Overexploitation syndrome, it must be stated that the ordinal assumptions made for M′ does not provide a beneficial structure. In addition to the risks of resource depletion and diminishing capital


121

Figure 3.6: State-transition graph of the overexploitation syndrome under ordinal assumptions. The fourth column in each state denotes the sign of δ. The light blue and red states in grey boxes indicate no-return sets and invariant sets as discussed in the text. The states outside the grey boxes form a cycle, where in the orange boxes overcapacities emerge.

which are already present in the more general case of M, there are additional “cascades” of problematic irreversible developments which cannot easily be classified as such in an early stage. For example, without the result depicted in Fig. 3.6, it is not straightforward to forecast a continuous resource overuse at a stage where K, R and L increase and extraction infrastructure expands faster than the resource regenerates. If the properties expressed by the ordinal assumptions hold for a particular system, there should be interventions to change that, e.g. by introducing incentives to value resource quality in the formation of profit expectations more strongly. If the assumptions do not hold for another system, it should be avoided that they enter into force. Of course, for a systematic assessment of management options, further ordinal assumptions have to be considered – a task which remains to be done. Here algorithmic solutions are currently under development.

3.3.3

The Dust-Bowl Syndrome: Consequence of a Non-adapted Industrial Agriculture

As a further example we examine the interaction of environmentally destructive agricultural practices. A paradigmatic case were the historic droughts of the 1930s transformed the Wheat Belt in the west and southwest of the U.S. into the so-called “Dust Bowl” - a dry landscape where dust storms prevailed. “Black blizzards” swept away the nutrient-rich topsoil of the region - like the storm on 9th May 1934, which transported approx. 350 million tonnes of dust from Montana and Wyoming via Dakota towards the east coast. The state-aided Dry Farming Program powered by the export drive to war-torn Europe had “prepared the ground” for the “Dust Bowl” phenomenon in the U.S.: through the massive use of machines (tractor, disc

122

K. Eisenack et al.

harrow, combine, etc.) the Great Plains were transformed into a monotonous “grain factory”, whose broken-up surface remained exposed to the weather and unprotected for a large part of the year (Worster ). This kind of interaction of state subsidies with a modern, capital intensive agricultural sector is an important facet of the so called Dust-Bowl syndrome as defined by the Wbgu (1995) which is in general characterized by a minimization of human labor input through the use of a wide range of machines on spacious, “cleared” agricultural areas and in “animal factories”. Attempts are made to maximize yield and capacity through • mono-cultivation of highly productive kinds of plants, • intensive livestock farming, • large quantities of pesticides and medicine, • intensive use of fertilizer and feed, • intensive irrigation. 3.3.3.1

Basic Interactions

The main symptoms of the corresponding soil damage profile are (for a more detailed description see Wbgu 2000): • great susceptibility to wind and water erosion as a consequence of the considerable exposition times of the ploughed-up soil, combined with the low degree of structuring of the agricultural landscape; • destabilization of turf and subsequent erosion through over-sized herds and overgrazing; • loss of fertility due to deep ploughing, elimination of harvest wastes and monotonous crop rotation; • reduction of soil drainage as a result of compaction by heavy agricultural machines; • chemical soil pollution via overfertilisation and contamination (pesticides). This causal pattern seems to be also relevant to the situation in Europe after World War II, where governmental subsidies within the framework of the European Economic Community were used to foster the modernization of the agricultural sector - which had a comparatively low productivity at this time. An important regulation tool was (and is) to guarantee high producer prices which should increase and stabilize the income of the farmers and allow to increase productivity by investments in mechanization and chemisation. After some success in the beginning, economically and environmental problematic consequences occurred: tax payers subsidized the over-production of different agricultural goods (e.g. the “butter


123

mountain” in the 1970s) and the capitalized agricultural production system generated the syndrome-typical environmental damages. To better understand this specific human-environment system we developed a qualitative system-analytical model which includes the most important variables and interactions as follows: Agricultural income depends on production and producer prices. Above a particular income threshold the farmer is able to invest in his productivity which in the short term will increase his production with a positive feedback on his income. This positive feedback loop is dampened by two effects: • when the total production satisfies the demand the guaranteed producer prices will be lowered by the regulating administration to avoid overproduction • with an increase in mechanization increasingly more money is needed to sustain and use this machinery In competition with this loop there is a negative feedback which works on intermediate time scales: as agricultural productivity depends also on the quality of natural resources (e.g. soil fertility and structure) and capital intensive farming often degrade these, either an increasing part of the income is necessary to compensate for these losses or the farmer will face long-term production losses despite his investments. Another important competition is between the wish of politicians and tax payers to reduce subsidized prices in case of overproduction on the one side and the efforts of the agrarian lobbyists to keep the income of their clientele high on the other. In general the success of the lobbyists depends of the importance of the agrarian sector in society and we will take here the total income of the farmers as an indicator. For a more detailed description of these mechanisms see the report of L¨ udeke and Reusswig (1999) on the Dust-Bowl syndrome in Germany. In Tab. 3.3 we summarize the so far introduced variables and Fig. 3.7 shows the discussed relations between them as discussed above. Variable G– Price Income Productivity Env– Damage Production Inf– Lobby

x1 x2 x3 x4 x5 x6

Definition politically guaranteed producer price for farmers income generated on the farm area or labor productivity of agriculture damage to environment and natural resources of the farm agricultural production influence of agrarian lobbyists on guaranteed producer price

Table 3.3: Variables used in the description of the regulated agrarian system.

124

K. Eisenack et al.

Figure 3.7: Cause-effect diagram (causal-loop diagram) for a (price)regulated agrarian system as, e.g., realized in the Eec/Eu since World War II. Arrows denote enforcing, bullets attenuating forces. Note the self-attenuation of productivity (e.g. by decay of machines), guaranteed prices (the higher they are the larger is the public pressure to reduce them) and the influence of the agrarian lobbyists (which has to be actively sustained by the agrarian sector).

3.3.3.2

Model Ensemble

The following equations (3.9) describe the causal-loop diagram of Fig. 3.7 in the structure of a system of ordinary differential equations (for the abbreviations of the variables, see Tab. 3.3). x˙ 1 = g1 (x1 , x5 , x6 ), x2 = g2 (x1 , x5 ), x˙ 3 = g3 (x2 , x3 , x4 ),

(3.9)

x˙ 4 = g4 (x3 ), x5 = g5 (x3 ), x˙ 6 = g6 (x2 , x6 ) . To adapt the model further to the European situation where the agrarian lobbyists were always able to avoid a significant reduction of subsidies we integrate this into the model. This reduces the solution space while loosing only unrealistic trajectories. Another simplification is possible, because in Europe productivity determines production much more than the spatial extension of farming. Including this into the model one gets: x˙ 1 = f1 (x3 ), x2 = f2 (x1 , x3 ), x˙ 3 = f3 (x2 , x3 , x4 ), x˙ 4 = f4 (x3 ),

(3.10)


125

From this, the respective model ensemble can be defined. With F1 = {f ∈ C 1 (R+ , R+ )| ∃ pr− s > 0 :

∀ x < pr− s : Dx f < 0 and x ≥ pr− s : f (x) = 0}

F2 = {f ∈ C 1 (R2+ , R+ )| ∀(x1 , x2 ) ∈ R2+ : Dx1 f ≥ 0, Dx2 f ≥ 0, f (0, x2 ) = f (x1 , 0) = 0}

1

H1 = {f ∈ C (R+ , R+ )| ∃ i− inv > 0 :

∀ x < i− inv : f (x) = 0 and ∀ x ≥ i− inv : Dx f > 0}

1

H2 = {f ∈ C (R+ , R+ )| ∃ ed− c > 0 :

∀ x < ed− c : f (x) = 0 and ∀ x ≥ ed− c : Dx f > 0}

(3.11)

H3 = {f ∈ C 1 (R+ , R+ )| ∃ pr− c > 0 :

∀ x < pr− c : f (x) = 0 and

F3′

= {f ∈ C

1

(R3+ , R+ )|

F3 = {f ∈ C

1

(R3+ , R+ )|

∀ x ≥ pr− c : Dx f > 0} ∀ (x1 , x2 , x3 ) ∈ R3+ :

Dx1 f ≥ 0, Dx2 f ≥ 0, Dx3 f ≥ 0, ∀ (x1 , x2 , x3 ) :

∃g ∈ F3′ , f1 ∈ H1 , f2 ∈ H2 , f3 ∈ H3 :

f (x1 , x2 , x3 ) = g(f1 (x1 ), f2 (x2 ), f3 (x3 )} F4 = {f ∈ C 1 (R+ , R+ )| ∃ pr− c > 0 :

∀ x < pr− c : f (x) = 0 and ∀ x ≥ pr− c : Dx f > 0}

where pr– s, pr– c, ed– c denote the relevant threshold values (landmarks). The model ensemble is defined by M = {f = (f1 f2 f3 f4 f5 )|f1 ∈ F1 , f2 ∈ F2 , f3 ∈ F3 , f4 ∈ F3 } with the respective solution operator (cf. also Section 3.3.2.2) S(x) = {x(·) ∈ C 1 (R4+ , R4+ ) |

x(0) = x, ∃f ∈ M∀t ∈ R+ :

x˙ 1 (t) = f1 (x3 (t)),

x2 (t) = f2 (x1 (t), x3 (t)), x˙ 3 (t) = f3 (x2 (t), x3 (t), x4 (t)), x˙ 4 = f4 (x3 (t))}.

(3.12)

126 3.3.3.3

K. Eisenack et al. Results

Using the Qsim algorithm, one gets all possible qualitative trajectories, which are summarized in Fig. 3.8.

Figure 3.8: All qualitative trajectories which are in accordance with the cause-effect diagram for the (price-) regulated agrarian system as given in Fig. 3.7. Each qualitative state is symbolized by a rectangle which is subdivided into four columns denoting the different variables with respect to their qualitative magnitude (see state definition in the upper left corner) and trend direction. A rhombus stands for an undefined direction, the bullet for constancy in time. The large arrows denote possible sequences of states. No-return sets are symbolized by the large red rectangles.


127

Here, state (a) in the lower left corner characterizes the immediate post-war situation in Europe: guaranteed prices start to increase, farm income starts also to increase but is still below the threshold i– inv necessary for investment in productivity increases. Productivity is below the value where significant resource damage occurs and the directly related production is below the demand, justifying further increase in guaranteed prices. The necessary successor state (b) reproduces the observed initial success of the regulation: Income exceeds i– inv and productivity starts to increase. But already state (c) which follows necessarily, shows an ambivalent situation: once the productivity has transgressed the threshold pr– c where significant costs occur to sustain it the further income development becomes undefined with respect to direction and magnitude, i.e. the system does no longer inherently guarantee a secure livelihood for the farmers. This state is closely connected with four other states which constitute a no-return set (Nrs1), i.e. when the trajectory leaves this set, it can never return to it as long as it governed by the assumed relations. State (d) which can follow (c) depict the most positive situation in Nrs1: stable and high income of the farmers, stable and high productivity at pr– c, which generates sufficient production and a stop of the increase of guaranteed prices. But this situation is not systemically stable: it may either return back into (c) or proceed into (e), depicting subsidized overproduction - the “butter mountain” case. As long the trajectory remains in Nrs1, productivity remains above (or at) pr– c and resource/environmental damage below ed– c. Unfortunately the system allows several ways of transgression into Nrs2 which is characterized by Env– Damage above ed– c, a situation where the reduced quality of the resource influences productivity negatively, resulting either in additional costs to compensate for this or in productivity losses. This generates a much more unstable situation for productivity than in Nrs1 while the income stability is similarly poor. So, for policy Nrs1 is preferable because at least the aim of productivity increase is realized while the objective of stabilizing the farmer’s income is not reached. After entering Nrs2 there is no way back to a secured high productivity situation. Instead environmental damage and increasing subsidies have to be expected and – even worse – state (f) can be reached: the total collapse of the agrarian system becomes possible, constituted by the breakdown of production, income and resources. Comparing this model–deduced structure of the state space with the actual observed situation of the agrarian system in Europe, it is probably located within Nrs2, i.e. only structural changes will allow the improvement of the situation. This is because the endogenous dynamics will end up in the same situation after temporary interventions which generate only a “jump” in the state space. The proposed model can serve as a means of policy assessment with respect to such structural changes which will certainly have to be more complex than the simple recipe of “de-regulation” which throws back the system into the unsolvable contradiction between short-term profit interests and mid- to long-term environmental impacts. Obviously an intelligent re-structuring of regulations is the way to go as, e.g., suggested by the Sru (1998).

128

3.3.4

K. Eisenack et al.

The Sahel Syndrome

The third example systematized in this manner is a pattern which is described by the Sahel syndrome (cf. Tab. 3.3.1). It addresses the field of the closely related environmental, economical, social and political aspects of smallholder farming in developing countries – a field where anthropogenic environmental change often feeds back rather rapidly on the socio-economic situation of the actors – and where the latter are confronted with strong constraints like social and economic marginalization, population pressure and fragile natural production conditions (L¨ udeke et al. 1999). 3.3.4.1

Basic Interactions

From several studies on the semantic aspects a formal model is teased out which describes the qualitative functional relationships between labor allocation into off-farm labor, resource conservation measures and short-term yield maximizing activities and natural degradation, income and poverty, market prices, access to resources, population growth and some further relevant variables. Figure 3.9 shows an aggregated synopsis of relations as for example stated by Leonhard (1989), Kates and Haarman (1992), Kasperson et al. (1995), and others dealing with the environment– poverty problem. This scheme is supposed to constitute “mental maps” (cause-effect diagram) (cf. Fig. 3.9) of the most important mechanisms underlying the dynamics of smallholder agriculture in an intermediate functional resolution. In the agricultural subsystem yield depends on the state of the resource and the directly yieldoriented labor investment, i.e. it increases with both factors and vanishes if one factor becomes zero. We subsequently call this relation “qualitative multiplication” (cf. Kuipers 1994). Under “yield oriented labor” we subsume frequent tillage and weeding, the shortening of fallow periods, intensive irrigation etc. All these measures can increase the annual yield in the short term but put pressure on the resource, leading to resource degradation by erosion, soil fertility losses, salinization etc. On the other hand we have agricultural activities which counteract these degradation trends as terracing, drainage ditches, hedge plantings, mulching etc. We include here not only activities for resource conservation, but also endogenous technological progress which increases the efficiency of the resource and obviously needs some engagement by the farmer, a process which – in our qualitative framework – can be represented by developing the resource (in that way we map Boserupian aspects of the dynamics). These two kinds of activities constitute the pool of agricultural labor – and decreasing one of them means increasing the other (under constant total agricultural labor). This simple relation is called “qualitative addition” (Kuipers 1994). The second realm of smallholder activities is the off-farm labor: here an offfarm income is generated via the hourly wage. This income, measured in units of agricultural produce, together with the obtained yield adds up to the total consump-


129

Figure 3.9: Mental map (causal-loop diagram) of smallholder agriculture. In contrast to Figs. 3.3 and 3.7 here potential side effects are introduced by a “trigger finger”. Furthermore qualitative nexuses are shown.

tion. The total available labor is divided into the labor on the own farm (as land manager) and off-farm labor. Furthermore the qualitative model shown in Fig. 3.9 considers population which, on the one hand, generates the total available labor and, on the other hand, divides the total consumption into per capita consumption, sometimes resulting in poverty. Marginalization of smallholders influences their access to resources as well as to wage labor (e.g. via ethnic discrimination). The total availability of wage labor and the hourly wage depend on trends in the national economy. How can such a mental map be used for any (weak) kind of prediction, which is the basis for any assessment of policy options? One way would be to quantify the variables and relations and to introduce an utility optimization hypothesis for the two decision problems considered in the qualitative model: the allocation of labor between on farm vs. off-farm and between short term yield and resource development (red symbols in Fig. 3.9). This kind of approach was chosen by many modelers (e.g. Barbier 1990; Barrett 1991; Grepperud 1997) – but both the optimization hypothesis

130

K. Eisenack et al.

Definition Resource quality (e.g. soil) Total available labor Off-farm labor Yield oriented labor Hourly wage Labor for development of R Yield Consumption per capita critical value for C

Variable R LT LW LY w LQ Y c ck

Table 3.4: Variables of the qualitative smallholder model.

and the possibility of adequate quantification are questionable. In particular under the rapid change of the conditions of smallholder agriculture in developing countries the argument that an optimization approach would model the result of a long lasting evolution process leading to an optimal adaptation of the actor’s strategy seems implausible. In contrast to these attempts we define qualitative behavior rules which are far less sophisticated: • the reallocation between on- and off-farm labor is performed according to the difference between present labor productivity of the agricultural and the offfarm activity • the reallocation between yield oriented labor and labor for the development/conservation of the resource is governed by the relation of present per capita consumption and a critical level of consumption, ck , below which conservational labor is reduced According to the “mental map” in Fig. 3.9 and the allocation rules stated above we get the following structure in terms of the variables defined in Tab. 3.4: dR = g1 (LY, LQ) dt dLT = g2 (t) dt ! Y dLW = g3 w − dt LY dv Y + w · LW = g4 , ck dt LT

(3.13) !

Due to LT = LW + LY + LQ, the number of relevant variables can be reduced to four. We chose (R, LT, LW, v), where v = LQ LY . The latter is appropriate as the competing influences of LY and LQ on the change of R are best represented by the

131


relation of the different work inputs. After this variable transformation we obtain the following set of ordinary differential equations dR dt dLT dt dLW dt dv dt

= f1 (v) = f2 (t)

(3.14)

= f3 (R, LT, LW, v) = f4 (R, LT, LW, v)

which belongs to the following model ensemble: M = {f = (f1 f2 f3 f4 )t ∈ C 1 (R4+ , R4+ ) |  0 0 4 ∀x ∈ R+ : sgn(J (f )(x)) =  − +

0 0 ? ?

0 0 ? ?

 + 0 }. ? −

(3.15)

Now the number of unclear signs in the Jacobian can be significantly reduced by introducing the assumption that the labor productivity of agriculture, Y /LY , decreases with increasing LY . This results strictly in:   0 0 0 + 0 0 0 0  sgn(J (f )(x)) =  − + − −}, + ? ? −

and in most reasonable cases:



 0 0 0 + 0 0 0 0  sgn(J (f )(x)) =  − + − −}, + − ? − 3.3.4.2

Results

In the following, we will discuss the case where the last question mark is “–”, which is valid when agriculture contributes significantly to household income. In Fig. 3.10 the resulting qualitative trajectories are shown in their most aggregated form. We obtain two disconnected graphs, characterized by constantly increasing or decreasing total labor force, LT , which reflects that no feedback of the system dynamics on this variable was considered. In both cases a persistent trend combination occurs, the simplest form of a locked set. In sub-graph (a) characterized by increasing LT ,

132

K. Eisenack et al.

Figure 3.10: Resulting trajectories, sequence of variables as defined in the text (R, LT, LW, v)

i.e. population growth, this locked set shows constantly decreasing resource quality, increasing engagement in off-farm labor and a decreasing fraction of the on-farm labor invested in sustaining or improving the resource. This means - according to the model assumptions - insufficient consumption per capita (below ck ). If an observed smallholder system shows this trend combination one can not expect that the situation will improve due to the endogenous mechanisms. Further inspection of graph (a) reveals that this locked set is a possible, but not a necessary outcome of the dynamics - an oscillatory behavior between states with increasing and decreasing resource quality is also in accordance with the model assumptions. A more detailed analysis of the result could probably identify dangerous trend combinations which could be precursors of the locked set, such generating “early warning indicators”. Sub-graph (b) is characterized by a decreasing population and here a desirable locked set occurs: once in a situation with increasing resource quality, decreasing off-farm engagement and increasing labor investment into the resource base, the endogenous mechanisms will stabilize this trend combination. In general these results emphasize the important role of the demographic development - only in case of decreasing population the considered smallholder system exhibits a stable and acceptable development path which is in this case based on the continuous improvement of the production system under control of the farmer. It should be stressed that decreasing LT means a decrease in labor force and a respective decrease in consumption. Population number reducing processes, like the disproportionate out-migration of younger people or those capable to work as well as the consequences of epidemics like Aids do clearly not fulfill this condition.

3.4

From Competition to Cooperation in Sustainable Resource Management - A Multi-Actor Approach

So far syndromes have basically been analyzed as single actor problems. In a social environment the interaction among multiple actors can lead to many possible types of interaction, ranging from conflict among all actors (all couplings negative) to cooperation among all actors (all couplings positive). Game theory analyses a variety of archetypes of interactions, with the zero-sum game and the prisoner’s dilemma game as well-known cases. These games play a role in natural resource management, in particular to understand the tragedy of the commons.


133

In the following, we describe the interaction between the reproduction of a natural resource stock R, the resource extraction (harvesting) Ei , profit Pi and the cost of extraction efforts Ci for actors i = 1, ..., n (cf. Scheffran 2000). The resource growth with extraction is represented by R˙ = r(R) −

X

Ei ,

i

which depends on the reproduction function r(R) of the resource stock and the resource extraction by all actors, combined in the extraction vector E = (E1 , ..., En ). The extraction function Ei = ei (R, Ci ) = γi RCi for each actor i depends on the resource stock R, the extraction costs (investments) Ci and the extraction efficiency γi of the extraction technology employed. Resource reproduction is represented here by a logistic function r(R) = rR(1−R/R+ ) where r is the reproduction rate and R+ is the maximum carrying capacity of the ecosystem for R. Net profit Pi = pEi − Ci of actor i is the income from selling the extracted resource Ei on the market at price p, diminished by extraction cost Ci . According to the demand-supply relationship the price declines with P extraction DEj p = ∂p/∂Ej < 0. We use the standard linear function p = a − b j Ej where a is the initial price for the extracted resource and b is the slope of the demand curve. Inserting the extraction function yields profit as a function of resource stock and extraction cost of all actors: Pi = pEi − Ci = ui −

with market price p = a − b

P

j

X j

vij Cj Ci ,

(3.16)

γj RCj , ui = aγi R − 1, vij = bR2 γi γj .

The dynamics of extraction cost is adjusted by a decision rule C˙ i = fi (R, C) that describes an actor’s response function fi : R × Rn → R to the state of resources and current costs of all actors. An individual target setter would select a decision rule fi = ki (Ci∗ − Ci ), moving towards a target cost Ci∗ (R, C) with response intensity ki . Another decision rule describes an actor that adjusts cost proportionate to the gradient of profit with regard to cost change, DCi Pi , i.e. cost is increased or decreased proportionate to its impact on profit. DCi Pi = 0 is satisfied for

Ci =

ui −

P

j6=i vij Cj

2vii

=: Ci∗ ,

which will be used as target cost Ci∗ of actor i following an optimizing decision rule. Then the dynamic interaction between actors and resources is fully described by the

134

K. Eisenack et al.

system of equations (i = 1, ..., n) R˙ = rR(1 − R/R+ ) −

n X

γi RCi =: g(R, C)

(3.17)

i=1

C˙ i = ki (Ci∗ − Ci ) =: fi (R, C) n X γj RCj γi RCi − Ci =: hi (R, C) Pi = a − b

(3.18) (3.19)

j=1

The couplings are given by the following set of P inequalities. Here we use Cγ = P ¯ C as the total effective costs and γ¯ = i ωi γi as the average extraction i γi Ci = γ P efficiency, weighted by the fractions ωi = Ci /C of total costs C = i Ci . X

DR g

= r(1 − 2R/R+ ) −

DCi g

= −γi R < 0, 2 1 − aRγi /2 > 0 for R < =: RiC , = ki 3 3 aγi bR γi = −ki < 0,

DR fi DCi fi

γj Cj > 0 for C
0 for C < DR hi = γi Ci a − 2bR j

DCi hi = ui −

X j

vij Cj − vii Ci > 0 for Ci
C˜ P and resolving the quadratic equation, we find 0 < R1 < R < R2 < R+ where the two limits R1 and R2 vary around R+ /2 with a range depending on a/(2br). Thus for R in this middle range, an increase in costs first leads to a negative resource coupling to profit and then to a negative coupling to resource growth. Closer to the resource boundaries it is the opposite. We study now two viability constraints (cf. also Chapter 2): 1. Ecological viability: The rate of change of the resource stock R˙ is seen as viable if it exceeds a critical threshold R− . R˙ = 0 leads to the equilibrium ¯ = R+ (1 − Cγ /r) which corresponds to a sustainable total cost C sus = R r(1 − R/R+ )/¯ γ . Below the threshold C ext = r/¯ γ there is no positive resource ¯ equilibrium R. The resource stock is to stay in the sustainable domain R− < R ≤ R+ , where R+ is the maximum carrying capacity for the resource stock R (upper limit) and R− is a lower limit to keep a “safety distance” from total extinction. The growth rate adapts to this lower limit with the rule R˙ ≥ α(R− − R) which implies that resource growth should be positive as long as the resource stock is below the limit R− and can be negative when resource stock is above the limit. The parameter α represents the required strength of adaptation. This ecological viability condition translates into a condition for actual change R˙ = rR(1 − R/R+ ) − γ¯ R C ≥ α(R− − R). This leads to the ecologically maximal sustainable cost limit C≤

r(1 − R/R+ ) + α(1 − R− /R) =: C R γ¯

(3.20)

which is positive for R− ≤ R ≤ R+ and may become negative outside of these limits. For R → R− or α → 0, threshold C R converges to C˜ R . 2. Economic viability: The flow of net profit for actors i = 1, ..., n should exceed a threshold Pi = pEi − Ci ≥ Pi− . The left-hand side is a quadratic function in both R and extraction costs Ci of actor i, depending on the extraction costs of all other actors Cj (j 6= i). On the right hand side, the lower

136

K. Eisenack et al. profit limit Pi− = Ki − Ki− should guarantee that the available accumulated profit (stock of savings) Ki could exceed a lower limit Ki− . This could be used for investing costs Ci . We discuss here the special case that these savings are at the lower limit Ki = Ki− which leads to Pi− = 0. In this case, the equation above reduces to (a−b¯ γ CR)γi R−1 ≥ 0 which generates the economic viability condition for total cost of actor i′ s profit C≤

1 aR − 1/γi = C˜ p − =: C Pi , 2 b¯ γR bγi R2

(3.21)

i.e. costs above C Pi are too high to sustain minimal profits. Because of C Pi < C˜ P , an increasing total cost will hit the economic non-viability thresholds of all actors before passing C˜ P which implies that increasing resources reduce total profits: the smaller γi and the less efficient actor i is in extracting the resource, the tighter this constraint becomes. This confirms that the most efficient actors have the best chance to maintain profit in this competitive environment, suggesting that the most efficient actor can beat all others by further increasing extraction costs until its own limit is reached. The economic viability threshold becomes negative for R < 1/(aγi ), i.e. resource stock is too low for a given efficiency and initial market price. In different words, the efficiency γi < 1/(aR) ≡ γ ∗ of actor i is too low to make positive profit. This is the same efficiency threshold that all actors need to pass to make profit. Total joint profit of all actors becomes positive for X P = Pi = (a − b¯ γ CR)¯ γ CR − C > 0. i

which results in the upper cost limit C < (a −

1 1 ) =: C P γ¯ R b¯ γR

(3.22)

This cost limit becomes negative for c¯ = 1/(¯ γ R) > a (which defines a resource P limit R ). c¯ is the average cost for extracting one resource unit. Thus, this condition means that the unit cost exceeds the initial unit price of the resource. Joint profit is maximized for DC P = 0 which results in the optimal joint cost C∗ =

a¯ γR − 1 . 2b(¯ γ R)2

Inserting this into the profit function determines maximum profit Pmax =

(a −

1 2 γ ¯R )

2b

≥ 0.

(3.23)


137

Thus, if actors invest their joint optimal extraction costs, joint profit is never negative. However, this is generally not possible within the given boundaries, in particular for very small γ¯ R which would require very high optimal costs which is not economically sustainable. On the other hand, very high γ¯ R would be not ecologically sustainable. Finally, we examine how total effective costs Cγ evolve, assuming ki ≡ k for all i = 1, ..., n: X X C˙ γ = γi C˙ i = γi ki (Ci∗ − Ci ) i

= k where

P

i

anR − i 1/γi n+1 − kCγ = k(C¯γ − Cγ ) 2bR2 2 P anR − i 1/γi ¯ Cγ = (n + 1)bR2

(3.24)

is the overall equilibrium. For γi ≡ γ for all actors this provides the joint cost ¯ equilibrium C¯ = bγ(aγR−1)n 2 R2 (n+1) which is close to the zero-profit condition. C is negative for γR < 1/a, reaches its maximum at γR = 2/a and then declines to zero for γR → ∞. For n = 1 we have C¯ = C ∗ , for large n we have C¯ → 2C ∗ , i.e. individual optimization leads to an equilibrium far above the joint optimum. So far we assessed the case of each actor acting according to its individual profits Pi , investment strategies Ci and extraction efficiencies γi and discussed their joint impacts on the resource stock which in return affects the joint profits and investments. This situation can be assessed using viability concepts (for details cf. Chapter 2). The state space structure (see below for an example) cannot guarantee that every trajectory starting from a state which meets the ecological and the economic viability constraint will meet these requirements forever. In general, the viability kernel is a proper subset of the set described by the constraints. Thus, individual competition may lead to outcomes which are not ecologically or economically viable, or both, such that regulation and cooperation are relevant to stabilize the interaction within viable limits: • One regulation strategy seeks to make resource extraction more costly, by substracting a tax τ Ei from profit proportionate to extraction. This corresponds to replacing the initial unit cost by a − τ in all the equations above which implies that the cost thresholds and profit are reduced accordingly which allows ¯ for a higher resource equilibrium R. • As long as each actor seeks to adjust its extraction cost to maximize its own profit P Pi , then the dynamics differs from seeking to maximize joint profit P = i Pi . To act jointly, the actors would match the joint decision rule C˙ = k(C ∗ − C) by adapting their individual decision rules C˙ i = ki (Ci∗ − Ci ),

138

K. Eisenack et al.

Figure 3.12: Special case for multi-actor resource competition in the cost-resource diagram, showing the main isoclines C¯ for constant total costs and C sus for constant resource stock. The arrows indicate the direction of the dynamics.

most obviously by selecting the response strength ki properly. This requires mutual adaptation and negotiation on allocating joint cost, maximizing joint profit and distributing it to the actors according to criteria of fairness, e.g. proportionate to efficiency or efforts of actors. A detailed assessment of such regulation measures is left to the future. Rather we demonstrate the general terminology here for a specific case. Example: To specify the different threshold conditions, we use a particular parameter setting (a = 1, b = 0.005, γ¯ = 0.005, r = 0.2, R+ = 1000, R− = 100, α = 0.5, n = 6). Note that it is not assumed that all actors are identical. However, the following results can be computed for the aggregate costs C. Figure 3.12 shows the two main isoclines C sus for R˙ = 0 and C¯ for C¯˙ = 0 as well as the directions of change outside of these curves. The various threshold curves defined in this Chapter are depicted in Fig. 3.13 which shows the full complexity of the problem. Most striking is that at R = RP three of the four cost curves intersect. This implies that below this resource threshold positive profit is not feasible and that total costs C¯ (individual optimization) and C ∗ (joint optimization) are negative which implies that there is no inventive to increase costs. The ecologically viable cost threshold C R exceeds them all because the higher the resource stock the higher the admissible resource reduction and cost, as a function of R− and α. As expectable we have C R = C sus for R = R− . The diagram also shows that for individual optimization the total costs C¯ are about twice as high as for joint optimization C ∗ . We have C sus > C ∗ for R < RC and else otherwise. It is also clear that the dynamics, given by the arrows,


139

Figure 3.13: Special case for multi-actor resource competition in the cost-resource diagram, showing the thresholds for ecological viability C R , economic viability C P , change of ¯ and threshold couplings C˜ R , C˜ P , RC , profit maximizing and equilibrium costs C ∗ and C, ext sus costs C and C .

exceeds C ∗ which implies that the joint optimum is not stable against individual action (it is not a Nash equilibrium). For C < C˜ R more resources spur resource growth until this threshold is exceeded and resource growth slows down to reach C sus and resource growth stops.

3.5

Discussion

In this Chapter we have shown how to deal with complex archetypal cause-effect patterns of global change. The derivation of the patterns (syndromes) allows already an improved awareness regarding the underlying problems, but also a structuring of information about complex situations. This is a precondition for anticipation of critical developments and any type of action useful to accomplish a turnaround to sustainability. The syndrome approach is a strong learning tool in this context, which can be applied to various stakeholder communities, raising awareness that improved conceptual and methodological concepts are needed to anticipate the development of the co-evolutionary dynamics of society-nature interactions. The detailed discussion in this Chapter have made clear that such an analysis poses several challenges:

140

K. Eisenack et al. • Generality: models should provide insights for single applications, but should also apply to a broader set of cases with general features in common. They should classify and subsume different instances, because communalities between different cases are important to obtain a global overview, to classify different instances, and to be the base for transferring best practices. • Uncertainty: models have to take account of various uncertainties. Many interactions of social-ecological systems are not known quantitatively, knowledge about the processes is often limited, there are data gaps or unpredictable future influences, e.g. depending on strategic political choice. Under such conditions, the modeler cannot discriminate between alternative quantitative models, but if urgency to solve a problem is high, the analysis and the management strategies should be robust. • Quantitative and qualitative knowledge: to understand social-ecological systems, knowledge from different disciplines and with different degrees of quantification has to be integrated. • Complexity: social-ecological systems tend to be composed of many tightly coupled, non-linear subsystems and interactions which are often difficult to disentangle. If this complexity cannot completely be resolved by models, adequate core mechanisms have to be identified. • Normativity: it is necessary to classify patterns of interactions as problematic or desirable. This involves value judgments where science can only contribute. However, normative knowledge has to be considered in a transparent way and research can contribute to the assessment and development of management practices.

The exemplarily discussed syndromes show that essential progress can be achieved which leads beyond the basic syndrome analysis. This is feasible by utilizing smart mathematical methods from physics and information sciences. Applying this strategy a formal syndrome description is possible. Furthermore it might by reasonable that a general theory of complex man-nature interactions can be developed by application of the introduced concepts. Qdes allow to abstract from single cases whilst the viability concepts is suitable to test potential system developments against normative settings from policy or society (cf. Eisenack et al. 2006a for an example). This is important, since a lot of management strategies are putted into action without a systematic analysis of potential side effects or potential failure factors. In this contribution we show, how relevant policy-relevant information can be deduced for complex problems even if we have only inhomogeneous knowledge on hand. For the Overexploitation syndrome, for instance, the most prominent result is the existence of boom-and-bust cycles. These make clear the urgent need to reconsider subsidies policies. Further several irreversibilities (no-return sets) allows to identify time horizons for concrete actions - otherwise safe limits can not be achieved. Similar results


141

are obtained for the Dust-Bowl syndrome implying that environmental impacts and sustainable productivity can only guaranteed by an intelligent restructuring of regulations in industrial agriculture. For the Sahel syndrome it becomes obvious that smallholder’s pressure on marginal land can be reduced only by implementing a suitable population policy. It is further shown by an multi-actor problems can be addressed adequately by combining game theoretic approaches with several other methods, as e.g. viability concepts and qualitative modeling. Our results causes optimism that essential progress can be achieved and that therefore environmental systems analysis can surmount arbitrariness, in particular, if we are focussing on policy relevant information.

3.6

Conclusion

In this paper we introduced a mathematical and a substantial framework which allows to provide a a systematic and cross-wide syndrome analysis. Archetypes are a suitable way of representing generalizable system features by reducing complex interaction to basic mechanisms. Parallel they cover a broad range of particular systems, i.e. a large class of uncertainties is considered at the same time. Furthermore they semi-formally represent qualitative knowledge. By operationalizing them as qualitative differential equations, also non-quantitative knowledge about trends and on ordinal scales can be included. Different normative assessments can be performed on this base using viability concepts. We further indicate ways to draw conclusions for the management of social-ecological systems. We have shown how one can derive systems knowledge which lies far beyond the common practice to utilize short-term observations for long-term planning and management, although only weak systems knowledge is available. Although qualitative modeling and viability theory cannot provide exact quantitative predictions and crisp strategies, alternative development paths and feasible management options can be explored and implemented in the daily practice of decision makers (cf. syndrome related quotations in this Chapter). Thus, the presented approach is not only a tutorial exercise, it paves the road to a closed theory-based analysis of complex man-environment interactions and for further innovative research.

Acknowledgments. Many colleagues from various disciplines and institutions have helped to develop the traditional “Syndrome Approach” on which the presented work is based. We gratefully acknowledge, in particular, the contributions made by the members of the German Advisory Council on Global Change (Wbgu). This work was supported by the Federal Ministry for Education and Research under grant numbers 03F0205B and 01LG9401.

142

K. Eisenack et al.

References Anderies, J. M., M. A. Janssen, and E. Ostrom (2004). A framework to analyze the robustness of socio-ecological systems from an institutional perspective. Ecology and Society 9 (1), 18. Aubin, J.-P. (1991). Viability Theory. Boston: Birkhäuser. Barbier, E. B. (1990). The farm-level economics of soil conservation: The uplands of Java. Land Economics 66 (2), 199–211. Barreto, P., C. Souza Jr., R. Nocuer´ on, A. Anderson, and R. Salom˜ ao (2006). Human Pressure on the Brazilian Amazon Forests. Belém: World Resources Institute. Barrett, S. (1991). Optimal soil conservation and the reform of agricultural pricing policies. Journal of Development Economics 36, 167–187. Boardman, J. T. (1995). Wholes and parts – a systems approach. IEEE Journal of Systems, Man, and Cybernetics 25 (7), 1150–1161. Cassel-Gintz, M. and G. Petschel-Held (2000). GIS-based assessment of the threat to world forests by patterns of non-sustainable civilization-nature interaction. Journal of Environmental Management 59, 279–298. Charles, A. (2001). Sustainable Fishery Systems. Oxford: Blackwell. Clancy, D. J. (1997). Solving complexity and ambiguity problems within qualitative simulation. Ph. D. thesis, University of Texas at Austin. Cury, P. M., C. Mullon, S. M. Garcia, and L. J. Shannon (2004). Viability theory for an ecosystem approach to fisheries. ICES Journal of Marine Science 62, 577–584. Davis, R., H. Shrobe, and P. Szolovits (1993). What is knowledge representation? AI Magazine (Spring), 17–33. Daw, T. and T. Gray (2005). Fisheries science and sustainability in international policy: s study of failure in the European Union’s common fisheries policy. Ecological Economics 36, 223–236. Dechter, R. (2003). Constraint Processing. San Mateo: Morgan & Kaufman. Eisenack, K. (2006). Model Ensembles for Natural Resource Management: Extensions of Qualitative Differential Equations Using Graph Theory and Viability Theory. Ph. D. thesis, Free University Berlin. Eisenack, K. and J. P. Kropp (2001). Assessment of management options in marine fisheries by qualitative modelling techniques. Marine Pollution Bulletin 43 (7-12), 215–224. Eisenack, K., J. Scheffran, and J. P. Kropp (2006a). Viability analysis of management frameworks for fisheries. Environmental Modelling and Assessment 11 (1), 69–79. Eisenack, K., H. Welsch, and J. P. Kropp (2006b). A qualitative dynamical modelling approach to capital accumulation in unregulated fishery. Journal for Economic Dynamics and Control . in press, online: DOI:10.1016/j.jedc.2005.08.004. Fao (Ed.) (2003). The state of world forests 2003, Rome. Food and Agriculture Organization of the United Nations.


143

Fao (Ed.) (2004). The state of world fisheries and aquaculture 2004, Rome. FAO Fisheries Dept., Food and Agriculture Organization of the United Nations. Freire, J. and A. Garcia-Allut (2000). Socioeconomic and biological causes of management failures in European artisanal fisheries. Marine Policy 24 (5), 375–384. Go˜ ni, R. (1998). Ecosystem effects of marine fisheries. Ocean and Coastal Zone Management 40, 37–64. Gregory, J. M., K. W. Dixon, R. J. Stouffer, A. J. Weaver, E. Driesschaert, M. Eby, T. Fichefet, H. Hasumi, A. Hu, J. H. Jungclaus, I. V. Kamenkovich, A. Levermann, M. Montoya, W. Murakami, S. Nawrath, A. Oka, A. P. Sokolov, and R. B. Thorpe (2005). A model intercomparison of changes in the atlantic thermohaline circulation in response to increasing atmospheric CO2 concentration. Geophysical Research Letters 32, L12703. Grepperud, S. (1997). Poverty, land degradation and climatic uncertainty. Oxford Economic Papers 49, 586–608. Hayek, F. A. (1973). Law, Legislation, and Liberty: Rules and Order. London: Routledge. Hilborn, R. and C. J. Walters (1992). Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertaintiy. London: Chapman and Hall. Hutchings, J. A., C. Walters, and R. L. Haedrich (1997). Is scientific inquiry incompatible with government information control? Can. J. Fish. Aquat. Sci. 54, 1198–1210. Jentoft, S. (2004). Institutions in fisheries: what they are, what they do, and how they change. Marine Pollution Bulletin 28, 137–149. Kapos, V., I. Lysenko, and R. Lesslie (2000). Assessing Forest Integrity and Naturalness in Relation to Biodiversity. Cambridge: Unep-World Conservation Monitoring Centre. Kasperson, J. X., R. E. Kasperson, and B. L. Turner II (1995). Regions at risk: Comparisions of Threatened Environments. Tokyo: United Nations University Press. Kates, R. W. and V. Haarman (1992). Where the poor live: are the assumptions correct? Environment 34, 4–11. Kelso, J. A. S. (1997). Dynamic Patterns. Cambridge: MIT Press. Kropp, J. P., A. Block, F. Reusswig, K. Zickfeld, and H. J. Schellnhuber (2006a). Semiquantitative assessment of regional climate vulnerability: The North Rhine – Westphalia study. Climatic Change. in press, online: DOI:10.1007/s10584-005-9037-7. Kropp, J. P., K. Eisenack, and J. Scheffran (2006b). Marine overexploitation: A syndrome of global change. In S. Sonak (Ed.), Multiple Dimensions of Global Environmental Change, New Dehli, pp. 257–284. TERI Press. Kropp, J. P., K. Eisenack, and J. Scheffran (2004). Sustainable marine resource management: Lessons from viability analysis. In C. Path-Wostl, S. Schmidt, A. E. Rizzoli, and A. J. Jakeman (Eds.), Complexity and Integrated Resources Management, Transactions of the 2nd Biennial Meeting of the International Environmental Modelling and Software Society, Manno, Switzerland. International Environmental Modelling and Software Society. Kropp, J. P., M. K. B. L¨ udeke, and F. Reusswig (2001). Global analysis and distribution of unbalanced urbanization processes: The Favela Syndrome. GAIA 10 (2), 109–120.

144

K. Eisenack et al.

Kropp, J. P. and H. J. Schellnhuber (2006). Prototyping broad-scale climate and ecosystem classes by means of self-organizing maps. In P. Agarwal and A. Skupin (Eds.), Self-Organising Maps: Applications in Geographic Information Sciences, New York. Wiley & Sons. Kropp, J. P., K. Zickfeld, and K. Eisenack (2002). Assessment and management of critical events: The breakdown of marine fisheries and the north atlantic thermohaline circulation. In A. Bunde, J. P. Kropp, and H. J. Schellnhuber (Eds.), The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes, Berlin, pp. 192–216. Springer-Verlag. Kuipers, B. (1994). Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. Artificial Intelligence. Cambridge: MIT Press. Lambin, E. and B. Mertens (1997). Human ecology remote sensing and spatial modelling of land-use changes in Afrika. LUCC Newsletter (2), 6–9. Leonhard, H. J. (1989). Environment and the Poor: Development Strategies for a Common Agenda. Oxford: Transaction Books. L¨ udeke, M., O. Moldenhauer, and G. Petschel-Held (1999). Rural poverty driven soil degradation under climate change. Environmental Modelling and Assessment 4, 315– 326. L¨ udeke, M. K. B., G. Petschel-Held, and H. J. Schellnhuber (2004). Syndromes of global change: The first panoramic view. GAIA 13 (1), 42–49. L¨ udeke, M. K. B. and F. Reusswig (1999). The Dust-Bowl Syndrome in Germany. A feasibility study on the formulation of system-analytical indicators for integrated strategies aiming at sustainable development in Germany, exemplified by the problems of intensive agriculture. PIK/HGF Report #100, Potsdam. Mea (2005). Ecosystems and Human Well-Being: Synthesis Report. Washington: Island Press. online: http://www.milleniumassessment.org/products.aspx. Mullon, C., P. Cury, and L. Shannon (2004). Viability model of trophic interaction in marine ecosystems. Natural Resources Modeling 17, 27–58. Munro, G. R. (1999). The economics of overcapitalisation and fishery resource management: A review. In A. Hatcher and K. Robinson (Eds.), Overcapacity, Overcapitalisation and Subsidies in European Fisheries, Portsmouth, pp. 7–23. University of Portsmouth: cemare. Nakićenović, N., J. Alcamo, G. Davis, B. de Vries, J. Fenhann, S. Gaffin, K. Gregory, A. Gr¨ ubler, T. Y. Jung, T. K. an W. K. L. Rovere, L. Michaelis, S. Mori, T. Morita, W. Pepper, H. Pichter, L. Price, K. Riahi, A. Roehrl, H. Rogner, A. Sankowski, M. Schlesinger, P. Shukla, S. Smith, R. Swart, S. van Rooijen, N. Victor, and Z. Dadi (2000). IPCC Special Report on Emission Scenarios. Cambridge: Cambridge University Press. Nepstad, D. C., A. Verissimo, A. Alencar, C. Nobre, E. Lima, P. Lefebvre, P. Schlesinger, C. Potter, P. Moutinho, E. Mendoza, M. Cochrane, and V. Brooks (1999). Large-scale impoverishment of Amazonian forest by logging and fire. Nature 398, 505–508. Pauly, D., V. Christensen, S. Guénette, T. J. Pitcher, R. Sumaila, C. J. Walters, R. Watson, and D. Zeller (2002). Towards sustainability in world fisheries. Nature 418, 689– 695.


145

Petersen, E. H. (2002). Economic policy, institutions, and fisheries development in the pacific. Marine Policy 26, 315–324. Petschel-Held, G., A. Block, M. Cassel-Gintz, J. P. Kropp, M. K. B. L¨ udeke, O. Moldenhauer, F. Reusswig, and H. J. Schellnhuber (1999). Syndromes of Global Change: A qualitative modeling approach to support environmental management. Environmental Modeling and Assessment 4 (4), 295–314. Polya, G. (1954). Mathematics of Plausible Reasoning: I Introduction and Analogy in Mathematics, II Patterns of Plausible Interference. Princeton: Princeton University Press. Potter, B. (2002). Predatory politics: group interests and managment of the commons. Environmental Politics 11 (2), 73–94. Power, T. (1996). Lost landscapes and failed economies: The search for a value of place. Washington D.C.: Island Press. Richardson, G. P. (1986). Problems with causal-loop diagrams. System Dynamics Review 2, 158–170. Scheffran, J. (2000). The dynamic interaction between economy and ecology - cooperation, stability and sustainability for a dynamic-game model of resource conflicts. Mathematics and Computers in Simulation 53, 371–380. Schellnhuber, H. J. (1998). Earth System Analysis: The Scope of the Challenge. In H. J. Schellnhuber and V. Wenzel (Eds.), Earth System Analysis: Integrating Science for Sustainability, Berlin. Springer-Verlag. 3 - 195. Schellnhuber, H. J., A. Block, M. Cassel-Gintz, J. P. Kropp, G. Lammel, W. Lass, R. Lienenkamp, C. Loose, M. K. B. L¨ udeke, O. Moldenhauer, G. Petschel-Held, M. Pl¨ ochl, and F. Reusswig (1997). Syndromes of Global Change. GAIA 6 (1), 19–34. Schellnhuber, H. J. and J. P. Kropp (1998). Geocybernetics: Controlling a complex dynamical system under uncertainty. Naturwissenschaften 85 (9), 411–425. Schellnhuber, H. J., M. K. B. L¨ udeke, and G. Petschel-Held (2002). The syndromes approach to scaling: Describing global change on an intermediate functional scale. Integrated Assessment 3 (2-3), 201–219. Smith, R. J., R. D. Muir, M. J. Walpole, A. Balmford, and N. Leader-Williams (2003). Governance and the loss of biodiversity. Nature 426, 67–69. Sru (1998). Umweltschutz: Erreichtes sichern - Neue Wege gehen. Stuttgart: MetzlerPoeschel. Der Rat von Sachverständigen f¨ ur Umweltfragen (SRU) Umweltgutachten 1998. Stainforth, D. A., T. Aina, C. Christensen, M. Collins, N. Faull, D. J. Frame, J. Kettleborough, S. Knight, A. Martin, J. Murphy, C. Piani, D. Sexton, L. A. Smith, R. A. Spicer, A. J. Thorpe, and M. R. Allen (2005). Uncertainty in predictions of climate response to rising levels of greenhouse gases. Nature 433, 403–406. Swart, R. J., P. Raskin, and J. Robinson (2004). The problem of the future: sustainability science and scenario analysis. Global Environmental Change 14, 137–146. Tsang, E. (1993). Foundations of Constraint Satisfaction. London: Academic Press.

146

K. Eisenack et al.

Turner II, B. L., W. C. C. and R. W. Kates, J. F. Richards, J. Tuchman-Mathews, and W. B. Meyer (Eds.) (1990). The Earth as Transformed by Human Action: Global and Regional Changes in the Biosphere over the Past 300 Years. Cambridge: Cambridge University Press. Un (1995). The United Nations Agreement for the Implementation of the Provisions of the United Nations Convention on the Law of the Sea of December 1982 relating to the Conservation and Management of Straddling Fish Stocks and Highly Migratory Fish Stocks (Agreement on High Seas Fishing). New York: United Nations Department of Public Information. V´ amos, T. (1995). A strategy of knowledge representation for uncertain problems: modeling domain expert knowledge with patterns. IEEE Transactions on Systems, Man, and Cybernetics 25 (10), 1365–1370. Wbgu (Ed.) (1994). World in Transition: Basic Structure of Global People-Environment Interactions, Bochum. German Advisory Council on Global Change: Economica Verlag. Annual Report 1993. Wbgu (1995). World in Transition: The Threat to Soils. Bonn: Economica. Annual Report German Advisory Council on Global Change to the Federal Government (WBGU). Wbgu (2000). World in Transition: Strategies for Managing Environmental Risks. Berlin: Springer-Verlag. Annual Report German Advisory Council on Global Change to the Federal Government. Wickham, E. (2003). Dead Fish and Fat Cats: A No-Nonsense Journey through our dysfunctional Fishing Industry. Vancouver: Granville Island Publ. Worster, D. Dust Bowl: D¨ urre und Winderosion im amerikanischen s¨ udwesten. In R. P. Sieferle (Ed.), Fortschritte der Naturzerst¨ orung, Frankfurt. Suhrkamp.

CHAPTER 4

Analyzing a Sustainability Indicator by Means of Qualitative Kernels ´lez, Francisco Ruiz, Cecilio Angulo, Luis Gonza `, Francisco Velasco, and Nu ´ria Agell Andreu Catala

4.1

Introduction

At the beginning of this new century, the environment and its sustainability have become increasingly vital issues for our society, above all as a means of enhancing quality of life. It was in this context that a meeting of European municipalities held in Hannover in February 2000 agreed on indicators for the measurement of sustainability and the first indicator selected was the satisfaction of citizens with their local community. The social indicator movement had begun in the 1950s with objective indicators like health, educational level, per capita income and life expectancy. At the time, it was felt that an improvement in objective circumstances would affect some aspects of quality of life. However, the limitations of the explanatory power of objective conditions soon became evident, and this led in the following decade to a second line of research, based on the use of more direct subjective indicators (Andrews and Withey 1976; Bradburn 1969; Michalos 1992), more closely linked to the individual whose quality of life was being assessed. Instead of inferring welfare from group characteristics, the individual was questioned directly regarding his or her level of well-being. Throughout 2002, the research group Grec1 , formed mainly by researchers from the Universitat Politècnica de Catalunya and Esade-Universitat Ramon Llull, in Spain, was engaged in a study commissioned by the Town Council of Vilanova i la Geltr´ u, a town with over 55,000 inhabitants on the subjective indicator “citizen satisfaction”. The work was commissioned for two reasons. Firstly, the independence of the public institution itself gave credibility to the results obtained from the reference questionnaires. Secondly, the research group concerned had wide experience in processing qualitative information. An initial descriptive study, successfully defended and presented to the Town Council, permitted a considerable volume of data to be collected. This information was basically qualitative, but with a high informative content on the perception citizens had of their town, Vilanova i la Geltr´ u. At this juncture, the Grec group and a working party from the Universidad de Sevilla, members of the Arca2 network, put forward the need to analyze this highly uncertain and subjective information using new artificial in1 2

Knowledge Engineering Research Group, http://www.upc.es/web/GREC Qualitative Reasoning Automation and Applications, http://www.lsi.us.es/arca

148

C. Angulo et al.

telligence techniques based on collective and interval reasoning. For this reason, main results in the Chapter are related to the clustering of data and the distances between clusters. This topic is of high relevance for sustainability issues, especially when data are heterogeneous (e.g. categorical, ordinal and numerical). New methods for identifying classes of similar cases are valuable to characterize patterns of global environmental change and to develop sustainability indicators. This document presents two totally new inference techniques for the fields of interval and qualitative analysis, oriented at sustainability data: kernels on a discrete structure (orders of magnitude) and interval kernels. These techniques are based on kernel methods and statistical learning methodology, both shown to be very effective in machine learning theory. Moreover, they make it possible to work on any data space, not necessarily numerical ones: bio-informatics, string-type kernels, etc. In order to apply these methodologies, numerical variables are first automatically grouped in intervals. Next, either qualitative variables are converted to interval values, or intervals obtained from numerical variables are converted to qualitative variables. Finally, the main task is performed: similarities in data are found by using kernels on either orders of magnitude or intervals. Behavior of both approaches is studied on several examples, and an initial application to the sustainability data of the Town Council of Vilanova i la Geltr´ u is performed. There is then a feasibility analysis by a soft-computing tool which allows the degree of citizen satisfaction to be qualitatively assessed.

4.2

Design of the Survey

The study on the global subjective indicator “citizen satisfaction”, carried out from December 2001 to November 2002, was divided into four main parts: design, fieldwork, information extraction and exposition of results. At the design stage it was decided to conduct stratified sampling by controlling the representativeness of people living in the town. Parts one and two were performed in three phases. The first entailed random sampling on phone numbers and surveys were administered to people answering the phone. When the first round of answers had been stored, the representativeness of the sample was analyzed. The second round of phone surveys had an adapted schedule and specific people search. Finally, the third phase was performed on the street looking for specific individuals. The questionnaire was based on a standard form provided by the Town Council of Vilanova i la Geltr´ u, a reference document for all European cities signing the Hannover Agreement in February 2000 that can be used to compare sustainability indicator 1 (citizen satisfaction). Only minor changes and additions to the questionnaire were allowed for the Town Council to achieve some qualitative subjective information. Changes in the order of the questions were also allowed. The “general data” section in the questionnaire was originally composed of the socio-demographic variables “gender”, “age”, “profession” and “per capita income”

Analyzing a Sustainability Indicator by Means of Qualitative Kernels

149

information. These are the only questions with objective information in the standard European questionnaire. In addition, the questionnaire provided a number of “subjective” questions with five possible linguistic label answers, expressing degree of satisfaction with the required item. Here, there were 10 questions about basic urban services, namely “natural resources protection”, “carefulness of public spaces”, “employment”, “cultural and leisure events”, “health services”, “educational services”, “public transportation”, “local governance” (planning and decision making processes), “public safety services”, and “housing”. Finally, an 11th question was added, a more synthetic and subjective issue dealing with the general convenience of the city as a good place to live and work in.

4.3

Artificial Intelligence for the Qualitative Treatment

Standard statistical treatment was used in the work commissioned by the Town Council. However, the researchers involved observed that there were possibilities of a direct qualitative treatment of information from an artificial intelligence viewpoint. With this in mind, an in-depth analysis of the work done was performed, and two aspects were selected as possible items for improvement by taking a direct qualitative perspective. The first, involving quantitative variables such as “age”, refers to selecting some automatic criteria to discretize to obtain interval values. The second, refers to a measure of similarity within qualitative variables and discretized quantitative variables, to be defined by using some ordered associate variable. The first item in this work has been analyzed using the Cum method (for details cf. Cochran 1992). In the study, the range of variables was initially split into several groups according to people expertise; next, a random selection of people in the municipality to be surveyed was performed. After all surveys have been collected, it had to be confirmed that information about initially-defined groups was well distributed. At this point, the Cum method was employed. The “age” variable was used to illustrate the validity of the chosen ranges for the variables in the survey. The age of the surveyed people varied from 16 to 92, and they had to be grouped into categories. The number of categories was limited to 4 or 5, since no relevant information was introduced when more categories were selected. The well-known statistical method Cum was used to build the categories for the age variable. This method makes a clustering of the initial values, minimizing the average of the standard deviations, constrained to all class marks being equally representative. The process is defined based on statistical sampling techniques, and a complete study can be found in Cochran (1992) and González and Gavilán (2001). When the Cum rule was applied to our data to obtain 4 categories in the age variable, intervals like those shown in Tab. 4.1 were obtained, very similar to the intervals initially defined by the experts. The natural statistical option was to consider equal-sized intervals, since this is easier to deal with. However, such an

150

C. Angulo et al.

approach is less illustrative of data frequencies. Using, a posteriori, our selection, only one category would have a different size, and the representativeness of the data was respected. Although more precise work should be carried out, the present Chapter is not mainly concerned with this aspect. Age

# Samples

% Sample

% Town

16–31

279

26.8

25.2

32–45

298

28.6

28.5

46–63

255

24.5

26.7

> 63

209

20.1

19.5

Total

1041

100

99.9

Table 4.1: Ranges obtained with the Cum method to build 4 categories in the age variable.

As regards the second item, the questionnaire, as usual, offered people 4 options in answering the questions: from “very unsatisfied” to “very satisfied”. Statistically, the numbers 1 to 4 replaced these answers. This allows their use as associate variables to assign a measure of similarity to the qualitative objective variable. What is the similarity between cab drivers and waiters? Are they more similar than cab drivers and airplane pilots? In order to respond to these questions auxiliary numerical variables must be designed. Salary, for instance, is one auxiliary variable that could be considered, but sedentary time at work could also be valid. At this point two conditioning factors must be pointed out. Firstly, that the main numerical features considered as auxiliary variables must be designed by the user or experts working on the particular problem. And secondly, that the features considered as auxiliary variables to define similarity between items vary according to the type of classification desired. Continuing with the example above: if salary is considered decisive in our classification, then it can be taken as an auxiliary variable. But this variable includes a wide range of values in the case of waiters, and the same also applies in the other cases. It would therefore be useful to assign an interval consisting of normal salary levels. Based on this idea, a specific treatment of intervals as qualitative measures must be developed. This is therefore the thesis of our approach: to deal with qualitative data in the form of intervals by translating qualitative variables into auxiliary interval variables, thus capturing the features relevant for our study. In the following Section, two approaches using kernel methods are presented employing a direct qualitative perspective. Kernel methods have been successfully used to treat bio-informatics information and in a text mining context. The first approach deals with interval distances, and the second uses orders of magnitude to deal with the qualitative information. In each of these two approaches a suitable distance, and therefore a measure of similarity, has been defined. This measure of similarity on intervals or order of magnitudes will allow the assignment of a measure of similarity on qualitative variables.


4.4

151

Interval Kernel for Qualitative Processing

Uncertainty in measurement of data from any system can arise from several sources: noise in the sensors, imprecision in the associate variable, or an indivisible treatment of the values range. A number of works and related research areas deal with the basic elements of this uncertainty from different perspectives. Taking as key definitions those of González et al. (2005), González (2002) and Skhölkopf (2000) of a similarity measure between probabilistic events, it is possible to build a distance between intervals by considering these events from a certain σ-algebra. However, to do so we need a perfectly-defined random variable, in order to calculate probabilities that allow distances to be determined. Hence the main problem with applying this approach directly is knowing the distribution of the random variable. There is another drawback to overcome: original distance between random events can consider relative size of the intervals in probabilistic terms, but its relative position on the real line is not considered. It is not possible to design a distance with this property into the events space, so the new interval distance must be directly defined on R. In order to provide a distance between intervals, a specific building design very common in statistical learning theory framework is used: the definition of kernel functions. For a general review of this theory see texts in Vapnik (1998) and Cristianini and Shawe-Taylor (2000). In essence, the objective when a kernel is built is to ensure the existence of a mapping φ defined from the original working set – perhaps not provided with any mathematical structure – to a vector space provided with a scalar product named feature space F. From this mapping φ, a kernel function k(·, ·) is defined, on pairs of elements from the original working set such as the dot product of their transformed pairs into the feature space, k(·, ·) = hφ(·), φ(·)iF . (4.1) where h·, ·iF denotes a scalar product on F. A similarity can be defined as a measure of correspondence between the intervals to be studied, that is, a function that, given two intervals I1 and I2 , returns a real number characterizing their similarity. So, a kernel function k(·, ·) allows similarities to be established between the original elements by using their representation in F. Therefore when the similarity is appropriately chosen, it is possible to define a distance between the original points. Mapping φ must be capable of capturing the main features of the elements in the original space to be considered for the similarity measure and its possible associate distance. Following the research from the intervals approach presented in González et al. (2004), the family of all the open intervals of finite dimension in the real line3 , 3 By default, work will be developed on open intervals, however the study can be translated in a natural way to closed intervals.

152 denoted by I, is

C. Angulo et al.

I = (c − r, c + r) ⊂ R : c ∈ R, r ∈ R+

where the Borel notation on the intervals is used, since the use of radius and centers of intervals is more amenable than using the ends of intervals. From the kernel machine research perspective, the natural choice for the map φ is that associating each interval with a vector in R2 . So, a mapping φ : I → R2 is considered as follows c c p11 p12 φ(I) = P = r r p21 p22 provided that P is a non singular 2 × 2 matrix. Note 4.4.1 Each interval defines a point in the plane. When P is the identity matrix, then the image of the application φ is the region (c, r) ∈ R2 : r > 0 for the Borel notation. The map φ, allows us to define a similarity kernel function and a distance between intervals as follows: Definition 4.4.2 Given two intervals I1 = (c1 −r1 , c1 +r1 ) and I2 = (c2 −r2 , c2 +r2 ). A kernel k and a distance d between intervals are defined as follows: c2 k(I1 , I2 ) = c1 r1 S r2 d2 (I

1 , I2 )

=

∆c ∆r

S

∆c ∆r

provided that P is a non singular 2 × 2 matrix assuring the injective mapping φ, S = P t P (a symmetrical and positive defined matrix), ∆c = c2 −c1 and ∆r = r2 −r1 . Matrix S is called the weight-matrix. Note 4.4.3 It is obvious that: k(I1 , I2 ) = hφ(I1 ), φ(I2 )i and application d(·, ·) is certainly a distance between intervals because map φ is injective (φ(I1 ) = φ(I2 ) ⇔ I1 = I2 for all I1 , I2 ∈ I) since P is a non-singular matrix; and d(I1 , I2 ) is defined as the quadratic norm of the vector φ(I1 ) − φ(I2 ) in the real plane, i.e. the distance defined from the kernel is a ℓ2 distance. Note 4.4.4 Through matrix P , weight assigned to the position of the intervals, c, and to their size, r, can be controlled.


153

1 0 , considering intervals I1 = (0, 2), I2 = 0 1 2 2 (1, 3) and I3= (0, 4), then it follows d (I1 , I2 ) = 1 and d (I1 , I3 ) = 2, whereas for 1 0 S = P tP = , d2 (I1 , I2 ) = 1 and d2 (I1 , I3 ) = 4, hence weighting allows that 0 3 ordination be changed in the set of the intervals I. Example 4.4.5 For S = P t P =

4.5

Kernel for Qualitative Orders of Magnitude Spaces

Within the artificial intelligence framework, a key factor in situations where one has to obtain conclusions from imprecise data is being able to use variables described via orders of magnitude. One of the goals of qualitative reasoning is precisely to tackle problems in such a way that the principle of relevance is preserved (Forbus 1988), and each variable involved in a real problem is valued with the required level of precision. If only a finite set of symbols or qualitative labels is considered, for example a finite partition of the real line, then a model is required to build a certain algebra or structure. Here we consider the absolute orders of magnitude model (Piera 1995). The absolute orders of magnitude model of granularity n, OM (n), is defined from a symmetric partition of the real line in 2n + 1 classes in the form Ni = [−ai , −ai−1 ), 0 = {0}, and Pi = (ai−1 , ai ], ai being positive numbers. Each label is named basic element and a set S1 is built, S1 = {Nn , Nn−1 , . . . , N1 , 0, P1 , . . . , Pn−1 , Pn } .

(4.2)

Once the partition that defines S1 is fixed, the quantity space S is designed by labels in the form I = [X, Y ] ∀ X, Y ∈ S1 , with X < Y , i.e. x < y ∀ x ∈ X and y ∈ Y . An order relation “to be more precise than” ≤P is defined in S. Definition 4.5.1 Given X, Y ∈ S, X is more precise than Y , (X ≤P Y ) if X ⊆ Y . From this relation the concept of basis of a qualitative label is defined. Definition 4.5.2 ∀X ∈ S − {0}, the basis of X is the set BX = {B ∈ S1 − {0} : B ≤P X}. Definition 4.5.3 Elements I, J ∈ S are q-equals, I ≈ J, if they have any common basic element. The pair (S, ≈) is called a qualitative space of absolute orders of magnitude, and taking into account that it has 2n + 1 basic elements, it said that (S, ≈) has granularity n. Finally, in order to transform numerical values into qualitative labels, it is useful to consider the qualitative expression of a real set A, denoted by [A] and that is defined by the most precise element of S that contains A.

154

4.5.1

C. Angulo et al.

An Explicit Feature Mapping from (OM(n))k

Following the method used in Cristianini and Shawe-Taylor (2000) to obtain a kernel over a discrete space, and in particular to define a kernel over the space S k , a mapping φ will be explicitly defined from the quantity space to a feature space F. Hence, the kernel will be obtained from this mapping and the inner product in F. Definition 4.5.4 Given R ∈ S1 , the map ΨR : S → S, such that ΨR (I) = [I ∪ R] = IR

(4.3)

is called an R-expansion. Definition 4.5.5 The map remoteness with respect to R, aR : S → N, is defined as aR (I) = CardS ([IR \I]) (4.4)

where CardS (·) is the number of basic elements in the R-expansion IR not included in I. The explicit feature mapping to obtain the kernel can be defined in the following way, φ : Sk → F (4.5) φ(I) = (φR (I))R∈S1 −{0} = (φNn (I) , . . . , φPn (I)) where φR associates any basic element R ∈ S1 −{0} on the space S k with an element in [0, 1]k in the form, (4.6) φR (I) = φR (I1 , · · · , Ik ) = λaR (I1 ) , · · · , λaR (Ik )

with λ ∈ (0, 1). The decay factor is used in each component to weight the remoteness between two elements in S. The map φR transforms each element in S k into an element in [0, 1]k , which reflects the remoteness of I’s components w.r.t. the basic element R. In this way, the components in I that are qualitatively equal to R take value 1 in the corresponding component, and less than 1 if they are not.

4.5.2

Building a Kernel on an Order of Magnitude Space

Once the explicit mapping φ has been defined on the space S k indicating the similarity of the I’s components w.r.t. the basic elements, the kernel is defined via the inner product in F. The explicit kernel expression is as follows, X aR (J1 ) aR (Jk ) aR (I1 ) aR (Ik ) i , λ ,··· ,λ k (I, J) = h λ ,··· ,λ R∈S1 −{0}

=

X

k X

R∈S1 −{0} i=1

(4.7)

λ

aR (Ii )+aR (Ji )


155

Once a kernel on an order of magnitude space is defined, a distance can be introduced as considered in the interval case: Definition 4.5.6 Let I and J be two elements in S k . The distance between the two labels is defined as: p k (I, I) + k (J, J) − 2k (I, J) d(I, J) = (4.8) p = hφ(J) − φ(I), φ(J) − φ(I)i

It can be easily shown that all the distance properties are fulfilled due to the injectivity of the φ map.

4.6

Application of the Kernel Distances

The sustainability indicator 1 surveyed was given a traditional statistical treatment form for delivery to the Town Council. However it will be demonstrated that qualitative treatment would improve and make easier the analysis of this data. In the two earlier Sections, two different approaches have been described for treating the qualitative information from both interval and order of magnitude perspectives. Now, a further application will be used to show that this innovative approach is useful and powerful for extracting information cached in the data and expressing it in a very effective manner for its later expert analysis. Let us consider x a variable taking values x1 , x2 , . . . , xk as not necessarily numerical. In order to understand the similarity between these qualitative values, one (or more than one) numerical auxiliary variable will be considered. We will first describe the case of a single auxiliary variable y. For each value xj of the x variable, the following map is defined: φ(xj ) = (µj , σj ), which represents the mean and standard deviation respectively of the auxiliary variable y measured over a set of items for which the variable x takes the value xj . This map associates to each value of x an interval centered on µj and radius σj . This interval will represent the range of values where it is more likely to find the variable auxiliary y when the value of x variable is xj . On the one hand, the above application allows us to associate to each values of variable x an interval, with which we can compute the interval distances defined in previous Sections. On the other hand, by considering a proper discretization of the intervals, it is possible to apply the distance on the orders of magnitude defined previously. This process can be easily extended to L auxiliary qualitative variables by φ(xj ) = (µ1j , σ1j , . . . , µLj , σLj )

(4.9)

where µij and σij are, respectively, the mean and standard deviation for the auxiliary variable i measured over the set of individuals for which its variable x is xj . For

156

C. Angulo et al. Age

1

2

3

4

µi

σi

16–29 30–44 45–64 > 64

0.004 0.007 0.003 0.010

0.051 0.027 0.024 0.016

0.651 0.537 0.479 0.335

0.294 0.429 0.493 0.639

3.235 3.388 3.462 3.602

0.553 0.577 0.564 0.578

Table 4.2: Table of frequencies, means and standard deviations for the auxiliary variable associated with the age of the people surveyed.

this reason it is going to be associated to each variable value x, L intervals, or a hyper-rectangle in the RL space. The dimension of the features space will then be 2L.

4.6.1

Application of the Interval Distance

As an applied example of the methodology given above, let us analyze the influence of the age variable own satisfaction. In this case, the variable x (“age”) is considered to be a qualitative variable defined by means of four labels: 16-29, 30-44, 45-64 and >64, as initially defined by the experts and used in the study. Note that intervals are almost the same, but not exactly the same as best representative intervals according to the Cum method. Note that, although the values taken by the age variable are numerical, this is considered qualitative because the numerical values do not have any significance with respect to degree of satisfaction. Town satisfaction is considered through the question “What is your general satisfaction with the town?” Answers to the question, considered as a variable, have four qualitative values: 1=“very unsatisfied”,. . . , 4=“very satisfied”. That is an ordinal description of these qualitative values, and therefore it can be assimilated to a numerical ordination, creating the auxiliary numerical variable y. Frequencies, means and standard deviations of the auxiliary variable are shown in Tab. 4.2 for each age group. Looking at the frequencies diagram (Fig. 4.1), the differences between the opinion of the citizens of Vilanova about their town according to their age can be appreciated. Table 4.2 and Fig. 4.1 are significant for an expert. However, it is not clear how to extract this similarity or dissimilarity from the given diagram. When the intervals approach is employed, the treatment is more Age

16–29

30–44

45–64

> 64

16–29

0

–

–

–

30–44

0.154

0

–

–

45–64

0.227

0.074

0

–

> 64

0.368

0.214

0.141

0

Table 4.3: Interval distance according to the categories in age variable of the asked people.


157

Figure 4.1: Polygons of frequencies for the categories of the age variable. Legend: X-label represents the values for the auxiliary variable “satisfaction level”. Y-label represents the relative frequencies for each “age” group and “satisfaction level” value.

amenable. Figure 4.2 displays the associate intervals as points in the feature space where the similarities can easily be seen. Finally, the numerical value of the interval distance between age categories is shown in the Tab. 4.3. Some conclusions can be extracted. Categories are similar, in general, when their representations in feature space are closer. Categories 16-29 and 45-64 are the most similar. So middle-aged and old people have a similar opinion about the town. In any case, it is interesting to note that the satisfaction improves with age. A similar treatment has been carried out where instead of just one auxiliary variable, a set of 10 variables is considered using the answer to partial satisfaction Age

16–29

30–44

45–64

> 64

16–29 30–44 45–64 > 64

0 0.439 0.538 0.916

– 0 0.338 0.640

– – 0 0.595

– – – 0

Table 4.4: Interval distance according to the categories in age variable of the asked people considering 10 auxiliary variables.

158

C. Angulo et al.

Figure 4.2: Intervals associated as points in the feature space. Legend: X-label represents interval center of the auxiliary variable “satisfaction level”, for each interval in “age”. Ylabel represents the radius of the same interval.

questions: satisfaction about the natural environment, about the built environment, about the health service, etc. The feature space in this case is R20 , and the distances are considered within this. Mean and standard deviation of each of the 10 variables are shown in Tab. 4.5 for each age group. By using the map where a point on R20 is associated to each age group and relevant kernel, the distances shown in Tab. 4.4 are obtained among the age groups. Although we have now considered more variables, we have found that relative distances between groups are similar to the previous case.

4.6.2

Application of the Absolute OM Distance

Let us now consider the discretization of the interval (1, 4) in six subintervals with length 0.5. They are labeled as N3 , N2 , N1 , P1 , P2 , P3 . Considering the ten variables described in the previous Section directly, Tab. 4.6 shows the answers for each question and age group. By using the process explained in Section 4.5.2, Tab. 4.7, shows the distances between age groups in function of the decay factor λ. In order to be able to compare the distances obtained by means of this method with those obtained in the previous Section, Tab. 4.8 shows these distances for λ=0.5

16–29 30–44 45–64 >64

µ σ µ σ µ σ µ σ

it 1

it 2

it 3

it 4

it 5

it 6

it 7

it 8

it 9

it 10

2.657 0.644 2.652 0.686 2.608 0.689 2.894 0.564

2.624 0.594 2.541 0.608 2.581 0.672 2.625 0.703

2.273 0.734 2.425 0.772 2.332 0.713 2.337 0.741

2.341 0.739 2.624 0.645 2.715 0.644 2.988 0.524

2.641 0.724 2.653 0.710 2.577 0.718 2.809 0.624

2.894 0.519 2.842 0.616 2.929 0.478 2.954 0.534

2.644 0.761 2.678 0.715 2.736 0.721 2.748 0.731

2.630 0.618 2.672 0.637 2.661 0.614 2.907 0.519

2.610 0.649 2.522 0.680 2.367 0.751 2.351 0.695

2.053 0.766 2.262 0.787 2.210 0.654 2.348 0.735

Table 4.5: Table of frequencies, means and standard deviations for the 10 auxiliary variables corresponding to partial satisfaction questions associated with the age of the people surveyed.


Age

159

160

C. Angulo et al.

item 1 item 2 item 3 item 4 item 5 item 6 item 7 item 8 item 9 item 10

16–29 [N1 , P3 ] [N1 , P1 ] [N2 , P1 ] [N2 , P1 ] [N1 , P2 ] [P1 , P2 ] [N1 , P2 ] [N1 , P1 ] [N1 , P2 ] [N2 , P1 ]

30–44 [N1 , P3 ] [N1 , P1 ] [N2 , P1 ] [N1 , P2 ] [N1 , P2 ] [N1 , P2 ] [N1 , P2 ] [N1 , P2 ] [N1 , P1 ] [N2 , P1 ]

45–64 [N1 , P3 ] [N1 , P1 ] [N2 , P1 ] [N1 , P2 ] [N1 , P2 ] [P1 , P2 ] [N1 , P2 ] [N1 , P2 ] [N2 , P1 ] [N2 , P1 ]

>64 [P1 , P3 ] [N1 , P2 ] [N2 , P1 ] [P1 , P2 ] [N1 , P2 ] [P1 , P2 ] [N1 , P2 ] [P1 , P2 ] [N2 , P1 ] [N2 , P1 ]

Table 4.6: Table of elements of S associated with each group age and item.

4.7

Conclusion

Many sustainability indicators, for example the satisfaction of citizens with their local community, are measured through surveys that include subjective questions. Comparison between answers from different individuals is therefore difficult. Moreover, clusters or groups must be designed so that the information is collected in a way that experts can deal with. Standard statistical treatment is a good starting point for dealing with information. However, the uncertainty associated with the answers and the qualitative design of categories for variables cannot be completely captured using this treatment. Artificial intelligence techniques from the qualitative reasoning and kernel

30–44 45–64 > 64 45–64

√ √

16–29 10λ2

5 − 10λ + − 10λ3 + 6λ4 − 2λ5 + λ6 √ 5 − 10λ + 10λ2 − 10λ3 + 5λ4

10 − 18λ + 16λ2 − 16λ3 + 9λ4 − 4λ5 + 3λ6 √

30–44

2 − 4λ + 4λ2 − 4λ3 + 3λ4 − 2λ5 + λ6

> 64

√

> 64

45–64 √ 2 5 − 10λ + 10λ − 10λ3 + 8λ4 − 6λ5 + 3λ6

7 − 14λ + 14λ2 − 14λ3 + 11λ4 − 8λ5 + 4λ6

Table 4.7: OM distances according to the categories in age variable of the asked people considering 10 auxiliary variables.

Analyzing a Sustainability Indicator by Means of Qualitative Kernels Age

16–29

30–44

45–64

> 64

16–29 30–44 45–64 > 64

0 1.256 1.250 1.867

– 0 0.800 1.500

– – 0 1.269

– – – 0

161

Table 4.8: OM distances according to the categories in age variable of the asked people considering 10 auxiliary variables and λ=0.5.

methods research areas used to manage the information from a qualitative or interval perspective are therefore explored in this work. The first of these, the kernel method approach, allows definition of a similarity measure, definition of a distance, and comparison of behaviors of the different groups according to a specified variable. The similarity measure obtained is very illustrative for later expert analysis. Work must be extended to deal with more than one variable by weighting each variables’s influence on the similarity measure. The relevance of the variables as features to be used – or the weighting assigned to the variables – when information is classified or clustered must be a compromise between the Ai technique used and expert advice. The second innovative approach also uses the kernel strategy. However, the input space is not an interval space, but an orders of magnitude space. An appropriate feature mapping is defined to measure the similarity of each element with the basic elements, and these are used to compare into the quantity space. Further work must be done on this line of research, in particular on reducing the cost in time for calculation of the kernel evaluation. Acknowledgments. This work has been partially supported by the grants ACC944-SEJ-2002 from the Junta de Andaluc´ıa (Spain), TIC2002-04371-C02-02 (Merito) from the Ministerio de Ciencia y Tecnolog´ıa of Spain and the Town Council of Vilanova i la Geltr´ u.

References Andrews, F. M. and S. B. Withey (1976). Social indicators of well being. New York: Plenum Press. Bradburn, N. M. (1969). The structure of psychological well being. Chicago: Aldine. Cochran, W. G. (1992). Técnicas de muestreo. México: Editorial Continental. Cristianini, N. and J. Shawe-Taylor (2000). An Introduction to Support Vector Machines and other Kernel-based Learning Methods. Cambridge: Cambridge University Press. Forbus, K. D. (1988). Common sense physics. Annals Revue of Computer Science, 197– 232.

162

C. Angulo et al.

Gonz´ alez, L. (2002). An´ alisis discriminante utilizando m´ aquinas n´ ucleos de vectores soporte. Funci´ on n´ ucleo similitud. Tesis doctoral, Universidad de Sevilla. Gonz´ alez, L. and J. Gavil´ an (2001). Una metodolog´ıa para la construcci´ on de histogramas. Aplicaci´ on a los ingresos de los hogares andaluces. XIV Reuni´ on ASEPELT-Spain. Gonz´ alez, L., F. Velasco, C. Angulo, J. Ortega, and F. Ruiz (2004). Sobre n´ ucleos, distancias y similitudes entre intervalos. Inteligencia Artificial 8 (23), 113–119. In Spanish. Gonz´ alez, L., F. Velasco, and R. Gasca (2005). A study of the similarities between topics. Computational Statistics 20 (3), 465–479. Michalos, A. C. (1992). Global report on students well-being. New York: Springer-Verlag. Piera, N. (1995). Current Trends in Qualitative Reasoning and Applications. Barcelona: CIMNE. Skh¨ olkopf, B. (2000). Statistical learning and kernel methods. Technical Report MSRTR-2000-23, Microsoft Research Limited. Vapnik, V. N. (1998). Statistical Learning Theory. New York: John Wiley & Sons.

CHAPTER 5

Constraint Satisfaction Problems for Modeling and Explanation in Sustainability Science Franz Wotawa

5.1

Introduction

Constraint satisfaction problems (Csps) provide means for a natural representation of models. In computer science Csps are used to represent the knowledge for solving different tasks including configuration, diagnosis, and the computation of explanations. A Csp comprises a set of variables and a set of constraints (see also Chapter 3 for a further application). Each constraint has a scope which is a subset of the variable set and provides a relationship between the variables in its scope. Hence, models of physical systems which usually comprise variables and their relationships can be directly represented by means of Csps. A solution of a Csp is an assignment of values to the variables such that no constraint is violated. In this article we introduce the basic concepts behind Csps, show how Csps can be solved efficiently, and present new directions in further improving the solution process. We use one example throughout this text for explaining the basic ideas. Moreover, we show how explanations for certain observations can be derived directly from the Csp models. These explanations can be used both for deriving decisions and explaining solutions, for example, in tutoring systems. Tutoring systems, especially for highly complex systems, are very important in sustainability science in order to ease the understanding of complex processes and relationships. Csps are in wide-spread use and many applications from very different areas have been published. Application areas of Csps for representing models of systems are bio-informatics (Backofen and Gilbert 2001; Krippahl and Barahona 2002), knowledge-based configuration (Stumptner et al. 1998; Fleischanderl et al. 1998), scheduling (Pape and Smith 1988), diagnosis (Sabin et al. 1994), user- and graphical interfaces (Marriott et al. 2003), and intelligent information systems (Torrens et al. 2002), among others. The reason for the usability of constraints is their capability for representing knowledge, e.g., system models, in a very natural way. Every mathematical formula can be viewed as a constraint. For example, the formula V = I · R stating Ohm’s Law defines a relationship between three quantities V , I, and R. Setting two of them to a value constrains the value of the third one to exactly one value. Such formulae always occur in physics and can be represented by means of Csps.

164

F. Wotawa

In order to illustrate the introduced concepts and definitions for Csps we make use of a small solar-panel water-pump system which is depicted in Fig. 5.1. The behavior of the system can be verbally explained as follows: On sunny days the solar panel is producing enough power to load the battery and to drive the pump. If it is cloudy, only the pump can be driven. At night and during rainy days the battery is used to drive the pump unless it is empty. The task of the pump is to transport water from the lake to the water tank. If the lake is empty, no water can be transported. Only rain refills the lake. The control unit handles the loading of the battery and the power supply for the pump. Note that we do not want to consider time for the model. Instead we are interested in representing possible states of the model elements. For example, the model should allow to derive that it is not possible to pump water from the lake to the tank during the night when the battery is empty. WEATHER CONDITION

C

TIME T

WATER TANK W SOLAR PANEL SP CONTROL UNIT CU

B BATTERY

M

MOTOR PUMP

L LAKE

Figure 5.1: A solar-panel water-pump example.

5.2

Basic Definitions

In this section we introduce the basic concepts and definitions of Csps. An introduction including a description of algorithms and improvements can be found in Dechter (1992), Mackworth (1987) and more recently in Dechter (2003) which provides a good starting point for studying Csps. A Csp is characterized by a set of variables V = {V1 , . . . , Vn }, each associated with its (not necessarily finite) domain Di , 1 ≤ i ≤ n, and a set of constraints C = {C1 , . . . , Ck }. Each of the constraints Cj has an associated corresponding pair (Xj , Rj ), where Xj ⊆ V is a set of variables, and Rj is a relation over Xj . Xj is called

CSPs for Modeling and Explanation

165

the scope of constraint Cj . For convenience we assume a function dom : V 7→ DOM that maps a variable Vi to its domain Di , a function scope : C 7→ 2V that maps a constraint to its corresponding scope, and a function rel : C 7→ RELAT ION S that maps constraints to their relations. For our example, we can easily describe the behavior of the system from Fig. 5.1 using a Csp. First every element of the system represents a constraint, e.g., the weather condition can be represented by a constraint C. Hence, for the system we have 8 constraints: S, T, SP, B, CU, M, L, W . In the next step we have to define the scopes of the constraints, and afterwards the domains of the variables. For the constraint associated with the weather condition we only have one variable c stating whether it is sunny, cloudy, or rainy. For the constraints representing the time of the day, we again have a single variable t with its corresponding domain {day, night}. The photovoltaic solar panel SP produces power which corresponds to a variable p. The variable p is high if the sun is shining during the day; p is at the nominal level if it is cloudy, and at the low level if it is rainy or during the night time. Therefore, the scope of SP is {c, t, p}. The control unit which corresponds to constraint CU uses the power from the solar panel and drives the motor. If the produced power is high, the battery is loaded. If it is low, the power of the battery is used to drive the motor. If produced power can only drive the pump the battery is in the wait state. Hence, the scope of CU is {p, b, m} where b is the variable that corresponds to the current state of the battery, and m is for driving the motor. The domain of m comprises two elements ‘stop’ and ‘work’. The battery B has four modes. It can be empty and no power can be supplied. It can be full and nominal power can be supplied, it can be in the loading mode where as well no power can be supplied, or it is in the waiting mode. The pump M pumps water from the lake L to the water tank W if m =work and the lake is not empty. Otherwise, there is no water transportation. The lake L can be empty only if it is not raining and it can be full. The state of the lake is represented by the variable f . The water flow to the pump is represented by x. The scope of L is {c, f, x}. The scope of the pump is {m, x, y} where the variable y represents the outward flow to the water tank. The water tank constraint W itself has two variables in its scope: y and its state l with the same domain as f . When assigning a unique value to each variable from a subset of V , we get an instantiation. We further say that an instantiation satisfies a given constraint Cj if the assignments which correspond to the scope of Cj are elements of the relation Rj of the constraint. Otherwise, we say that the constraint Cj is violated. In our example, the instantiation c = cloudy, t = day, p = low satisfies the constraints C and T but violates the constraint SP because (cloudy,day,low) is not an element of the relation of SP . The notation of satisfaction and violation naturally leads to the definition of a solution for a given Csp. A solution is an instantiation of all variables V such that all constraints are satisfied. Such an instantiation is also called a legal or locally consistent instantiation. Note that usually there exists not just one

166

F. Wotawa C(c) :

c B(b) : b sunny full cloudy empty rainy load wait

T (t) :

SP (c, t, p) :

c sunny cloudy rainy sunny cloudy rainy

t day day day night night night

p high nom low low low low

L(c, f, x) :

CU (p, b, m) :

p high nom low low

b load wait full empty

m work work work stop

M (m, x, y) :

W (y, l) :

t day night

c f x sunny full flow sunny full no flow sunny empty no flow cloudy full flow cloudy full no flow cloudy empty no flow rainy full flow rainy full no flow m work work stop

x y flow flow no flow no flow no flow no flow

y l flow full no flow full no flow empty

Table 5.1: The Csp model for the solar-panel water-pump example from Fig. 5.1.

solution for a given Csp. Hence, solving a Csp may have different interpretations. It may refer to all possible solutions or simply state that only a single solution should be computed. It can also mean that the best solution with respect to given objectives or that a local instead of a global solution has to be searched for. A local solution is an instantiation which comprises a subset of the variables that satisfies the corresponding constraints. One (global) solution for our example Csp is the following instantiation: c = sunny, t = day, p = high, b = load, m = work, f = full, x = flow, y = flow, t = full, which assigns an unique value to all variables. The interpretation of this solution is that during sunny days the motor is working, the battery is loaded and water is pumped from the lake to the tank if the lake is full.


167

The given solution of our example is not the only one, because all different behaviors of the system are modeled. In practice someone might be interested in all possible behaviors in a specific situation, e.g., all behaviors when the sun is shining during the day. Such situation can be itself represented by constraints. Someone has only to specify observations or desired properties of a system. For example, we can easily add the observations that the sun is shining and the tank is empty, by specifying a constraint OBS(c, t) with only one relation (sunny,empty). Alternatively, we could add two constraints. One specifying a value for variable c and the other a value for t. Obviously, all solutions of the extended Csp have to satisfy the new constraints and the old one. Thus, some prior solutions are no longer valid when adding additional constraints. However, both the model of the system and properties or observations are represented using the same basic technique and there is no need to change the way to compute solutions. Therefore, we concentrate on computing solutions for Csps in the following section. More specific we concentrate on general methods for computing solutions and discuss improvements, including decomposition methods. Afterwards, we introduce a more specific algorithm that allows to derive explanations for systems that are modeled using Csps and discuss related modeling issues. Although, Csps are not restricted to finite relations, we only consider finite relations from here on. Csps where at least some of the constraints are defined over an infinite domain can be solved using interval arithmetic (Hyvönen 1992) or equation solver, e.g., the Gaussian elimination method. However, some of the techniques described in this article, e.g., decomposition methods, can be used for all kind of Csps.

5.3

Computing Solutions

A simple method for computing a solution for a given Csp is to try every possible variable assignment and check the constraints. If no constraint is violated, the assignment is a legal instantiation and we find one solution. However, this procedure is not feasible because it requires the explicit search in the whole search space with a size of O(D n ) where D is the size of the largest domain and n is the number of variables. A better and more effective way of computing solution is to use backtracking search. Although, this method has the same worst-case complexity O(D n ), it is fast enough in practice and can be improved by using heuristics. For backtracking we assume an order for the variables. We start with the first variable and assign a provisional value. We further assign provisional values to the successive variables as long as the constraints are satisfied. For this purpose we only have to consider constraints where all variables have an assigned value. If one constraint is violated we backtrack to the variable that has been assigned a value in the last step and choose another value. If there is no value, we have again to track back to the previous variable and so on. If there is no further value to assign for the first variable, there is no solution. Otherwise, the procedure stops when all variables have been

168

F. Wotawa

backtrackCSP(CSP,VO,I) 1. If V O is empty, then return the current variable assignment I as result. 2. Otherwise, let v be the first element of V O. 3. For all values x ∈ dom(v) of the currently selected variable v do: (a) Add the assignment v = x to the set of current assignments I. (b) Check all constraints where variables have a value assignment in I. If at least one constraint is violated, remove v = x from I. Otherwise, do the following: i. Call backtrackCSP(CSP,VO\{x},I) recursively and store the result in r. ii. If r = ◦ / , then remove v = x from I. Otherwise, return r.

4. Return ◦ /.

Figure 5.2: Backtracking algorithm for Csps.

assigned a value that satisfies all constraints and we finally find a solution. The backtracking algorithm is given in Fig. 5.2. The algorithm uses the Csp, an ordered collection of variables, and a set of variable assignments as input. It returns a solution, i.e., a legal instantiation, if there is one, and the empty set ◦ / if no solution can be found. When initiating the backtrack search the algorithm has to be called by backtrackCSP(CSP,VO, ◦ / ). For example, consider our Csp from Fig. 5.1 together with the additional constraint OBS(c, t, f ) = (sunny, day, full) stating that the sun is shining, it is daytimes, and the lake has water. Moreover, we assume the following variable ordering: {c, b, t, p, m, f, x, y, l}. Using this assumptions we get the following trace when applying the backtracking algorithm. c = sunny b = full t = day p = low Constraint SP violated p = nom Constraint SP violated p = high m = work Constraint CU violated m = stop Constraint CU violated t = night ... b = load t = day p = low Constraint SP violated


169

p = nom Constraint SP violated p = high m = work f = full x = flow y = flow l = full Solution found What we see is that there is heavy backtracking once enough variables have been assigned a value which allows for checking more interesting constraints like SP or CU . Moreover, we see that some decisions may have a strong impact on search. For example, in the second line assuming b = f ull guides the search in a branch where no solution exists and, therefore degrading the running time performance. Beside optimizations regarding the used data structures, there are three ways for improving the running time performance of the backtracking algorithm. Firstly, the used variable ordering has an huge impact. Secondly, it is not always necessary to go through all values of a specific domain. For example, if one constraint restricts the possible value of a variable to one value, e.g., OBS restricts c to be sunny, then it is not necessary to consider other values. Finally, backtracking can be improved by detecting dead ends as soon as possible and by going back several steps in the search tree. In backtrackCSP only one step backtracking is performed. The question of variable orderings has been evident in the Csp community for a very long time (Freuder 1982; Dechter and Pearl 1989). The proposed heuristic methods have in common that they prefer tightly constraint variables. Hence, it is a good idea to choose the variable first that occurs in the highest number of constraints. As an effect, the search space for selecting values for other variables is reduced. Another good heuristics is to prefer variables with a small domain. Variables, where only a unique value can be assigned, should be instantiated first. Once a variable has been selected for instantiation, we have to choose a value. This value should be a value that maximizes the number of possible options for future assignments. This heuristics can help to restrict the number of dead ends search a specific algorithm has to visit before finding a solution (Dechter and Pearl 1988). A different idea for Csp algorithms that have been introduced after analyzing the backtrack behavior can be summarized as consistency algorithms (Mackworth 1977) but the underlying ideas can also be used to improve backtracking. For example, when locking at constraint OBS, we easily see that c must be ‘sunny’, t must be ‘day’, and f must be set to ‘full’. All other assignments contradict the constraint OBS. We can formalize this idea by saying that all elements of a variable’s domain must be used in at least one constraint. If this is the case for a Csp, then it is called node consistent. Every Csp can easily be made node consistent by removing all domain values, that are not used in any constraint. Formally, we write Di = Di ∩ {x|(. . . , x, . . .) ∈ Rj (. . . , Vi , . . .)}. We see that our example Csp from Fig. 5.1 together with constraint OBS is not node consistent. However, we can

170

F. Wotawa

easily make this Csp node consistent by setting dom(c) = {sunny}, dom(t) = {day}, dom(f ) = {full}. In the next step, we further restrict the domain of other variables. Because of dom(c) = {sunny} and constraint SP , we know that variable p can only be ‘high’ or ‘low’ but never ‘nom’. Hence, we further reduce the following domains dom(p) = {low,high} and dom(b) = {load, full, empty}. This kind of consistency is called arc consistency and can be formalized as follows. Given two variables Vi , Vj . A Csp is arc consistent if for every assignment of a value to Vi which satisfies some constraints there must also exist a value for Vj that satisfies these constraints (among the two variables). Again, we can transform every Csp to an equivalent but arc consistent Csp by applying the following rule: dom(Vi ) = dom(Vi ) ∩ {x|∃y ∈ dom(Vj ) : (. . . , x, . . . , y, . . .) ∈ Rk (. . . , Vi , . . . , Vj , . . .)}. The idea of consistency can be extended to the general case. We say that a Csp is i-consistent if for any set of i − 1 variables along with values for each variable satisfying all constraints where the variables are in their scope, there exists a value for any ith variable, such that the i values together satisfy all the constraints among the i variables. In our example establishing 3-consistency reduces the domains of variable p to the single value ‘high’ because c =‘sunny’ and t =‘day’ is only together consistent with p =‘high’. When subsequently enforcing consistency we restrict the domain of b to {load} and dom(m) to {work}. Node and arc consistency are only special cases of i-consistency with i=1 and 2, respectively. Successively ensuring i-consistency leads to a solution if all variable domains can be reduced to sets with one element. A procedure that makes use only of consistency is not effective in general since ensuring i-consistency is exponential in i. i-consistency allows to improve backtracking by reducing the variables’ domain. Hence, it seems to be a good choice to make use of i-consistency in a pre-processing step before calling the backtracking algorithm. Some experimental results of the trade-off between pre-processing and backtracking have been published in Dechter and Meiri (1989) and Haralick and Elliott (1980).

5.4

Decomposition Methods

A different approach to solving Csps are methods which consider structural properties of a given Csp. It is well known that Csps can be represented by a hypergraph, i.e., a graph with vertices and arcs where each arc connects two or more vertices. In the hypergraph representation the variables of a Csp are the vertices and the constraints, in particular their scopes, correspond to arcs. In Fig. 5.3 the hypergraph representation of our Csp model (Fig. 5.1) is given. The question is now, why do consider structural properties? The answer can be given when having again a closer look at the backtracking algorithm. The running time performance of this algorithm heavily depends on the Csp, the given variable ordering, and the variables’ domains. In general backtracking works fine in case of a

171

CSPs for Modeling and Explanation SP c

C

L

T

p

t

B

f

CU

b

M x

y

m

W l

Figure 5.3: A hypergraph for the Csp of Fig. 5.1.

limited number of backtrack operations. The optimal situation would be backtrackfree search. All of the previously introduced heuristic methods deal with a reduction of backtracking during search for a solution. The relationship between the structure of a Csp and backtracking can be easily explained. Consider a Csp with 3 variables {x, y, z} and the following constraints {C1 (x, y) = {(0, 0), (1, 1)}, C2 (y, z) = {(0, 0), (1, 1)}, C3 (z, x) = {(1, 0), (1, 1)}}. When now setting x = 0, all constraints are satisfied. This holds again when setting y = 0. But, when assigning z the value 0 constraint C3 is violated. In this case, a one-step backtrack does not help because setting y to one violates constraint C2 and we have to backtrack to variable x. The reason for this behavior is that the corresponding hypergraph is cyclic and that a wrong decision at a node in the cycle may only be detected when walking along the cycle again and revisiting this node as happened for variable x. If a Csp is acyclic, the situation is different. Choosing a value for a variable has only a local influence on the connected variables but cannot lead to a situation where it has an influence on itself. The above observations can be summarized as follows. For tree-structured Csps, i.e., Csps where their corresponding hypergraph is acyclic, the computation can be done in a backtrack-free manner and in linear time (Freuder 1982; Mackworth and Freuder 1985; Yannakakis 1981). In the rest of this Section, we first show how to check whether a hypergraph is acyclic or not, second introduce a Csp solver for acyclic systems, and finally, show how general Csps can be converted to acyclic Csps without changing any possible solution of the original Csp. This conversion procedure is important because it leads to a structural property that is a good indicator of the complexity of solving a given Csp. A simple algorithm that proves acyclicity of hypergraphs is given in Fig. 5.4, where V denotes the set of vertices and E the set of arcs. The acyclicHG algorithm returns True if the hypergraph is acyclic and False, otherwise. The running

172

F. Wotawa

acyclicHG(V ,E) 1. Repeatedly apply the following operations until they can be applied: (a) Delete a vertex that occurs only in one arc. (b) Delete an arc that is contained in another arc. 2. If no vertex remains, then return True. Otherwise, return False. Figure 5.4: Testing acyclicity of hypergraphs.

time is quadratic in the size of the hypergraph. In Tarjan and Yannakakis (1984) the authors presented a linear time algorithm for testing acyclicity of hypergraphs. The hypergraph in Fig. 5.3 is cyclic because constraints L, SP, CU , and M cannot be further eliminated. Hence, the following algorithm for computing solutions in a backtrack-free manner cannot be applied to our example Csp without any modifications. The backtrack-free algorithm works on hypertrees rather than on (acyclic) hypergraphs. An acyclic hypergraph can easily be converted to a hypertree. Every arc of the hypergraph is mapped to a vertex of the hypertree. Two vertices of the hypertree are connected if they share a variable in the original hypergraph. If the hypergraph is cyclic, we need a decomposition method, which will be described later, to convert it to a hypertree. Once we have a hypertree, we choose one vertex as the root vertex of the tree. The Csp algorithm works from the leafs of the tree to the root. In every step the relation that is associated with a constraint, i.e., a vertex n of the hypertree, is reduced by using the relations of its children. For example, assume the following set of constraints: {C1 (x, y) = {(1, 0)}, C2 (u, v) = {(0, 1)}, C3 (y, u) = {(0, 0), (0, 1), (1, 0)}} which can be represented by a hypertree where C3 is the root vertex, and C1 , C2 are its children. Obviously, the tuples (0, 1) and (1, 0) can be deleted from C3 ’s relation because the 1 is neither possible for y nor for u when considering constraints C1 and C2 , respectively. The operation we use to delete tuples from a relation of a parent vertex is the semi-join operation. When all relations from the leafs to the root are reduced using the semi-join operation,1 the only thing to remain is to extract a solution. This is done by selecting a value for all variables at the root vertex and than going from the root back to the leafs and successively selecting values for variables that have not been selected previously. Finally, this algorithm stops with one instantiation that satisfies all constraints. It is worth noting, that after the first part of the algorithm when we reach the root, we can determine whether there exists a solution or not. If 1

A semi-join operation on relations is defined using a projection function and the join operation and can be efficiently implemented. In particular the semi-join on R and S is a join of R and S where the result is projected only on variables from R. A semi-join removes the tuples in R that have no corresponding tuple (via the variables that are used in R and S) in S.


173

solveTreeCSP(HT ) 1. Upward computation: Starting from the leafs to the root apply the following rule: (a) Let n be a vertex where all of its children have been processed before. (b) For all children m apply the semi-join operation on the relation of m and the relation of n in order to delete tuples that are not justified by tuples of m. (c) If the remaining relation of n is empty, then return nil to indicate that there is no solution. 2. Solution extraction: From the root to the leaf recursively apply the following rules: (a) Let n be a vertex that has not been processed before but where its parent has been processed. (b) Remove all tuples from rel(n) that are not valid with respect to the previous variable assignments. (c) From the remaining tuples choose values for variables that have not been assigned a value before. (d) Return the set of variable assignments as a solution. Figure 5.5: Solving tree-structured Csps.

the set of tuples is empty for the relation at the root, then there exists no solution. Otherwise, there exists a solution which can be extracted as described above. The algorithm for solving tree-structured Csps is summarized in Fig. 5.5. The input of this algorithm is a hypertree representation of the Csp. The algorithm returns a solution if one exists and nil, otherwise. In order to apply the backtrack-free search algorithm on Csps, it remains to show how general Csps can be decomposed to their tree-structured equivalents. In literature there are several approaches, including biconnected components (Freuder 1985), tree clustering (Dechter and Pearl 1989), hinge decomposition (Gyssens, Jeavons, and Cohen 1994), and most recently hypertree decomposition (Gottlob et al. 1999a; Gottlob et al. 1999b; Gottlob et al. 2000) among others. They have in common the search for vertices or arcs in Csp graph representations, that potentially separate the graph into tree-structured sub-graphs. During the decomposition some constraints are put together to form a larger constraint in the decomposed tree. Usually, the number of constraints that have to be put together is called the width of the original Csp. Since, joining constraints together has an impact on memory and running time performance, a small width indicates that the Csp can be solved

174

F. Wotawa

in a reasonable amount of time. Because Gottlob et al. (2000) proved that hypertree decomposition is the currently most optimal decomposition methods with respect to the width, we only introduce the basic ideas behind this decomposition method. For more details regarding hypertree decomposition and an algorithm we refer the reader to Gottlob et al. (1999b). Moreover, there is some work (Gottlob et al. 2002) which deals with the combination of different decomposition methods to improve their running time behavior. However, running time of decomposition is not as important as computing a solution because the decomposition can be computed once and used many times. We introduce hypertree decomposition using our Csp from Fig. 5.1 and its hypertree representation from Fig. 5.3. This hypergraph is cyclic because of constraints SP, CU, M, L. In order to remove this cycle we have to combine some of the constraints, say for example, SP and W . The hypergraph without these constraints is cyclic. The only thing left is the construction of the corresponding hypertree. For this purpose we start with the sum-constraint SP, M as the root of the hypertree. The constraint CU and L that lie on the cycle of the hypergraph share variables with the sum-constraint. Hence, we have to introduce arcs from these constraints to the root. Using similar arguments for the remaining constraints we finally get one hypertree which is depicted in Fig. 5.6. Note that this hypertree is not unique. The relation of the sum-constraint is defined by joining the relations of its comprising constraints. In this example the width, in particular the hypertree width, of the Csp is 2 because 2 constraints have to be joined in order to eliminate all cycles. SP, M

W

T

L

C

CU

B

Figure 5.6: A hypertree of the hypergraph from Fig. 5.3.

More formally, a hypertree of a hypergraph H is a triple (T, χ, λ), where T = (N, E) is a rooted tree with vertices N and arcs E, χ and λ are labeling functions which associate two sets χ(p) ⊆ var(H) and λ(p) ⊆ arcs(H) to each vertex p ∈ N . Hence, λ associates constraints to be joined and χ associates a set of variables to the vertices of T . We further define χ for a subtree T ′ = (N ′ , E ′ ) of T as S χ(T ′ ) = v∈N ′ χ(v), and for any p ∈ N , Tp the subtree of T rooted at p. Given the definitions the hypertree decomposition of a hypergraph H is defined as a Hypertree HD = (T, χ, λ) where T = (N, E) which satisfies the following

175

CSPs for Modeling and Explanation conditions: 1. For each arc h ∈ arcs(H), there exists a p ∈ N such that var(h) ⊆ χ(p).

2. For each variable Y ∈ var(H), the set {p ∈ N |Y ∈ χ(p)} induces a connected subtree of T . 3. For each p ∈ N , χ(p) ⊆ var(λ(p)). 4. For each p ∈ N , var(λ(p)) ∪ χ(Tp ) ⊆ χ(p). Moreover, we say that h ∈ arcs(H) is strongly covered in HD if there exists a vertex p ∈ N such that var(h) ⊆ χ(p) and h ∈ λ(p). A hypertree decomposition HD of a hypergraph H is a complete decomposition of H if every arc of H is strongly covered in HD. Note that it is always possible to make an incomplete decomposition complete by adding new vertices to the decomposition. In our example hypertree, the χ and λ values for vertex SP, M are: χ(SP, M ) = {c, t, p, m, x, y}, λ(SP, M ) = {SP, M }. The relation of the sum-constraint is specified in the following table: c sunny cloudy rainy sunny cloudy rainy sunny cloudy rainy sunny cloudy rainy sunny cloudy rainy sunny cloudy rainy

t day day day night night night day day day night night night day day day night night night

p high nom low low low low high nom low low low low high nom low low low low

m work work work work work work work work work work work work stop stop stop stop stop stop

x flow flow flow flow flow flow no flow no flow no flow no flow no flow no flow no flow no flow no flow no flow no flow no flow

y flow flow flow flow flow flow no flow no flow no flow no flow no flow no flow no flow no flow no flow no flow no flow no flow

This table is substantially larger than the sum of the tables for both constraints. In particular the join operation of SP and M in this example is equivalent to the Cartesian product because the constraints share no variables. The advantage now is that we can use the backtrack-free search algorithm for computing solutions.

176

5.5

F. Wotawa

Searching for Explanations

Explanations for a specific behavior of a system are very important, not only for understanding the system’s behavior under certain conditions, but also for taking appropriate actions once a misbehavior has been detected. For example, if we observe that the water quality of a lake or river has not proved satisfactory, we are interested in the causes of this observations. Once, the cause has been fixed, appropriate counter-measures can be taken. This process requires a deep understanding and knowledge of the relationships between the system’s components and the capability of explaining observations based on the available knowledge. Research in artificial intelligence and more specific in model-based reasoning (Mbr) has dealt with the use of models for explaining certain observations for several years. In model-based diagnosis (Reiter 1987) a formal framework for deriving diagnoses, i.e., set of components that explain observations, was introduced. More recently, the same techniques have been applied to tutoring systems that deal with physical systems (de Koning and Bredeweg 1998; de Koning et al. 2000). In this section we introduce the ideas behind the TREE* algorithm (Stumptner and Wotawa 2001) that allows for computing explanations of a system’s behavior that is represented as a Csp. For a formal introduction and empirical results we refer to the literature. The only requirement for using TREE* is that the Csp is tree-structured. However, in the previous section we presented a method for converting a general Csp to its tree-structured variant. In Stumptner and Wotawa (2003) the authors describe the coupling of TREE* and the hypertree decomposition in detail. Other algorithms for computing explanations from Csps are available, e.g., SAB (Fattah and Dechter 1995). However, in Stumptner and Wotawa (2001) it has been shown experimentally that TREE* outperforms SAB. The basic computation procedure of TREE* is similar to solveTreeCSP (Fig. 5.5). In contrast to solveTreeCSP TREE* starts at the root vertex and successively removes all tuples of the constraint relations that contradict some given observations. Once, the leafs of the tree are reached, TREE* goes upward the tree and summarizes the explanations that correspond to the tuples in the constraint relations. We use an example to introduce the TREE* algorithm. Consider our solarpanel system from Fig. 5.1. For simplicity we reduce this model to the constraints C, T , and SP . This Csp is obviously acyclic. In order to derive explanations of the behavior of the system’s components, i.e., time, weather conditions, and the solar-panel, we have to make the state of the components explicit. In particular, we have to explicitly distinguish between the behavior of a system under a given state. For example, if the weather condition is in a good state, then the sun is shining. Or if the solar panel is in a good condition, then the behavior is described by the table from Fig. 5.1. Otherwise, regardless of time and weather condition, the solar panel produces no power, which is represented by the value low. The state of the components itself can be represented as an additional variable for each constraint.

177


Hence, it can be easily integrated in our Csp model description. In the following table we specify the behavior of our example Csp with 3 constraints {C, T, SP }: C(c, d(C)) :

SP (c, t, p, d(SP )) :

c d(C) T (t, d(T )) : sunny {{good(C)}} cloudy {{avg(C)}} rainy {{bad(C)}} c sunny cloudy rainy sunny cloudy rainy sunny cloudy

t day day day night night night day day

p high nom low low low low low low

t day night

d(T ) {{day(T)}} {{night(T)}}

d(SP ) {{}} {{}} {{}} {{}} {{}} {{}} {{ab(SP)}} {{ab(SP)}}

In the above tables the state is represented by an additional column d(x) for every constraint x ∈ {C, T, SP }. Because TREE* uses this column for computing one or more explanations this column must contain one set of sets, where each set is an explanation, i.e., a conjunctive sentence of predicates. If the empty set is element of d(x), then this row of the table, i.e., the tuple, is said to represent the correct behavior. Otherwise, the tuple represents a behavior that is specified by the predicate(s) of the set. For example, the predicate ab in constraint SP says that the solar panel is behaving abnormal. For explaining TREE* we assume that SP is the root vertex of the hypertree with two children T and C. TREE* starts at the root and removes all tuples that are in contradiction with the given observation(s). For example, if we assume that the produced power is low, then the first two rows of the relation are deleted from SP . In the next step TREE* is recursively called on the children, and again tuples contradicting a given observation are removed. In our case we assume no additional observations and the relations of T and C remain the same. After reaching the leafs, we go upward and compute explanations for every tuple in SP by joining them with the tuples of the leafs. For example, consider the first row in the relation of SP . The value of c is sunny and the value of t is day. The value of d(SP ) at row 3 is now given by d(SP ) at row 3 combined with d(C) at the row where c is rainy and d(T ) at the row where t is day. Hence, we get {{bad(C), day(T )}} as an explanation which can be interpreted as follows: If we have a bad weather condition during the day, the power production of the solar panel is low. This interpretation is not the only one. For the given observations we finally receive the following results:

178

F. Wotawa SP (c, t, p, d(SP )) :

c rainy sunny cloudy rainy sunny cloudy

t day night night night day day

p low low low low low low

d(SP ) {{bad(C),day(T)}} {{good(C),night(T)}} {{avg(C),night(T)}} {{bad(C),night(T)}} {{ab(SP),good(C),day(T),}} {{ab(SP),avg(C),day(T)}}

When we add an additional observation, e.g., it is during the day, the number of explanations is reduced to 3. One is saying that it is raining, and the other two indicates that the solar panel is not working correctly.

5.6

Conclusion

In this paper we introduced the concept of constraint satisfaction problems in the context of sustainability science. Csps provide means for a very natural representation of models. Moreover, general techniques can be used for computing solutions from a given Csp. Hence, we can focus on modeling issues rather than on computational issues. In this paper, we focused on providing the basic concepts and algorithms for Csps giving a first impressions on how to use these techniques in practice. Moreover, we introduced an algorithm that allows for computing explanations from tree-structured Csps. The basic principles of compiling general Csps to their tree-structured equivalent are also discussed.

References Backofen, R. and D. Gilbert (2001). Bioinformatics and constraints. Constraints 6 (2–3), 141–156. de Koning, K. and B. Bredeweg (1998). Using GDE in Educational Systems. In Proceedings of the European Conference on Artificial Intelligence (ECAI), Brighton, pp. 279–283. de Koning, K., B. Bredeweg, J. Breuker, and B. Wielinga (2000). Model-based reasoning about learner behavior. Artificial Intelligence 117, 173–229. Dechter, R. (1992). Encyclopedia of Artificial Intelligence, Chapter Constraint networks, pp. 276–285. New York: Wiley & Sons. Dechter, R. (2003). Constraint Processing. San Mateo: Morgan Kaufmann. Dechter, R. and I. Meiri (1989). Experimental evaluation of preprocessing techniques in constraint satisfaction problems. In Proc. 11th Int. Joint Conf. on Artificial Intelligence, pp. 271–277. Dechter, R. and J. Pearl (1988). Network-based heuristics for constraint-satisfaction problems. Artificial Intelligence 34, 1–38.


179

Dechter, R. and J. Pearl (1989). Tree clustering for constraint networks. Artificial Intelligence 38, 353–366. Fattah, Y. E. and R. Dechter (1995). Diagnosing tree-decomposable circuits. In Proc. 14th Int. Joint Conf. on Artificial Intelligence, pp. 1742–1748. Fleischanderl, G., G. Friedrich, A. Haselb¨ ock, H. Schreiner, and M. Stumptner (1998). Configuring large-scale systems with generative constraint satisfaction. IEEE Intelligent Systems 13 (4), 59–68. Freuder, E. C. (1982). A sufficient condition for backtrack-free search. Journal of the ACM 29 (1), 24–32. Freuder, E. C. (1985). A Sufficient condition for backtrack-bounded search. Artificial Intelligence 32 (4), 755–761. Gottlob, G., M. Hutle, and F. Wotawa (2002). Combining hypertree, bicomp, and hinge decomposition. In Proc. European Conf. Artificial Intelligence (ECAI), Lyon. Gottlob, G., N. Leone, and F. Scarcello (1999a). Hypertree Decomposition and Tractable Queries. In Proc. 18th ACM SIGACT SIGMOD SIGART Symposium on Principles of Database Systems (PODS-99), Philadelphia, pp. 21–32. Gottlob, G., N. Leone, and F. Scarcello (1999b). On Tractable Queries and Constraints. In Proc. 12th International Conference on Database and Expert Systems Applications DEXA 2001, Florence. Gottlob, G., N. Leone, and F. Scarcello (2000). A comparison of structural CSP decomposition methods. Artificial Intelligence 124 (2), 243–282. Gyssens, M., P. G. Jeavons, and D. A. Cohen (1994). Decomposing constraint satisfaction problems using database techniques. Artificial Intelligence 66, 57–89. Haralick, R. M. and G. L. Elliott (1980)). Increasing tree search efficiency for constraint satisfaction problems. Artificial Intelligence 14, 263–313. Hyv¨ onen, E. (1992). Constraint reasoning based on interval arithmetic: The tolerance propagation approach. Artificial Intelligence 58, 71–112. Krippahl, L. and P. Barahona (2002). PISCO: Solving protein structures with constraint programming and optimization. Constraints 7 (3-4), 317–331. Mackworth, A. (1987). Constraint satisfaction, Chapter Encyclopedia of Artificial Intelligence, pp. 205–211. New York: John Wiley & Sons. Mackworth, A. K. (1977). Consistency in networks of relations. Artificial Intelligence 8, 99–118. Mackworth, A. K. and E. C. Freuder (1985). The complexity of some polynomial network consistency algorithms for constraint satisfaction problems. Artificial Intelligence 25, 65–74. Marriott, K., P. Stuckey, V. Tam, and W. He (2003). Removing node overlapping in graph layout using constrained optimization. Constraints 8 (2), 173–207. Pape, C. L. and S. F. Smith (1988). IFIP, Temporal Aspects in Information Systems, Chapter Management of temporal constraints for factory scheduling, pp. 159–170. Reiter, R. (1987). A theory of diagnosis from first principles. Artificial Intelligence 32 (1), 57–95.

180

F. Wotawa

Sabin, D., M. C. Sabin, R. D. Russell, and E. C. Freuder (1994). A constraint-based approach to diagnosing distributed software systems. In Proc. 5th Int. Workshop on Principles of Diagnosis, New York. New Paltz. Stumptner, M., G. Friedrich, and A. Haselb¨ ock (1998). Generative constraint-based configuration of large technical systems. AI EDAM 12 (4), 207–320. Stumptner, M. and F. Wotawa (2001). Diagnosing tree-structured systems. Artificial Intelligence 127 (1), 1–29. Stumptner, M. and F. Wotawa (2003). Coupling CSP decomposition methods and diagnosis algorithms for tree-structured systems. In Proc. 18th Int. Joint Conference on Artificial Intelligence (IJCAI-03), Acapulco, pp. 388–393. Tarjan, R. E. and M. Yannakakis (1984). Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal of Computing 13 (3), 566–579. Torrens, M., B. Faltings, and P. Pu (2002). Smartclients: Constraint satisfaction as a paradigm for scaleable intelligent information systems. Constraints 7 (1), 49–69. Yannakakis, M. (1981). Algorithms for acyclic database schemes. In C. Zaniolo and C. Delobel (Eds.), Proc. of the International Conference on Very Large Data Bases (VLDB81), Cannes, pp. 82–94.

Part III

Stochastic and Fuzzy Approaches

CHAPTER 6

Catastrophic Risk Management using Spatial Adaptive Monte Carlo Optimization ¨nther Fischer, and Michael Obersteiner Tatiana Ermolieva, Gu

6.1

Introduction

The possibility of more frequent extreme natural disasters dominates the discussions of current global changes. A shift in the overall hazard exposure due to the occurrence of more frequent extreme events will lead to more frequent economic and social shocks at national and regional levels with consequences to the global economy. Under such conditions, the lack of proper risk management may significantly affect sustainability. An integrated catastrophic risk management approach, as presented in this Chapter, offers solutions for developing coherent, comprehensive and robust policy responses. Coherence is needed when integrating different perspectives, e.g., of economists, development planners, natural scientists and disaster managers; and comprehensiveness in order to identify policy gaps between the existing measures in place compared to those needed to ensure economic development that is robust against shocks from possible future catastrophes. The presented approach uses the Adaptive Monte Carlo (stochastic) optimization procedure (for further methodologies applicable for optimzation problems cf. Chapters 7 and 9) that allows the analysis of a variety of complex interactions between decisions and risks in order to find robust optimal portfolios of risk management measures for decreasing regional vulnerability with regard to economic, financial, and human losses. The approach is illustrated with a case study of earthquake risk management. Global change induced by anthropogenic activities, rapid and increasingly powerful technological innovations, creation of densely populated locations with high concentration of industries and infrastructure, is increasing the vulnerability of modern societies thereby making them more exposed to human–made and natural disasters. MunichRe reports that the losses from natural and human–made catastrophes are rapidly increasing: within the last three decades the direct damages from natural disasters alone have increased nine–fold (MunichRe 1999). It is estimated that within the next 50 years more than a third of the world’s population will live in seismically and volcanically active zones (Nrc 1999). One of the main reasons for this is current land use practices, which often ignore location–specific risks: analysis of insurance companies shows that because of economic growth in hazard– prone areas, damages due to natural catastrophes have grown at an average annual

184

T. Ermolieva et al.

rate of 5 percent (Froot 1997). The growth of losses due to catastrophes is also likely to be further exacerbated by the increasing frequency and severity of weather related extreme events due to global climate warming (Milly et al. 2002; Palmer and Raisanen 2002). Impacts of catastrophes cannot be properly evaluated on aggregate levels. For example, aggregate worldwide economic impacts of global warming may even be considered as beneficial whereas some regions and even countries may be wiped out. The consequences of disasters have spatial profiles and determine prospects for sustainable regional agricultural and rural development depending on the mitigative and adaptive capacities available to cope with them (Fischer et al. 2002). While the developed world has the financial and technological means to cope with extreme events, for many developing countries the impacts of catastrophes may cause major economic shocks and disruptions. Catastrophic risk management is a complex transdisciplinary problem requiring knowledge of environmental, natural, financial and social systems. The burden on these systems is unevenly distributed, debatable in scope, and not yet well matched to the policy making apparatus. Costly losses involve various agents and stakeholders: individuals, governments, farmers, producers, consumers, insurers, investors, etc., their perception of catastrophes, goals and constraints with respect to these rare but high impact events. The scarcity of historical data is an inherent feature and a main challenge in dealing with rare catastrophes and new strategies. Purely adaptive, learning–by–doing approaches may be very expensive and dangerous. Thus, catastrophic risks create new scientific problems requiring integrated approaches, new concepts and tools for risk–based land use planning and risk management. The role of models enabling the simulation of possible catastrophes and estimating potential damages and losses becomes a central task for designing mitigation and adaptation programs. In this Chapter we discuss an integrated framework, which enables the analysis of the spatial and temporal heterogeneity of various agents (stakeholders) induced by mutually dependent losses from extreme events. The model addresses the specifics of catastrophic risks: the lack of information, the need for long–term perspectives and geographically explicit models, and the multi–agent decision making structure. The model combines geographically explicit data on the distribution of capital stocks and economic values in infrastructure and land use in a region with a stochastic model generating magnitudes, occurrences, and locations of catastrophes. Using advanced stochastic optimization techniques, the model, in general, allows the analysis of robust optimal portfolios of ex–ante (land use, structural mitigation, insurance) and ex–post (adaptation, rehabilitation, borrowing) measures for decreasing regional vulnerability measured in terms of economic, financial, and human losses as well as in terms of selected welfare growth indicators. The approach is illustrated in a case study of earthquake risks in the Tuscany region of Italy. A Monte Carlo catastrophe model generates location specific random losses from an infinite number of

Catastrophic Risk Management

185

catastrophic earthquake scenarios. An embedded stochastic optimization procedure analyzes spatial and temporal heterogeneities of locations (individuals), insurers, and the government guiding the design of fair and sustained loss sharing programs. Discussions of similar issues regarding catastrophic risks can be found in (Amendola et al. 2000; Amendola et al. 2000; Ermoliev et al. 2000a; Ermoliev et al. 2000b; Ermolieva et al. 2000). As shown in these studies, the technique implied in the model may be important to governments, both central and local, to better negotiate risks and to make decisions on the allocation of properties, agricultural investments, and the implementation of mitigation and adaptation strategies. Also, it is important for insurance companies to make decisions on contracts, premiums, and re–insurance agreements in the case of highly dependent exposures. It is also useful for disaster planning agencies, because it provides a more realistic evaluation of the spatial and temporal distribution of the potential losses. Section 6.2 introduces conceptual risk management models for coping with catastrophic events such as floods, earthquakes, windstorms, etc. It illustrates the necessity to capture the main feature of catastrophes — their abruptness in time and space similar to “spikes” that cannot be properly treated or modeled on “average”. Section 6.3 discusses the main features of an integrated geographic information system (Gis)–based catastrophe management model that, in the absence of historical data, simulates samples of dependent potential losses. Traditionally, insurance and finance quantify extreme events in monetary units. The catastrophe model deals with events, which are non–quantifiable in this sense, and with multivariate distributions of extreme values, i.e., with cases that are not treated within the conventional extreme value theory. This Section also discusses the shortcomings of the standard “if–then” scenario analyses for catastrophic risk management. Section 6.4 describes a spatial and dynamic stochastic optimization model developed for the evaluation of a catastrophe loss–spreading program (Ermoliev et al. 2000a; Ermolieva et al. 2000) and its possible extensions for designing risk reduction measures. This Section also outlines general ideas of the Adaptive Monte Carlo Optimization (Amco) proposed in (Amendola et al. 2000; Ermoliev et al. 2000a; Ermoliev et al. 2000b; Ermolieva et al. 2000) to overcome the difficulties of a pure “if–then” approach. In order to stabilize the loss spreading (insurance) program the model uses economically sound risk indicators such as expected over–payments by “individuals” (cells of risk–prone areas) and an expected shortfall of the insurance. These indicators are used together with so–called stopping times to orient the analysis towards the most destructive scenarios. Section 6.5 specifies the model for a case study of catastrophic earthquakes. Numerical experiments are based on real and modified data from earthquake risk studies in the Tuscany region of Italy (Amendola et al. 2000). This Section illustrates that the “if–then” type of analyses, when based only on the intuition (opinions) of “stakeholders”, may easily fail to produce robust strategies in the case of highly interdependent multivariate distributions of catastrophic losses. Section 6.6 summarizes the conclusions gained in designing catastrophe management

186

T. Ermolieva et al.

models and their application in major case studies.

6.2

A Simple Risk Management Model

Catastrophic events such as floods, earthquakes, and windstorms occur abruptly in time and space as “spikes” that cannot be properly modeled on “average”. The following risk management models address this “abruptness” feature of catastrophes. Let us consider a simple model of growth under abrupt shocks, which is a stylized version of the insurance business (see, e.g., Borch 1962; Daykin et al. 1994; Grandell 1991). The main variable of concern is the risk reserve r t at time t: r t = r0 +π t −At , t ≥ 0, where π t , At are aggregated premiums and claims, and r0 is the initial risk PN (t) reserve. The process At = k=1 Sk , where N (t), t ≥ 0 denotes a random number of claims in interval [0, t] (e.g., a Poisson process) with N (0) = 0, and {Sk }∞ 1 is a sequence of independent and identically distributed random variables (claims) — in other words, replicates of a random variable S. In this model, the inflow of premiums π t pushes r t up, whereas the random outflow At pushes r t down. The main problem of catastrophic management is to avoid the situation when t r drops abruptly below the “vital” level (ruin) in our example, equal to 0. It is only possible with a certain probability Ψ = P r t ≤ 0 for some t, t > 0 . The

rt

t

- A p + t = r0 r t

pt

r0

At t

t

Figure 6.1: Trajectory of the risk reserve rt subject to the random process of claims.

deterministic approach is very simplified, as illustrated by the following calculations. Assume that N (t), Sk , are independent, N (t) has intensity α, i.e., E {N (t)} = αt, and π t = πt, π > 0. Then the expected profit over the interval [0, t] is (π − E(S)α) t, and the expected profit increases in time for π − E(S)α > 0. Thus, the practical deterministic model ignores complex interdependencies among the timing of claims (temporal clustering), their sizes, and the subsequent possibility of ruin, r t ≤ 0. In this formulation the random process r t is replaced by a linear function in t, r t = r0 + (π − αES) t. The difference π − αES is the “safety loading”. It follows from

187


the strong law of large numbers that π t − At /t → [π − αES] with the probability of 1. Therefore, in the case of positive safety loading, π > αES, we have to expect that the real random profit π t − At for a sufficiently large t would also be positive under the appropriate choice of premium π = (1 + ρ)αES, where ρ is the “relative safety” loading ρ = (π − αES)/αES. However, this holds only if ruin does not occur before time t. As illustrated in Fig. 6.2, despite the fact that sustained growth of risk reserves r t is guaranteed on average, the ruin of the real growth process may occur before sustained growth takes off. In other words, the substitution of the complex jumping process by a simple deterministic model (showing “robust” sustained growth) may lead to unforeseen collapses (surprises). Only a stochastic model is able to estimate the demand in such purely financial risk management measures as borrowing, contingent credits, or governmental bonds. It is also possible to reduce the severity of the distribution of claims by various loss reduction mitigation measures. However, all this is possible only by analyzing the probability of ruin Ψ. In general, various decision variables affect Ψ. The claim size S depends on the coverage of the insurer operating on geographically distinct locations. Important decision variables are r0 , φ, and re–insurance arrangements. The reduction of Ψ to acceptable levels can be viewed as the so–called chance constraint stochastic optimization problem (see Gilber and Gouy 1998; Prekopa 1995). The complexity is associated with the jumping process At with analytically intractable dependencies of At on decision variables, which requires specific stochastic optimization (Sto) methods.

t

r

t

t

t

r

+ = r0

p

-A

(p r = r0 + t

a ES ) t

r0 t

t

Figure 6.2: Expected and real growth of the risk reserve. There is an exit scenario due to an extreme event at time τ , which depletes the safety loading.

188

6.3

T. Ermolieva et al.

Stochastic Integrated Catastrophic Risk Management Model

Section 6.2 briefly outlined some methodological complexities of the catastrophe management model. This Section aims at discussing them with more elements. A model for integrated catastrophe management is a multidisciplinary product combining knowledge of physics, chemistry, engineering, mathematical, financial, social and other scientific disciplines. It consists, in principle, of three major sub–models: a catastrophe module, an engineering vulnerability module, and an economic multi– agent module. A catastrophe module simulates a natural phenomenon based on the knowledge of the event and scientific equations and variables describing it. For example, for a hurricane model variables are the radius of the maximum winds, or forward speed of the storm. For an earthquake model that simulates the shaking of the ground critical variables are the epicenter location, magnitude of the earthquake, and the Gutenberg–Richter law or attenuation characteristics. For a flood, these are precipitation curves, water discharge, river characteristics, etc. The catastrophe models used in Iiasa’s case studies are based on Monte Carlo dynamic simulations of geographically explicit catastrophe patterns in selected regions (a discussion of these models is beyond the scope of this Chapter but can be found in Amendola et al. 2000; Baranov et al. 2002; Ermoliev et al. 2000b; Rozenberg et al. 2001; Walker 1997). A catastrophe model, in fact, compensates for the lack of historical data on the occurrence of catastrophes in locations where the effects of catastrophes may have never been experienced in the past. The engineering module is used to estimate the damages that may result from catastrophes. Shaking intensities, duration of standing water, water discharge speed or wind speeds are what engineering modules take from the catastrophe modules to calculate potential damages. The engineering modules use vulnerability curves taking into account the age of the building and the number of storeys, to estimate the damages induced by the simulated disaster. The economic multi–agent model used in our case studies is a stochastic dynamic welfare growth model (see, e.g., Ermoliev et al. 2000b). This model maps spatial economic losses into gains and losses of agents: central government, a mandatory catastrophe insurance (a catastrophe pool), an investor, “individuals” (cells or regions), producers (farmers), etc., depending on implemented loss mitigating and sharing policy options. Gis–based modeling of catastrophes and vulnerability coupled with multi–agent models, though still limited in use, is becoming increasingly important: to governments and legislative authorities for better comprehension, negotiation and management of risks; to insurance companies for making decisions on the allocation and values of contracts, premiums, re–insurance arrangements, and the effects of mitigation measures; to households, industries, farmers for risk–based allocation of


189

properties and values. These models are gaining popularity in various scientific communities involved in global change and sustainability research. In the case of the insurance industry, for example, it is possible to simulate different patterns of catastrophes in a region as they may happen in reality and analyze their impacts on the stability of the companies or the industry (Daykin et al. 1994; Ermoliev et al. 2000b; Ermoliev et al. 2000a; Ermolieva et al. 2000).

A catastrophe can ruin many agents if their risk exposures are not appropriate. To design safe catastrophic risk management strategies it is necessary to define location specific feasible decisions based on potential losses generated by a catastrophe model. Some of these decisions reduce the frequencies (likelihood) and magnitudes of catastrophic events (say, land use decisions) and redistribute losses and gains on local and international levels (say, pools, insurance, compensation schemes, credits, borrowing). Different catastrophe scenarios in general, lead to different decision strategies. The number of alternative decisions may be very large. Thus, for a region with only 10 possible sub–locations and 10 alternative sizes of insurance coverage, the number of possible combinations is 1010 . The straightforward “if–then” evaluation of all alternatives by calculating the damages of location specific values may easily exceed 100 years.

The important question is how to bypass these limitations of “if–then” analysis and find such combinations of strategies that would be the “best” against all possible catastrophes. By Ermoliev et al. (2000a) it was shown that the search for “robust” optimal decisions can be done by incorporating stochastic spatial adaptive Monte Carlo optimization into catastrophic modeling. This enables the design of desirable robust solutions without evaluating all possible alternatives. Schematically, the model with an embedded optimization procedure is presented in Fig. 6.3. Starting with some initial setting, policy variables are input into the “catastrophe model”. The “catastrophe model” generates catastrophes and induced direct and indirect damages. The efficiency of the policies is evaluated with respect to performance indicators (block “indicators”) of the agents, e.g., insurers, insured, governments, etc. If these do not fulfill the requirements, goals and constraints, they are further adjusted in the block “adaptive feedbacks”. In this manner it is possible to take into account complex interdependencies between damages at different locations, available decisions and resulting losses and claims. A crucial aspect is the use of appropriate risk indicators (measures), e.g., to avoid bankruptcies of agents. Catastrophic losses often have multi–mode distributions and therefore the use of mean values (e.g., expected costs and profits) may be misleading. The model described in the next Section satisfies these requirements and it emphasizes the need for the collective management of catastrophic risks.

190

T. Ermolieva et al.

Outcomes

Adaptive Feedbacks

POLICY VARIABLES

Catastrophe

Direct

Indirect

Model

Losses

Losses

Indicators

Figure 6.3: Schematic representation of the models: adaptive Monte Carlo.

6.4

Spatial Adaptive Monte Carlo Optimization

The insurance model in Section 6.2 has a rather simplified illustrative character. In reality, damages and claims depend on geographical patterns of catastrophes, clustering of property values in the region, available mitigation measures and regulations, and the spread of insurance coverage among different locations. For this purpose, the model should be geographically explicit (see Ermoliev et al. 2000b; Ermoliev et al. 2000a; Ermolieva et al. 2000 and the references therein). A main task in connection with catastrophic risk management is the design of an appropriate loss mitigation and sharing program. A number of crucial questions arise within this analysis, such as: What are optimal ways to alter location–specific risk profiles by modification or/and reinforcement of structural measures such as dikes, reservoirs, irrigation systems, etc? How can the existing situation be adapted, in particular, land use practices, reallocation of capital, etc., to the existing risk profile? What are optimal financial strategies for mitigation and adaptation? Where is the balance between ex–ante “here–and–now” and ex–post “wait–and–see” decisions? Reallocation of properties away from risk prone areas is an important option, but not feasible, however, for many regions (or feasible only within the long–term horizon). These and other questions emphasize the need for collective efforts in managing catastrophes and involving multiple agents (stakeholders), governments, both local and central, planners, insurers, re–insurers, investors, firms, farmers, and individuals. Heterogeneity of the agents induced by mutually dependent losses from extreme events is a key feature of the integrated catastrophe management model. The analysis of possible gains and losses of the agents due to different arrangements of the loss sharing program takes into account the frequency and intensity of hazards, the stock of capital at risk, its structural characteristics, and different measures (in particular, engineering, financial) of vulnerability. Evaluation of appropriate structural mitigation and financial loss sharing measures should, in general, take into account the potential synergy of these two sets of actions: the preventive mitigation measures applied today reduce financial losses and adaptation costs tomorrow. Because of the complex interplays of these effects, we now concentrate only on a financial loss sharing program.

191


The insurance model of Section 6.2 is modified as follows. The study region is subdivided into sub–regions or locations j = 1, 2, . . . , m. Locations may correspond to a collection of households, a zone with similar seismic activity, to a watershed, etc. They may be constituted by a collection of grid cells for meaningful representation of the simulated patterns of events in space and time. We assume that for each location j an estimation Wj of the property value or “wealth” of this location exists, which includes values of houses, lands, factories, etc. Suppose that n agents, i = 1, . . . , n, (insurers, governments, re–insurers, investors) are involved in the loss sharing program. They may have contracts with locations and partially cover their losses. Each agent i has an initial fund or a risk reserve Ri0 that in general depends on magnitudes of catastrophic events. Assume that the planning horizon covers t = 0, 1 . . . , T − 1 time intervals. The risk reserve Rit at each t is calculated according to the following formula: Rit+1 = Rit +

m X j=1

where i = 1, 2, . . . , n, q t =

n

X t t , (q t ) − ctij (q t ) − Ltj (ωt )qij πij

(6.1)

j∈εt (ωt )

o t , i = 1, n, j = 1, m , q t is the coverage of a comqij ij

t (q t ) is the premium from contracts characterpany i in location j at time t, πij n o t t = 1), and ct (q t ) ized by coverage qij (full coverage of losses corresponds to qij ij

are transaction costs or administrative, running or other costs. Ltj (ωt ) is the loss (damage) at j caused by a catastrophe ωt at time t. Random catastrophic events ω = (ω0 , . . . , ωT −1 ) may affect a random number of different locations. In general, a catastrophic event at time t is modeled by a random subset εt (ω) of locations j and its magnitude in each j. The losses Ltj (ωt ) depend on the event ωt , mitigation measures (e.g., dikes against flooding or specific building codes in seismic areas), and vulnerability of property values in j. t and π t (q t ) allow the characterization of the differences in risks at Variables qij ij different locations. It is assumed that all agents may cover different fractions of t interconnect the processes catastrophic losses from the same location. Variables qij t (q t ), ct (q t ) with each other. Inflows of premiums push of Rit , i = 1, 2, . . . , n (e.g., πij ij t trajectories of Ri up, whereas claims and transactions costs push them down. In the case of a catastrophe, a location j faces losses (damages) Ltj . Individuals t from company i when such a loss occurs, at this location receive compensation Ltj qij t t and pay insurance premiums πij (q ). If Wj0 is the initial wealth (property value), then the location’s j wealth at time t + 1 equals: Wjt+1 = Wjt +

n X i=1

t t (Ltj qij − πij (q t )) − Ltj ,

t = 0, 1, . . . .

(6.2)

Equations 6.1 and (6.2) represent rather general processes of accumulation. Depending on the interpretation, they describe the accumulation of reserve funds, the

192

T. Ermolieva et al.

dynamics of contamination, or the processes of economic growth with random disturbances (shocks), the reserves of the insurance company at time t, the gross national product of a country or the accumulated wealth of a specific region. In more general cases, when catastrophes may have profound effects on economic growth (can “move” the economy, see, for example, Manne and Richels 1992; Nordhaus 1993), this specification can be generalized to an economic–demographic model (MacKellar and Ermolieva 1999) representing both the movements of individuals and the capital accumulation processes within the economy. Let us denote the decision variable by a vector x, which includes components of coverage q and feasible mitigation measures. For each insurer (agent) i consider a stopping time τi for process Rit (x, ω), i.e., a random variable with integer values t = 0, T . The event {ω : τi = t} with fixed t corresponds to the decision to stop process Wit (x, ω) after time t. of τi may be the time Examples of the ruin before a t given time T :τi (x, ω) = min T, min t : Ri (x, ω) < 0, t > 0 (in which case τi is a rather complex implicit function of x) or the time of the first catastrophe, t = τ . Assume that each agent, i, and location, j, maximize their “wealth” at t = τ , i.e., they are concerned with the resilience against possible catastrophes. In general, the notion of wealth at t requires an exact definition (see, e.g., Ginsburg and Keyzer 1997), as it must represent in a sense the whole probability distribution of R"iτ , Wiτ . The performance # of insurance depends on whether the accumulated fund h i τ m P P t t P τ . Thus, insurers πij (q ) − ctij (q t ) is able to cover claims Lτj (ωτ )qij t=1

j=1

j∈ετ (ωτ )

will maximize their wealth, which depends on the (random) balance of income and payments:   τ m X X X t τ  ϑτi = (q t ) − ctij (q t )  − πij . Lτj (ωτ )qij t=1

j=1

j∈ετ (ωτ )

The stability of an insurer is determined by the probability of the event E1 = {ϑτi < 0} .

(6.3)

Individuals (locations) maximize their wealth, which depends on whether the amount of premiums that they pay to the insurer does not exceed the compensation of losses at time t = τ : τ X τ τ τ πij (q k ) − Lτj (ωτ )qij . vj = t=0

Therefore, the “financial” stability of locations depends on the probability of the event E2 = {νiτ < 0} . (6.4)

Inequalities (6.3)–(6.4) define important events, constraining the choice of decision variables. The probability of events (6.3)–(6.4), i.e., underpayments to insurers and

193


over–payments by individuals, determine the stability (resilience) of the scheme. This can be expressed in terms of the probabilistic constraint P [E1 , E2 ] 6 p,

(6.5)

where p is a desirable probability threshold of the program’s failure (default) that occurs, say, only once in 100 years. Constraint (6.5) is similar to an insolvency constraint (Ermoliev et al. 2000a), a standard for regulations of the insurance business (see also Section 6.2). In stochastic optimization (Ermoliev and Wets 1988; Ermolieva et al. 2000), it is known as the so–called chance constraint. Note, however, that this constraint does not account for the attained values of E1 and E2 . The main goal in setting up the insurance scheme P can now be formulated as the minimization of expected total losses F (x) = E (1 − qij )Lτj including uncovered (un–insured) ij

losses by the insurance scheme subject to chance constraint (6.5), where vector x, in the most simple example, consists of the components πij and qij . Constraint (6.5) imposes significant methodological challenges even in cases when τ (x) does not depend on x and events 6.3–6.4 are defined by linear functions of decision variables (see discussion in Ermoliev and Wets 1988, p. 8, and in Ermoliev et al. 2000b; Ermoliev et al. 2000a). This constraint is of “black–and–white” character, i.e., it accounts only for a violation of (6.3)–(6.4) but not for the magnitude of violation. There are important connections between the minimization of F (x) subject to highly nonlinear and possibly discontinuous chance constraints (6.5) and the minimization of convex functions, which have important economic interpretations. Consider the following function: X X E max 0, νjτ E max {0, ϑτi } + β G(x) = F (x) + α (6.6) i

j

where α, β are positive parameters. It is possible to prove (see general results in Ermoliev et al. 2000b; Ermoliev et al. 2000a) that for large enough α, β the minimization of function G(x) generates solutions x with F (x) approaching the minimum of F (x) subject to 6.5 for any given level p. The minimization of G(x), as defined by eq. (6.6), has a simple economic interpretation. Function F (x) comprises expected direct losses associated with the insurance program. The second term quantifies the expected shortfall of the program to fulfill its obligations; it can be viewed as the expected amount of ex–post borrowing with a fee α needed for this purpose. Similarly, the third term can be interpreted as the expected ex–post borrowing with a fee β needed to compensate over–payments. Obviously, large enough fees α, β will tend to preclude the violation of (6.3)–(6.4). Thus, ex–post borrowing with large enough fees allows for a control of the insolvency constraints (6.5). In our model, for the adaptive search of optimal robust solutions we apply stochastic quasi–gradient methods (adaptive Monte Carlo optimization). These methods can be used for the maximization of the non–smooth function G(x) by using

194

T. Ermolieva et al.

only sample performance functions f (q, π, ω), G(x) = Ef (q, π, ω), X X X f (q, π, ω) = (1 − qij )Lτj (ωτ ) + α max {0, ϑτi (ωτ )} + β max 0, νjτ (ωτ ) . ij

i

j

Detailed descriptions of the method can be found in (Ermoliev and Wets 1988; Ermoliev et al. 2000b; Ermoliev et al. 2000a). Here, we illustrate its application rather schematically only with respect to some components of x, namely, for decisions on contracts q = {qij }. A sequence of approximate solutions q 0 , q 1 , . . . is generated directly by using statistical estimates (stochastic quasi–gradients for non–smooth function G(x)) of grad W (q, ω) without approximating W (q) by an explicit function. The adaptive search procedure is defined as follows. Let q 0 be an initial guess, and q k is the approximate solution after k steps. Then q k+1 = Pr j(q k + ρk ξ k ) , k = 0, 1, . . . ,

(6.7)

where ρk > 0 is (q 0 , q 1 , . . . q k )–measurable random variable (“step–size” multiplier 0 1 k k 0 1 k depending k on0 (q1 , q , .k.. q ))), ξ is k(q , q , . . . q )–measurable random vector such that E ξ |q , q , . . . q − gradW (q ) → 0, k → ∞. Stochastic quasi–gradients ξ k are often defined at each step k = 0, 1, . . . by using only one independent sample ω k . The symbol Pr j(y) defines the projection of y onto a feasible set Q, i.e., it is the point from Q minimizing the distance to y n o Pr j(y) = arg min kz − yk2 : z ∈ Q . The projection of q k + ρk ξ k (calculation of q k+1 ) is a very fast operation when it starts from q k . The current approximate decision variables q k = {qijkt } are adaptively adjusted according to feedback (6.7). Since function W (q) is not concave even for t (·), ct (·) (due to, e.g., dependence of τ on x), the random sequence q k convex πij ij generated according to (6.7) may not converge to a global solution. The choice of step–size–multipliers ρt in (6.7) satisfies conditions ρt > 0,

∞ X t=0

ρt = ∞,

∞ X t=0

ρ2t < ∞.

For example, ρt = Ct /t, where 0 6 C t 6 Ct 6 C t < ∞ ensures the convergence of k G(q ) to a local maximum value with probability 1 in practically all important cases. The random adjustment mechanism (6.7) has the ability to bypass local solutions. Global convergence can be achieved by introducing “shocks” when the sequence q k shows a steady–state tendency. The important feature of the functions t (q t ), ct (q t ) and τ independent of x, this function is con(6.6) is that for convex πij ij cave in contrast to the direct use of (6.5). This was achieved by a special choice of risk indicators in the definition of (6.3)–(6.5).


6.5

195

Case Studies: Earthquake Risks Management

In this Section we simplify the model of Section 6.4 to illustrate its applicability using modified but realistic data from a case study of seismic risks in the Tuscany region of Italy. The main question in connection with these catastrophes is by whom and in what way should the losses be covered? Until recently, losses from natural catastrophes were mainly absorbed by the victims and their governments (Ermoliev and Wets 1988; Froot 1997; Linnerooth-Bayer and Amendola 2000). The insurance industry (and investors) absorbs only a small portion of the losses. Currently, as losses increase, governments are concerned with escalating costs for disaster prevention, response, compensation to victims, and public infrastructure repair. It is important to create incentives for individuals and local governments to increase their responsibility in dealing with extreme events and their impacts and mitigation. Local governments may be more effective in the evaluation and enforcement of loss–reduction and loss–spreading measures, but this is possible only through location–specific analysis of potential losses, of the mutual interdependencies of these losses, and of the sensitivity of location–specific losses to new land use and other risk management strategies. An important consideration for national insurance strategies is linking private insurance with mitigation measures to reduce losses. Insurers, however, are reluctant to enter markets that expose them to the risk of bankruptcy. In the Usa, for example, many insurers pulled out of catastrophic risk markets in response to their large losses from natural catastrophes in the last decade (Iso 1994). In Italy, a law for integrating insurance in the overall risk management process was proposed only in late 1997 (within the Design of Law 2793: “Measures for the Stabilization of Public Finance”). This opened a debate on various policy options for national insurance strategies. The aim of this Section is to analyze a possible insurance program, which is “robust” against earthquakes and spreads the loss burden among multiple agents (e.g., private insurance, the financial market (contingent credit), the government, and individuals), taking into account their individual goals and constraints with respect to the catastrophes. The program allows for the transfer of some share of the loss via ex–post arrangements (borrowing). In our example, we use a catastrophe model based on a random earthquake and damage generator (see Amendola et al. 2000; Baranov et al. 2002; Ermoliev et al. 2000b; Rozenberg et al. 2001). Its main idea is the application of the Gutenberg– Richter law connecting the magnitudes of events with the random timing of their occurrence for a given set of seismic faults. The generator is not aimed at prediction, rather its purpose is to create a tool generating potential earthquake scenarios, which can be easily embedded into a stochastic optimization (Adaptive Monte Carlo) model discussed in Section 6.4 for the analysis of robust strategies. The earthquake and damage generator creates samples from multidimensional damage distributions

196

T. Ermolieva et al.

of catastrophe losses, based on results of modeling geophysical processes, geophysical data and a model of vulnerability of buildings in the pilot region. Geophysical data are given by a map of geo–tectonic structure of the region (Petrini 1995). Historical data are available as a map of seismic activity in the region and a map of maximum observed macro–seismic activity (http://www.dstn.pcm.it/ssn/). Figure 6.4 presents one of the earthquake scenarios. Specific attenuation characteristics and other data on the region were provided by the Seismic Risk Research Institute (Petrini 1995). The generator, in fact, can be easily adapted to other regions, incorporating different event distributions, non–Poissonian catastrophic processes, as well as micro zoning within modeled locations. For analyzing of loss management strategies and according to the general model of Section 6.4, the Tuscany region was subdivided into locations (grid–cells) – cf. Fig. 6.5 – which were further aggregated into M ≈ 300 sub–regions, meaningful for decision making and corresponding to the number of municipalities. For each municipality j the number and types of buildings, their vulnerability, and number of built cubic meters represented the so–called estimate of “wealth” Wj . Simulated in time and space, earthquakes ω0 , . . . , ωt occurred at different municipalities, inside or outside the region, had random magnitudes and, therefore, affected a random number of municipalities. The economic losses of destroyed cubic meters of buildings were defined as the cost for their reconstruction. Obviously, the reinforcement of a building’s environment would reduce the losses. This fact was adequately reflected in the model, which allowed the interplay between mitigation measures and risk-sharing mechanisms (insurance, re–insurance, financial markets) to be studied. The risk sharing and spreading scheme analyzed in this Chapter involves pooling of risks through mandatory insurance (this may be a pool of companies or the government itself acting as an insurer) based on location–specific exposures, partial compensation to the victims by the central government, and a contingent credit to the pool. This scheme encourages the accumulation of regional capital to better “buffer” against the volatilities of international re–insurance markets. We assumed that the insurance company covers a fraction qj , 0 6 qj 6 1 (e.g., qj = 0.75) of losses. The government compensates a portion ν. Thus, in the case of excessive losses the state could be severely affected. Let Lτj be random losses at municipality j at time t = τ when a catastrophe occurs. The accumulated P mutual catastrophe fund at time τ including the proportional compensation ν Lτj by the government amounts to j P P P τ πj + ν Lτj − qj Lτj . Thus, in this model we assume that the compensation j

j

j

to victims by the government is paid through the mandatory insurance. The stability of the insurance program depends on whether the accumulated mutual fund together with the government compensation is able to cover claims, i.e., on the probability of event: X X X e1 = τ πj + ν Lτj − qj Lτj > 0 . (6.8) j

j

j


Figure 6.4: Earthquake generator.

197

198

T. Ermolieva et al.

Figure 6.5: Geographically–explicit subdivision of the Tuscany regions in Italy.

199


The stability also depends on the willingness of individuals to accept premiums, i.e., on the location–specific probability of over–payments: e2 = τ πj − qj Lτj > 0, Apart from compensation ν

P j

j = 1, . . . , m.

(6.9)

Lτj , the government arranges a contingent credit y

with a fee ϕ to improve the stability of the mandatory insurance (pool) by transforming event (6.8) into (6.10): X X X e3 = τ πj + ν Lτj − qj Lτj + y − τ ϕy > 0. (6.10) j

j

j

Constraints (6.3) and (6.4) describe the burden sharing within the program. Here, we assume that the mandatory insurance pays the fee τ qy and receives a credit y, whereas the government repays the credit with the interest rate γ y, γ > 1. The choice of premiums drives the process of insurance supply–demand: if premiums are too high, (event (6.9) occurs too often) the willingness to participate in the program will be low. On the other hand, if premiums are not sufficient to guarantee inequality (6.8) with some acceptable probability, the stability of the mandatory insurance cannot be sustained. Alternative policy options for the choice of premiums can be proposed by Stakeholders (i.e., insurers, local governments, individuals). Some of them may correspond to the standard actuarial approaches to calculate the premiums based on the expected losses. In the outlined numerical experiments we consider the following three rules: 1. Premiums based on the average damage over all of the municipalities (solidarity principle, making less exposed locations pay premiums equal to those more severely exposed). 2. Location–specific premiums based on average damage in a particular municipality, i.e., risk–based premiums. The use of average losses may be misleading in the case of heavily tailed distributions that are typical for catastrophic losses. The stochastic optimization risk management model outlined in Section 6.4 allows calculation of premiums taking into account sustainability indicators of insurers and the state, constraints on individual incomes, willingness to overpay premiums, etc. These model–based robust premiums were stipulated as the third policy option: 3. Premiums that fairly equalize the risk of instability for the insurance company (the insurer may become bankrupt only once in 1 000 years) and the risk of premium overpayment for exposed municipalities (municipalities overpay premiums only once in 100 years).

200

T. Ermolieva et al.

Figure 6.6: (a) Distribution of municipality specific premiums, textitoption 2 (per m3 building volume/municipality in %). (b) Distribution of “fair” premiums, option 3 (per m3 building volume/municipality in %).

Besides premiums, the solution of the model also provides optimal policy options on location–specific coverage and the amount of governmental compensation and the contingent credit. Here we present results only relating to premium policies. This is the most interesting case for analyzing catastrophic exposures since optimal robust policies can be significantly different from the standard actuarial approaches. Thus, Figs. 6.6–6.7 show considerable differences between options 1, 2 and option 3. For option 1, where the burden of losses is equally distributed over all locations, the simulation of catastrophic losses showed that the annual premium is equal to the flat rate of 0.02 monetary units (m.u.) per cubic meter of building. For option 2, the distribution of municipality specific premiums based on average damage in each municipality (or according to the municipality specific risk) is shown in Fig. 6.6a. There are a prevailing number of municipalities (about 220) that have to pay 0.02–0.03 m.u., which is close to the flat rate of 0.02, as in option 1. About 20 municipalities are at no risk at all (0 rate). Municipalities that are more exposed to risk have to pay 0.04 and higher rates (more than 50 municipalities). Figure 6.7a shows the distribution of the insurers’ reserve (cumulated at τ within 50 years) for options 1, 2 premiums. The probability of insolvency, i.e. when the risk reserve accumulated until the catastrophe is not enough to compensate incurred losses, is also shown. This distribution, in fact, generates the demand for contingent credit. There is a rather high probability of “small” insolvency (values -90, -40 occurred 190 and 90 times out of 500 simulations). High over–payments (more than 500 m.u.) occurred in about 10 per cent of the simulations. The size of insolvency would represent the cost to the government of covering the losses uncovered by the pool (e.g., by using a possibility to transfer a fraction of the losses to international financial markets, as analyzed in Ermolieva et al. 2000). Figure 6.6b shows the distribution of premiums for option 3, i.e., robust optimal premiums. According to


201

Figure 6.7: (a) Distribution of insurer’s reserve, options 1, 2 (thousands m.u., 50 years). (b) Distribution of insurers’ reserve, option 3 (thousands m.u., over 50 years).

this principle, most of the municipalities (190) have to pay close to the flat rate of 0.02–0.03 m.u./m3 of a building. Rates of 0.04 and higher have to be paid by about 100 municipalities. In this case, the highest premium rate is 0.5, which is much lower when compared to the highest rate of 1.2 of option 2. The distribution of the insurer’s reserve in Fig. 6.7b also indicates the improvement of the insurer’s stability — the frequency of insolvency is considerably reduced. Figure 6.8 provides an illustrative comparison of the three premium options analyzed. For each municipality, it shows the optional premiums to be paid at the flat premium rate of 0.02, the option 2 municipality specific rate, and the “fair” robust

202

T. Ermolieva et al.

Figure 6.8: Comparison of options: municipality specific, “fair” and flat (0.02) premiums.

premium of option 3. Many municipalities in all three options have to pay a premium rate, close to the flat rate (0.015–0.03). For quite a number of municipalities in option 2, the rate significantly exceeds the flat rate. For these municipalities, special attention should be given to whether they are able to pay such high premiums. Option 3 allows taking individual constraints on over–payments into account and working out efficient premiums both for the insurer and the municipalities.

6.6

Conclusion

Managing catastrophic risks is a complex multidisciplinary problem that requires joint efforts of multiple agents and appropriate risk–based approaches to the design of efficient and sustained land use practices, loss sharing and mitigating programs. The risks are characterized by the lack of appropriate knowledge on their occurrence and magnitudes, heavily tailed distributions of losses that come abruptly in time and space. They are rare and hardly predictable (therefore, often unaccounted for), costly and unevenly distributed. They cannot be treated on average, and require specific risk measures. In this Chapter we demonstrate that coping with catastrophes should rely on comprehensive model–based approaches combining natural, engineering, financial and social systems. The model can be regarded as a mitigation tool to analyze the risks and reduce or share them. The types of models presented here are capable of representing a variety of stakeholder groups in an agent–based setting. They allow the analysis of various ex–ante burden sharing arrangements while paying attention to other measures of the disaster cycle (mitigation, preparedness, response, recovery) and assign responsibilities as well as analyze trade–offs in close cooperation with stakeholders.


203

In Section 6.5 we illustrate the applicability of the discussed approach for a seismic prone area. The same type of approach is applicable to a variety of catastrophic risk management situations, e.g., floods, windstorms or even outbreaks of diseases. The key issue in these cases is the specification of catastrophe models and the risk reduction and risk spreading decisions. While the risk spreading (financial) part of the model may be similar for quite different natural and human–made catastrophes, the catastrophe model and the risk reduction (mitigation) measures may be quite different for different case studies. In most cases a key issue is the change of prevailing land use practices, as a causative factor on the one hand, or as a risk–spreading adaptation and risk–reducing mitigation measure on the other, e.g., deforestation, afforestation, spatial patterns of settlements, livestock allocation and densities, etc., towards more resilient, risk–based policies.

References Amendola, A., Y. Ermoliev, and T. Ermolieva (2000). Earthquake risk management: A case study for an Italian region. In Proceedings of the Second Euro Conference on Global Change and Catastrophe Risk Management: Earthquake Risks in Europe, Laxenburg. International Institute for Applied Systems Analysis. Amendola, A., Y. Ermoliev, T. Ermolieva, V. Gitits, G. Koff, and J. Linnerooth-Bayer (2000). A systems approach to modeling catastrophic risk and insurability. Natural Hazards Journal 21 (2-3), 381–393. Baranov, S., B. Digas, T. Ermolieva, and V. Rozenberg (2002). Earthquake risk management: Scenario generator. Technical Report Interim Report IR-02-025, International Institute for Applied Systems Analysis, Laxenburg. Borch, K. (1962). Equilibrium in a reinsurance market. Econometrica 30 (3), 424–444. Daykin, C., T. Pentikainen, and M. Pesonen (1994). Practical Risk Theory for Actuaries. Monographs on Statistics and Applied Probability, Volume 53. London: Chapman and Hall Ltd. Ermoliev, Y., T. Ermolieva, G. MacDonald, and V. Norkin (2000a). Insurability of catastrophic risks: The stochastic optimization model. Optimization Journal 47, 51–265. Ermoliev, Y., T. Ermolieva, G. MacDonald, and V. Norkin (2000b). Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks. Annals of Operations Research 99, 207–225. Ermoliev, Y. and R. Wets (Eds.) (1988). Numerical Techniques of Stochastic Optimization: Computational Mathematics, Berlin: Springer-Verlag. Ermolieva, T., G. Fischer, and M. Obersteiner (2000). Integrated modeling of spatial and temporal heterogeneity induced by catastrophic events. Technical Report Interim Report IR-03-023, International Institute for Applied Systems Analysis, Laxenburg. Fischer, G., M. Shah, and H. van Velthuizen (2002). Climate change and agricultural vulnerability. Report commissioned by the United Nations for the Johannesburg Summit, United Nations, Johannesburg. Available on the Internet: http://www.iiasa.ac.at/Research/LUC/JB-Report.pdf.

204

T. Ermolieva et al.

Froot, K. (1997). The limited financing of catastrophe risk: and overview. Technical report, Harvard Business School and National Bureau of Economic Research. Gilber, C. and C. Gouy (1998). Flood Response and Crisis Management in Western Europe: A Comparative Analysis, Chapter Flood Management in France. Berlin: Springer-Verlag. Ginsburg, V. and M. Keyzer (1997). The Structure of Applied General Equilibrium Models. Cambridge: MIT Press. Grandell, J. (1991). Aspects of Risk Theory. Springer Series on Statistics: Probability and Its Applications. New York: Springer-Verlag. Iso (1994). The impact of catastrophes on property insurance. Technical report, Insurance Service Office (Iso), New York. Linnerooth-Bayer, J. and A. Amendola (2000). Global change, catastrophic risk and loss spreading. The Geneva Papers on Risk and Insurance 25/2. MacKellar, L. and T. Ermolieva (1999). The Iiasa social security project multiregional economic-demographic growth model: Policy background and algebraic structure. Technical report, International Institute for Applied Systems Analysis, Laxenburg. Manne, A. and R. Richels (1992). Buying Greenhouse Insurance: The Economic Costs of CO2 Emission Limits. Cambridge: Mit Press. Milly, P., R. Wetherald, K. Dunne, and T. Delworth (2002). Increasing risk of great floods in changing climate. Nature 415, 512–514. MunichRe (1999). Climate change and increase in loss trend persistence. Press release, MunichRe, Munich. Nordhaus, W. (1993). Rolling the Dice: An optimal transition path for controlling greenhouse gases. Resource and Energy Economics 15, 27–50. Nrc (1999). National Disaster Losses: A Framework for Assessment. Washington: National Academy Press. Palmer, T. and J. Raisanen (2002). Quantifying the risk of extreme seasonal precipitation events in a changing climate. Nature 415, 512–514. Petrini, V. (1995). Pericolosita’ sismica e prime valutazioni di rischio in Toscana (seismic hazards and preliminary risk evaluation in Tuscany). Technical report, Cnr/Irrs, National Research Council/Seismic Risk Research Institute, Milan. Prekopa, A. (1995). Stochastic Programming. Dordrecht: Kluwer Academic Publishers. Rozenberg, V., T. Ermolieva, and M. Blizorukova (2001). Modeling earthquakes via computer programs. Interim report IR-01-068., International Institute for Applied Systems Analysis, Laxenburg. Walker, G. (1997). Current developments in catastrophe modelling. In N. Britton and J. Oliver (Eds.), Financial Risks Management for Natural Catastrophes, Brisbane. Griffith University.

CHAPTER 7

Genetic Algorithms and Their Applications in Environmental Sciences Sue Ellen Haupt

7.1

Introduction

The genetic algorithm (Ga) is a useful tool for solving problems requiring search and optimization. It has found wide acceptance in many fields, ranging from economics through engineering. In the environmental sciences, some disciplines are using Gas regularly as a tool to solve typical problems; while in other areas, they have been barely assessed for use in research projects. The purpose of this paper is to introduce the elements of Gas, review their applications to environmental science problems, to present a couple of simple examples of how a Ga might be applied to a problem, and to suggest how they may be useful in future research directions. The genetic algorithm is an optimization tool that mimics natural selection and genetics. The parameters to be optimized are the genes, which are strung together in an array called a chromosome. A population of chromosomes is created and evaluated by the cost function (sometimes know as an objective function), with the ‘most fit’ chromosomes being kept in the population while the ‘least fit’ ones are discarded. The chromosomes are then paired so they can mate, combining portions of each chromosome to produce new chromosomes. Random mutations are imposed. The new chromosomes are evaluated by the objective, or cost function and the process iterates. Thus the parameter space is explored by combining parts of the best solutions as well as extending the search through mutations. The trade-offs involved in selecting population size, mutation rate, and mate selection techniques are discussed in more detail below. The key to using Gas in environmental sciences is to pose the problem as one in optimization (for an example how genetic algorithms can be applied also in complex methodological frameworks, see Chapter 9 in this monograph). Many problems are quite naturally optimization problems, such as the many uses of inverse models in the environmental sciences. Other problems can be manipulated into optimization form by careful definition of the cost function, so that even non-linear differential equations can be approached using Gas (Haupt 2003). Gas are well suited to many optimization problems where more traditional methods fail. Some of the advantages they have over conventional numerical optimization algorithms are that they:

206

S. E. Haupt • Optimize with continuous or discrete parameters, • Do not require derivative information, • Simultaneously search from a wide sampling of the objective function surface, • Deal well with a large number of parameters, • Are well suited for parallel computers, • Optimize parameters with extremely complex objective function surfaces, • Provide a list of semi-optimum parameters, rather than a single solution, • Work with numerically generated data, experimental data, or analytical functions.

These advantages outweigh the Gas’ lack of rigorous convergence proofs and slow convergence for some problems. In the following Sections we give a short overview of how the Ga works, briefly review some of the ways that Gas have been used in environmental science, and present a couple example applications that demonstrate the strength of the Ga.

7.2

Introduction to Genetic Algorithms

John Holland is often referred to as the ‘father of genetic algorithms.’ He developed this brand of genetic programming during the 1960’s and 1970’s and his work is described in his book (Holland 1975; Holland 1992). His student, David Goldberg, popularized the method by solving a difficult problem involving the control of gaspipeline transmission for his dissertation (see Goldberg 1989). Since that time, they have been applied to a wide variety of problems. Many applications encode the parameters as binary strings. This approach is particularly useful when parameters represent a list of different possibilities rather than real numbers. Many of the applications in environmental science have been done using binary encoding. Here, we emphasize the use of continuous parameter Gas, which represent the parameters using real numbers. The discussion follows the flow chart of Fig. 7.1. The real numbers are put in an array known as a chromosome. The Ga begins with a population of chromosomes which are fed to the cost function for evaluation. The fittest chromosomes survive while the highest cost ones die off. This process mimics natural selection in the natural world. The most fit (lowest cost) survivors mate. The mating process combines information from the two parents to produce offspring. Some of the population experiences mutations. The mating and mutation operators introduce new chromosomes which may have a lower cost than the prior generation. The process iterates until an acceptable solution is found.

Genetic Algorithms and Their Applications

207

Random Initial Population of Continuous Parameters

Natural Selection Mating Mutation Convergence Check done Figure 7.1: Flow chart of a continuous genetic algorithm.

7.2.1

Creating the Population

The first step of a continuous parameter genetic algorithm is creating the population of chromosomes. The array of real parameters is known as a chromosome, chromosome = p1 p2 · · · pα · · · pNpar (7.1)

where the pi are the parameters and there are a total of Npar parameters. The parameters are simply floating point numbers. The encoding function keeps track of which gene represents which parameter and assuring they are within given bounds. A population of such chromosomes is created using a random number generator so that the chromosome arrays are gathered together in a two dimensional matrix. Once the chromosomes have been created, their cost or fitness is evaluated by the cost or objective function, which is very problem specific. The lowest cost chromosomes (Nkeep ) remain in the population while the higher ones are deemed less fit and die off. The reduced population is then the portion of the population available for mating.

7.2.2

Choosing the Mates

There are a variety of methods to pair the chromosomes for mating. Some popular methods are reviewed by Haupt and Haupt (1998). Here, we discuss pairing the

208

S. E. Haupt

chromosomes according to numerical rank. For this pairing methodology, after the cost function evaluation, the chromosomes are sorted in order from lowest cost to highest. That is, the nth chromosome will have a probability of mating of: Pn =

Nkeep − n + 1 NP keep

(7.2)

k

k=1

Then the cumulative probabilities are used for selecting which chromosomes mate.

7.2.3

Crossover

Once two parents are chosen, some method must be devised to produce offspring which are some combination of these parents. Many different approaches have been tried for crossing over in continuous parameter genetic algorithms. Adewuya (1996) reviews some of the methods thoroughly and several interesting methods are demonstrated by Michalewicz (1992). Haupt and Haupt (1998) give an overview of some popular methods. The method used here is a combination of an extrapolation method with a crossover method. We wanted to find a way to closely mimic the advantages of the binary genetic algorithm mating scheme. It begins by randomly selecting a parameter in the first pair of parents to be the crossover point. α = roundup {random × Npar } We will let

parent1 = pm1 pm2 · · · pmα · · · pmNpar parent2 = pd1 pd2 · · · pdα · · · pdNpar

(7.3)

(7.4)

where the m and d subscripts discriminate between the mom and the dad parent. Then the selected parameters are combined to form new parameters that will appear in the offspring: pnew1 = pmα − β [pmα − pdα ] (7.5) pnew2 = pdα − β [pmα − pdα ] where β is also a random value between 0 and 1. The final step is to complete the crossover with the rest of the chromosome as before: offspring1 = pm1 pm2 · · · pnew1 · · · pdNpar (7.6) offspring2 = pd1 pd2 · · · pnew2 · · · pmNpar

If the first parameter of the chromosomes is selected, then only the parameters to the right of the selected parameter are swapped. If the last parameter of the chromosomes is selected, then only the parameters to the left of the selected parameter are swapped. This method does not allow offspring parameters outside the bounds set by the parent unless β is greater than one. In this way, information from the two parent chromosomes is combined in a way that mimics the crossover process during meiosis.


7.2.4

209

Mutations

If care is not taken, the genetic algorithm converges too quickly into one region of the cost surface. If this area is in the region of the global minimum, that is good. However, some functions have many local minima and the algorithm could get stuck in a local well. If we do nothing to solve this tendency to converge quickly, we could end up in a local rather than a global minimum. To avoid this problem of overly fast convergence, we force the routine to explore other areas of the cost surface by randomly introducing changes, or mutations, in some of the parameters. A mutated parameter is replaced by a new random parameter. Choosing the Ga parameters is more of an art than a science. The best mutation rate is dependent on the population size and both of these vary with the type of problem. Haupt and Haupt (2000) find that, in general, relatively low population sizes (on the order of 10’s) combined with relatively large mutation rates (on the order of 0.1 to 0.4) result in fewer evaluations of the cost function required for convergence. Many real world optimization problems have multiple objectives. Multiple objectives can be handled by weighting and adding the fitness from each objective. Multi-objective optimization does not typically have a single optimum solution relative to all objectives. Instead, there are a set of optimal solutions, known as Pareto-optimal or non-inferior solutions. A Pareto Ga attempts to find as many Pareto-optimal solutions as possible, since all these solutions have the same cost. Some of the applications discussed below use Pareto Gas.

7.3

Uses of Genetic Algorithms in Environmental Science Problems

There is a recognized need for better methods of optimization and search in the environmental sciences. For instance, many different problems involve fitting a model to observed data. Sometimes the data is a time series while other times it is an observed environmental state. Often, some general functional forms are known or surmised from the data. Frequently, the goal is to fit model parameters to optimize the match between the model and the data. Practitioners often go the next step and use the model to make predictions. The need for new tools involving artificial intelligence (Ai) techniques, including genetic algorithms, is noted by Hart et al. (1998) among others. One example of fitting a model to observed data using a Ga is reported by Mulligan and Brown (1998). They use a Ga to estimate parameters to calibrate a water quality model. They used non-linear regression to search for parameters that minimize the least square error between the best fit model and the data. They found that the Ga works better than more traditional techniques plus noted the added advantage that the Ga can provide information about the search space, enabling them

210

S. E. Haupt

to develop confidence regions and parameter correlations. Some other work related to water quality includes using Gas to determine flow routing parameters (Mohan and Loucks 1995), solving ground water management problems (McKinney and Lin 1993; Rogers and Dowla 1994; Ritzel et al. 1994), sizing distribution networks (Simpson et al. 1994), and calibrating parameters for an activated sludge system (Kim et al. 2002). In fact the use of Gas and more general evolutionary algorithms have become quite prevalent in the hydrological area and there are a plethora of applications in the literature. Most recently, it has progressed into development of very efficient methods for finding Pareto fronts of optimal multi-objective criteria to best estimate parameters of hydrological models (Vrugt et al. 2003). Managing groundwater supplies has found Ai and Gas useful. Peralta and collaborators have combined Gas with neural networks and simulated annealing techniques to combine the advantages of each. Aly and Peralta (1999a) used Gas to fit parameters of a model to optimize pumping locations and schedules for groundwater treatment. They then combined the Ga with a neural network (Nn) to model the complex response functions within the Ga (Aly and Peralta 1999b). Shieh and Peralta (1997) combined simulated annealing (Sa) and Gas to maximize efficiency and applied parallel nature of the Ga. More recently, Fayad (2001) together with Peralta used a Pareto Ga to sort optimal solutions for managing surface and groundwater supplies, together with a fuzzy-penalty function while using an artificial neural network (Ann) to model the complex aquifer systems in the groundwater system responses. Hilton and Culver (2000) have also used Gas to optimize groundwater remediation design. Another example is the successful application of a Ga to classification and prediction of rainy day versus non-rainy day occurrences by Sen and Oztopal (2001). They used the Ga to estimate the parameters in a third order Markov model. McCullagh and Bluff (1999) combined Gas with neural networks for rainfall estimation. An example from geophysics is determining the type of underground rock layers. Since it is not practical to take core samples of sufficient resolution to create good maps of the underground layers, modern techniques use seismic information or apply a current and measure the potential difference which gives a resistance. These various methods produce an underdetermined multi-modal model of the Earth. Fitting model parameters to match the data is regarded as a highly non-linear process. Genetic algorithms have found recent success in finding realistic solutions for this inverse problem (Jervis et al. 1996; Jervis et al. 1996; Sen and Stoffa 1992b; Sen and Stoffa 1992a; Chunduru et al. 1995; Chunduru et al. 1997; Boschetti et al. 1995; Boschetti et al. 1996; Boschetti et al. 1997; Porsani et al. 2000), among others. Minister et al. (1995) find that evolutionary programming is useful for locating the hypocenter of an earthquake, especially when combined with simulated annealing. Another inverse problem is determining the source of air pollutants given what is known about monitored pollutants. Additional information includes the usual combination (percentages) of certain pollutants from different source regions and

211


predominant wind patterns. The goal of the receptor inverse models is to target what regions, and even which sources contribute the most pollution to a given receptor region. This process involves an optimization. Cartwright and Harris (1993) suggest that a genetic algorithm may be a significant advance over other types of optimization models for this problem when there are many sources and many receptors. Loughlin et al. (2000) combined Gas with ozone chemistry air quality models to allocate control strategies to avoid exceeding air quality standards. A demonstration of this type of model appears in Section 7.5. Evolutionary methods have also found their way into oceanographic experimental design. Barth (1992) showed that a genetic algorithm is faster than simulated annealing and more accurate than a problem specific method for optimizing the design of an oceanographic experiment. Porto et al. (1995) found that an evolutionary programming strategy was more robust than traditional methods for locating an array of sensors in the ocean after they have drifted from their initial deployment location. Finally, Charbonneau (1995) gives three examples of uses of a genetic algorithm in astrophysics: modeling the rotation curves of galaxies, extracting pulsation periods of Doppler velocities in spectral lines, and optimizing a model of hydrodynamic wind. Hassan and Crossley (2002) have used Gas to configure constellations of satellites.

7.4

Example 1 – Fitting a Non-linear Inverse Model

Many of the applications reviewed above use a Ga to fit parameters to a model based on data. We choose to demonstrate the utility of the Ga on a specific inverse problem. In particular, we create our own non-linear time series data using the predator-prey model (also known as the Lotka-Volterra equations), namely: dx dt dy dt

= ax − bxy = −cy + dxy

(7.7)

where x is the number of prey and y the number of predators. The prey growth rate is a while the predator death rate is c. Parameters b and d characterize the interactions. Equations (7.7) were integrated using a fourth order Runge Kutta with a time step of 0.01 and parameters a=1.2, b=0.6,c= 0.8, and d=0.3. The time series showing the interaction between the two appears as Fig. 7.2a. This time series serves as the data for computing the inverse models below. The phase space plot is Fig. 7.2b where we see the limit cycle between the predators and the prey. A standard linear least squares model fit would be of the form: st = Ls + C

(7.8)

where s is a vector incorporating both x and y, L is a linear matrix operator, and C is the additive constant. This simple linear form is easily fit using standard analytical

212

S. E. Haupt

Figure 7.2: (a) Time series showing predator and prey variations over time according to equation (7.7). (b) State space showing predator-prey interactions.

techniques to minimize the least square error between the model and data. The least squares fit to the linear model appears in Fig. 7.3a. We note that the agreement is quite poor, as one would expect given that the system (7.7) is highly non-linear. With no non-linear interaction available, the number of prey grows while the number of predators remains stationary. To obtain a more appropriate non-linear fit, we now choose to model the data with a non-linear model: st = N sT s + Ls + C.

(7.9)

We now allow non-linear interaction through the non-linear third order tensor operator, N . Although one can still find a closed form solution for this non-linear problem, it involves inverting a fourth order tensor. For problems larger than this simple two-dimensional one, such an inversion is not trivial. Therefore, we choose to use a genetic algorithm to find parameters which minimize the least square er-

Figure 7.3: Lotka-Volterra model: (a) Least squares time series fit to predator-prey model. (b) Time series of predator-prey interactions as computed by the genetic algorithm.


213

ror between the model and the data. The Ga used an initial population size of 200, a working population size of 100, and a mutation rate of 0.2. A time series of the solution as computed by the Ga appears in Fig. 7.3b. Note that although the time series does not exactly reproduce the data, the oscillations with a phase shift of roughly a quarter period is reproduced. The wavelength is not exact and the amplitudes grow in time, indicating an instability. This instability is likely inherent in the way that the model is matched. However, the reproduction of such a difficult non-linear system is amazing given the comparison to traditional linear models. The state space plot appears in Fig. 7.4a. Once again, the limit cycle is not actually reproduced. The non-linear model instead appears unstable and slowly grows. However, the comparison with the linear least squares model resulted in merely a single slowly growing curve (not shown). The Ga non-linear model was able to capture the cyclical nature of the oscillations. Finally, Fig. 7.4b shows the convergence of the Ga for a typical run of fitting the non-linear model (7.9) to the data. Note that due to their random nature, the results of the Ga are never exactly the same. In particular the convergence plots will differ each time. However, it is amazing how the results are so reliable.

Figure 7.4: (a) The predator-prey relation in state space as computed by the non-linear model with parameters fit by the Ga. (b) Evolution of the minimum cost for the Ga fit to the non-linear model parameters.

7.5

Example 2 – Inverse Modeling of Air Pollution Data

An interesting inverse problem is when one has air pollution data at a receptor and wishes to identify the source of that data given some information about a variety of potential emission sources. One example of this class of problems is air pollution receptor modeling. In this type of problem, data exists on air pollution concentrations at a particular monitoring station. The characteristics of surrounding

214

S. E. Haupt

potential sources of pollution are known and so is some basic information about the usual transport from those sources and meteorological data matched in time to the receptor data. The problem is then to attribute the weighted average percentage of collected pollutant to the appropriate sources. Our example problem involves apportioning the contribution of local sources of air pollution in Cache Valley, Utah to the measured pollutants received at a monitoring station owned by the Utah Department of Air Quality. This demonstration problem uses sixteen sources surrounding the receptor. Of course, the spread and direction of pollutant plumes are highly dependent on wind speed and direction in addition to other meteorological variables. Cartwright and Harris (1993) used a genetic algorithm to apportion sources to pollutant data at receptors. They began with a chemical mass balance (Cmb) model of the form M ·S =R

(7.10)

where M is the source profile matrix, which denotes the effective strength of pollutant from a given source at the receptor; S is the fraction of the source that contributes to the concentration at the receptor, the unknown apportionments; and R is the concentration of each pollutant measured at a given receptor. In theory, these matrices may be the necessary size to incorporate as many sources, receptors, and pollutants as necessary. In our problem, we demonstrate the technique with a single pollutant at a single receptor. Cartwright and Harris (1993) chose to use a uniform dispersion in all directions, with a decrease of concentration with distance according to a r −2.5 power law, where r is the distance from the source to the receptor. Here, we instead choose to use wind data together with the dispersion law as found in Beychok (1994). Q exp C= uσz σy 2π

−y 2 2σy2

−(zr − He )2 −(zr + He )2 exp( ) + exp( ) 2σz2 2σz2

(7.11)

where: C= concentration of emissions at a receptor, (x, y, zr ) = Cartesian coordinates of the receptor in the downwind direction from the source, Q = source emission rate, u = wind speed, He = effective height of the plume centerline above ground, and σy , σz = standard deviations of the emission distribution in the y and z directions, respectively. Note that there are a myriad of assumptions hidden behind the problem. First, we assume that the wind speed and direction are constant over the entire time period. Although we know a priori that this assumption is poor, it is balanced by the assumption of Gaussian dispersion in a single direction. Thus, although a plume of pollutants may meander throughout the time period, we only care about the weighted average statistical distribution of the concentrations. Next we are forced to assume a constant emission rate, in this case an average annual rate. Of course, the hourly rate is much different. The major difficulty is in estimating


215

reasonable average values for the dispersion coefficients, σy and σz . Again, we must assume a weighted average over time and use parameters computed as σ = exp I + J(ln(x) + K(ln(x))2 (7.12)

where x is the downwind distance (in km) and I, J, and K coefficients dependent on the Pasquill stability class from a lookup table (Beychok 1994). The Pasquill stability class depends on wind speed, direction, and insolation and can be determined from look up tables (Beychok 1994). For this demonstration problem, we assumed neutral stability (class D). Equations (7.10) and (7.11) together with two look-up tables and (7.12) convert the source emission rates into the elements of the source matrix, M which indicate an expected average concentration due to each source for a constant emission rate and actual hourly average wind data. This process is repeated for each source at each time. The measured concentrations are also time averaged (in this case over a three day period) and go into the matrix, R. The goal is to solve for the fractions, S, that apportion the pollution to each source. That is where the genetic algorithm comes in. If R and S were constant one-dimensional vectors, one could easily solve for S. However, the need to sum the matrix times the factors hourly for differing meteorological conditions precludes a simple matrix inversion. The chromosomes of the Ga in this case represent the unknown elements of matrix S. The cost function is then the difference between the pollutant values at the receptor and the summation of the hourly concentrations predicted for each source as computed from the dispersion model (7.11) times the apportionment factors supplied by the Ga. The receptor model was run using actual meteorological data for July 10-12, 2002 and comparing predicted weighted average concentrations of Pm10 (particulate matter less than 10 micrometers in diameter) taken on July 12, 2002. The dispersion coefficients were computed assuming a Pasquill stability class D. Ten runs of the Ga were done using a population size of 12, mutation rate of 0.2, and crossover rate of 0.5. The fractions in the unknown vector, S, were normalized to sum to 1. Runs were made for up to 1,000 iterations. Overwhelmingly, the factors converged on the heaviest weighting of source 13, the Utah State University heating plant, with a weighting averaging about 0.9. Note that this does not necessarily imply that it contributed the most pollutant, but rather that its average emission rate, when dispersed according to (7.11) using actual wind data, must have a heavier weighting to account for the monitored Pm10. The second highest weighted source was number 9, a local construction company. The point of this exercise is to demonstrate that the Ga is a useful tool for problems like this which require including another model, in this case a dispersion model, to evaluate the cost of a function. In spite of the large number of times that (7.11) was evaluated, it still only required a couple of minutes time on a Pentium 4 PC to run 1,000 generations. Coupling Gas with simulations is becoming a more popular way to do searches. The work of Loughlin et al. (2000) coupled a full air

216

S. E. Haupt

quality model with a Ga to design better control strategies to meet attainment of the ozone standard while minimizing total cost of controls at over 1,000 sources. This type of coupled model application may be the wave of the future.

7.6

Conclusion

We have seen that genetic algorithms are not only an effective way of solving optimization and search problems, but they can also be rather fun to apply. They have begun to find their way into applications in the many disciplines of environmental science, but their strengths have only begun to be tapped. We have demonstrated here how versatile these algorithms are at finding solutions where other methods often fail. We saw that for a simple two-dimensional non-linear system describing predator-prey relations, the Ga was able to fit the parameters of a non-linear model so that the attractor was much better produced than by a traditional linear least squares fit. Although the match is not perfect, the non-linear Ga model captured the essence of the dynamics. In addition, the air pollution problem demonstrated that, even when another full model is inherent in the cost function, a Ga can be used to find the best fit for source apportionment. Here, we have only presented the very basics of one species of continuous parameter genetic algorithm and demonstrated some applications to environmental optimization and search problems. Some of the references of Section 7.3 have used other types of Gas in highly imaginative ways and combined them with other techniques. The use of Gas is growing. In some fields, such as groundwater work, they are becoming a common tool. For other areas, they have not yet been well investigated. In yet others, we are still dealing with preliminary work on toy problems; yet Gas may prove useful to making huge steps forward in modeling some phenomena. For instance, the simple example of Section 7.4 might be a precursor of fitting parameters of full climate models. Linear fits give a good first fit to climate model data. (Branstator and Haupt 1998) demonstrated that a simple linear stochastic model can reproduce responses to forcing better than a traditional linearized quasigeostrophic model that includes specific dynamics. But the linear techniques fall apart on highly non-linear problems. That is where the genetic algorithm can be useful. If we can formulate a non-linear form for a climate model in ways parallel to the Section 7.4 problem, we can use the Ga to estimate the coefficients. Even the highly non-linear Lorenz equations in the strange attractor regime can be fit to a quadratically non-linear model using this technique (Haupt 2003). One can imagine that as we attempt to model larger non-linear systems, techniques from artificial intelligence, such as genetic algorithms, could become useful to determining stochastic fits where the dynamical theory becomes intractable. The hope is that this Chapter has whet the reader’s appetite and that the Ga will find its way into other interesting problems. Although Gas are becoming mainstream in some areas, they have not yet been applied in others. Our goal is to inspire other environmental scientists to try the Ga on a variety of problems.


217

References Adewuya, A. A. (1996). New methods in genetic search with real-valued chromosomes. Aly, A. H. and R. C. Peralta (1999a). Comparison of a genetic algorithm and mathematical programming to the design of groundwater cleanup systems. Water Resources Research 35 (8), 2415–2425. Aly, A. H. and R. C. Peralta (1999b). Optimal design of aquifer clean up systems under uncertainty using a neural network and a genetic algorithm. Water Resources Research 35 (8), 2523–2532. Barth, H. (1992). Oceanographic experiment design II: Genetic algorithms. Journal of Oceanic and Atmospheric Technology 9, 434–443. Beychok, M. (1994). Fundamentals of Stack Gas Dispersion (3rd. ed.). Irvine. Boschetti, F., M. C. Dentith, and R. D. List (1995). A staged genetic algorithm for tomographic inversion of seismic refraction data. Exploration Geophysics 26, 331– 335. Boschetti, F., M. C. Dentith, and R. D. List (1996). Inversion of seismic refraction data using genetic algorithms. Geophysics 61, 1715–1727. Boschetti, F., M. C. D. R., and List (1997). Inversion of potential field data by genetic algorithms. Geophysical Prospecting 45, 461–478. Branstator, G. and S. E. Haupt (1998). An empirical model of barotropic atmospheric dynamics and its response to tropical forcing. Journal of Climate 11, 2645–2667. Cartwright, H. M. and S. P. Harris (1993). Analysis of the distribution of airborne pollution using genetic algorithms. Atmospheric Environment 27A, 1783–1797. Charbonneau, P. (1995). Genetic algorithms in astronomy and astrophysics. The Astrophysical Journal Supplement Series 101, 309–334. Chunduru, R. K., M. K. Sen, and P. L. Stoffa (1997). Hybrid optimization for geophysical inversion. Geophysics 62 (4), 1196–1207. Chunduru, R. K., M. K. Sen, P. L. Stoffa, and R. Nagendra (1995). Non-linear inversion of resistivity profiling data for some regular geometrical bodies. Geophysical Prospecting 43, 979–1003. Fayad, H. (2001). Application of neural networks and genetic algorithms for solving conjunctive water use problems. Ph. D. thesis, Utah State University, Logan. Goldberg, D. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. New York: Addison-Wesley. Hart, J., I. Hunt, and V. Shankararaman (1998). Environmental management systems a role for AI? In Binding Environmental Sciences and AI. ECAI 98. Hassan, R. A. and W. A. Crossley (2002). Multi-objective optimization for communication satellites with a two-branch tournament genetic algorithm. J. Spacecraft Rockets (Sept.). Haupt, R. L. and S. E. Haupt (1998). Practical Genetic Algorithms. New York: John Wiley & Sons.

218

S. E. Haupt

Haupt, R. L. and S. E. Haupt (2000). Optimum population size and mutation rate for a simple real genetic algorithm that optimizes array factors. Applied Computational Electromagnetics Society Journal 15 (2). Haupt, S. E. (2003). Genetic algorithms in geophysical fluid dynamics. AMS Conference on Artificial Intelligence. Hilton, A. B. C. and T. B. Culver (2000). Constraint handling for genetic algorithms in optimal remediation design. J. Water Res. Planning Mgt. 126 (3), 128–137. Holland, J. (1975). Adaptation in Natural and Artificial Systems. Ann Arbor: The University of Michigan Press. Holland, J. (1992, July). Genetic algorithms. Scientific American, 66–72. Jervis, M., M. K. Sen, and P. L. Stoffa (1996). Prestack migration velocity estimation using nonlinear methods. Geophysics 60, 138–150. Kim, S., H. Lee, J. Kim, C. Kim, J. Ko, H. Woo, and S. Kim (2002). Genetic algorithms for the application of activated sludge model #1. Water Science and Technology 45 (4-5), 405–411. Loughlin, D. H., S. R. Ranjithan, J. W. Baugh Jr., and E. D. Brill Jr. (2000). Application of genetic algorithms for the design of ozone control strategies. J. Air & Waste Managem. Assoc. 50, 1050–1063. McCullagh, J. B. C. and K. Bluff (1999). Genetic sythesis of a neural network for rainfall and temperature estimations. Aust. J. Intl Inf. Proc. Syst.. McKinney, D. C. and M.-D. Lin (1993). Genetic algorithm solution of ground water management models. Water Resources Research 30 (6), 3775–3789. Michalewicz, Z. (1992). Genetic Algorithms + Data Structures = Evolution Programs. Berlin: Springer. Minister, J.-B. H., N. Williams, T. G. Masters, J. F. Gilbert, and J. S. Haase (1995). Application of evolutionary programming to earthquake hypocenter determination. In Evolutionary Programming, pp. 3–17. Proc. 4th Ann. Conf. Evolutionary Programming. Mohan, S. and D. P. Loucks (1995). Genetic algorithms for estimating model parameters. In Integrated Water Resour. Plng. For the 21st Century, Cambridge. Proc. 22nd Annu. Conf. ASCE. Mulligan, A. E. and L. C. Brown (1998). Genetic algorithms for calibrating water quality models. J. of Environmental Engineering, 202–211. Porsani, M. J., P. L. Stoffa, M. K. Sen, and R. K. Chunduru (2000). Fitness functions, genetic algorithms and hybrid optimization in seismic waveform inversion. J. Seismic Explor 9, 143–164. Porto, V. W., D. B. Fogel, and L. J. Fogel (1995, June). Alternative neural network training methods. IEEE Expert Syst . Ritzel, B. J., J. W. Eheart, and S. Rajithan (1994). Using genetic algorithms to solve a multiple objective groundwater pollution containment problem. Water Resources Research 30 (5), 1589–1603.


219

Rogers, L. L. and F. U. Dowla (1994). Optimization of groundwater remediation using artificial neural networks with parallel solute transport modeling. Water Resources Research 30 (2), 457–481. Sen, M. K. and P. Stoffa (1992a). Genetic inversion of avo. Geophysics: The Leading Edge of Exploration, 27–29. Sen, M. K. and P. L. Stoffa (1992b). Rapid sampling of model space using genetic algorithms: Examples from seismic waveform inversion. Geophys. J. Int. 108, 281–292. Sen, Z. and A. Oztopal (2001). Genetic algorithms for the classification and prediction of precipitation occurrence. Hydrological Sciences 46 (2), 255–268. Shieh, H.-J. and R. C. Peralta (1997). Optimal system design of in-situ bioremediation using genetic annealing algorithm. In In Ground Water: An Endangered Resource, pp. 95–100. 27th Annual Congress of the International Association of Hydrologic Research. Simpson, A. R., G. C. Dandy, and L. J. Murphy (1994). Genetic algorithms compared to other techniques for pipe optimization. J. Water Resour. Plng. and Mgmt. 120 (4), 423–443. Vrugt, J. A., H. Gupta, L. Bastidas, W. Bouten, and S. Sorooshian (2003). Effective and efficient algorithm for multiobjective optimization of hydrologic models. Water Resour. Res 39 (8), 1214.

CHAPTER 8

Using Fuzzy Logic to Quantify Climate Change Impacts on Spawner–Recruitment Relationships for Fish from the North–Eastern Pacific Ocean Din G. Chen and James R. Irvine

8.1

Introduction

Major changes to marine ecosystems resulting from global climate variation pose enormous challenges to those responsible for the assessment and management of fishery resources. The successful implementation of policies that will enable sustainable fisheries, especially during periods of climate change, generally requires a good understanding of relationships between numbers of spawners and subsequent recruitment. Non–traditional analytical techniques, including artificial intelligence (Ai) methodologies, offer significant advantages over traditional techniques in the analyses of these relationships, and hence can improve our understanding of how climate change may affect fish abundance. In this Chapter we apply an Ai method, fuzzy logic, to various sets of fish spawner–recruitment time–series data (further applications of the fuzzy concept can be found in Chapter 9). Using fuzzy logic, we incorporate environmental changes in our analysis and thereby model the uncertain and poorly defined impacts of environmental regimes. Different from traditional approaches, fuzzy logic utilizes a continuous membership function that provides us with a rational basis to categorize spawner–recruitment data. We apply this approach to various populations of Pacific salmon (Oncorhynchus spp.), herring (Clupea harengus pallasi), and halibut (Hippoglossus stenolepis) from the north–eastern Pacific Ocean. Fuzzy logic models consistently outperformed traditional approaches as measured by several diagnostic criteria. Because fuzzy logic models address uncertainty better than traditional approaches, they have the potential to improve our ability to understand factors influencing spawner–recruitment relationships, and thereby manage fisheries more effectively. There is an extensive and developing literature dealing with the consequences of climate change on fish and fisheries. However, most authors rely on traditional statistical approaches when evaluating climate impacts. Non–traditional approaches, including artificial intelligence (Ai) methodologies, sometimes offer significant advantages and thereby can help us improve our understanding of relationships between climate changes and fish abundance (cf. Chapter 3 for further methods applied in the field of fisheries).

222

D.G. Chen and J.R. Irvine

Fuzzy logic is one Ai approach that is well–suited to the types of analyses undertaken by many researchers in the fields of global change and sustainability science. While fuzzy set theory, first described by Zadeh (1965) has revolutionized computer technology (McNeil and Freiberger 1993), it remains under–utilized in environmental and social fields, especially in areas other than the Orient. By mathematically representing uncertainty, fuzzy logic formalisms are an approach for dealing with the imprecision intrinsic to many environmental and social problems. Using fuzzy logic models, environmental variables such as temperature and salinity can be assigned a degree of membership to a particular regime (e.g. warm or cold, salty or not salty, positive or negative). Our goal is to demonstrate the utility of fuzzy logic models in the evaluation of climate change. We will draw largely from our own work on fish populations from the north–east Pacific Ocean. We hope that by demonstrating some of the benefits of fuzzy logic compared to traditional statistical models, we will convince more scientists to consider applying fuzzy logic and other Ai techniques when they investigate global change. We shall restrict our examples to analyses of relationships between spawning stock and recruitment, the most important and generally most difficult problem in the biological assessment of fisheries (Hilborn and Walters 1992).

8.2

Ricker Stock–Recruitment Model

Various models are available to examine relationships between spawners and recruitment (Quinn II and Deriso 1999), including the well–known Ricker stock– recruitment (Sr) model (Ricker 1975): Rt = St · exp (a − bSt + εt ) ,

(8.1)

where St is spawner abundance or biomass in year t, and Rt is the resulting recruitment. The parameters a and b have biological interpretations; a measures productivity and b measures density dependence. The error εt is assumed to be normally distributed as N 0, σ 2 . Ricker curves typically rise from the origin to a maximum, then bend over and decline as spawning levels increase, indicating reduced recruitment at higher spawner abundance. The greater the a parameter, the steeper the slope near the origin and the larger the b parameter, the more quickly density independence takes effect. If b is zero, model (8.1) is the so–called density– independent Sr model, which indicates constant survival. This Ricker model (Eq. (8.1)) can be extended to incorporate environmental variables (Hilborn and Walters 1992; Chen and Ware 1999; Chen and Irvine 2001) producing the Ricker climatic model: Rt = St · exp (a − bSt + γ Evt + εt ) ,

(8.2)

Fuzzy Logic to Assess Climate Change and Spawner Recruitment Relations

223

where Evt is a vector of environmental variables. After log transformation, models (8.1) and (8.2) become: Rt = a − bSt + εt (8.3) yt = log St yt = log

Rt St

= a − bSt + γ Evt + εt

(8.4)

We can estimate the model parameters a, b and γ by multiple linear regression and test the significance of b in model (8.3) as well as the significance of incorporating environmental variables γ in model (8.4). If γ is not statistically significant, we conclude the environmental variable (Ev) has no significance on the Ricker model (8.3).

8.3

Crisp Regime Stock–Recruitment Model

Stock recruitment analysis typically assumes there is no temporal pattern in the relationship between stock and recruitment. This is often not a valid assumption since many researchers have demonstrated a significant effect of climate on survival (e.g. McGinn 2002). When a rapid change in fish abundance and climate occurs, this is referred to as a regime shift (Beamish and Noakes 2002). Since environmental conditions affect Sr relationships (Koslow et al. 1986), fishery scientists would like to know if Sr relationships vary in different environmental regimes. More specifically, do the parameters a and b differ among regimes? Although the Ricker climatic model (8.2) or (8.4) is an improvement over the Ricker model (8.1) or (8.3) since it incorporates Ev, the climatic model does not allow us to test for regime impacts, and therefore does not allow us to answer this question. To illustrate how to test whether a and b vary among regimes, we start with a simple example and assume that there are only two regimes, 1 and 2, which could correspond to regimes that are warm and cold, positive and negative, salty and less salty, windy and less windy etc. We wish to determine whether Sr relationships differ between regimes 1 and 2: yt = a1 − b1 St in regime 1 (8.5) yt = a2 − b2 St in regime 2 where a1 and a2 measure fish stock reproductive performance at low stock sizes, and b1 and b2 represent density–dependence in pre–recruitment survival rates during regimes 1 and 2 respectively. If there is no difference between a1 and a2 , or b1 and b2 , then the Sr relationship does not appear to vary between regimes, which means that the environmental effect is not significant. If on the other hand there is a difference, we wish to evaluate the significance of Ev.

224


Suppose that the Ev is the sea surface temperature (Sst). A traditional approach for fishery researchers might be to partition the Sr data based on the mean long–term Sst using a crisp classification. When Sst is above average (i.e. Ev > 0.5), the regime 1 is “warm” (Fig. 8.1a, the dashed line) and the Sr data associated with these “warm” years are assigned to this “warm” regime 1. Otherwise, the Sr data are assigned to the “cold” regime 2 (Fig. 8.1b, the dashed line). The Ricker model (8.3) could then be fitted to the two categories of Sr data to estimate the associated Sr parameters (e.g. Fig. 8.2).

Membership value

1.0 (a) Regime 1

(b) Regime 2

0.5

0

0

0.5 EV (e.g. SST or PDO)

1.0 0

0.5 EV (e.g. SST or PDO)

1.0

Figure 8.1: Illustration of fuzzy membership function (Fmf) for the environmental variable (Ev) corresponding to fuzzy sets in two regimes with the Ev re–scaled between 0 and 1. Assuming that regimes 1 and 2 correspond to “warm” and “cold”, the dashed lines in (a) and (b) illustrate the traditional crisp thinking that it is “warm” if Sst is greater than the time series average (Ev > 0.5) and it is “cold” otherwise. The solid line in (a) and (b) illustrate the fuzzy membership function.

However, this approach has limitations, for instance it is easy to misclassify those years close to the long–term time series mean. As well, potentially useful information may be ignored since environmental information is not used in model fitting (the Sr data from the “warm years” are not used in fitting the Sr model to the data from “cool years” and vice versa). In general, categorical classification approaches, such the dichotomous “warm” vs. “cool”, “positive” vs. “negative”, and “good” vs. “bad” impose subjective breakpoints on a continuously varying factor. Fuzzy logic allows us to implement a non–dichotomous, multi–valued approach that can lead to improved Sr analysis. Fuzzy reasoning is particularly well–suited to investigating whether stock productivity or density dependent patterns differ among regimes (Mackinson et al. 1999; Chen et al. 2000; Chen 2001).

8.4

Fuzzy Logic Stock–Recruitment Model

Using fuzzy if–then rules, a fuzzy logic model mimics the qualitative aspects of human intelligence without requiring precise quantitative analyses. Fuzzy modeling,


225

Figure 8.2: Stock–recruitment data for 5 populations of Pacific salmon, Pacific herring and Pacific halibut. Solid lines and •’s are Ricker Sr model fits and data correspond to the “cold Sst” or “negative Pdo” years and dashed lines and ◦’s correspond to the “warm Sst” or “positive Pdo” years. Actual numbers of salmon spawners and recruits are 1,000 times numbers given.

first explored systematically by Takagi and Sugeno (1983), has found numerous practical applications (Kandel 1992). The approach offers substantial advantages and improvements compared to methods traditionally used to analyze ecological data (Lek et al. 1995; Mackinson et al. 1999; Chen 2002). A fuzzy logic model (fuzzy inference system or fuzzy controller) is composed of (i) the knowledge base, which contains a number of fuzzy if–then rules and a database to define the membership functions of the fuzzy sets used in the fuzzy rules, and (ii) the fuzzy reasoning or decision making unit that performs the inference operations based on the rules. Two operations are performed in fuzzy logic modeling. First, a fuzzification operation compares the input variables with the membership functions (Fig. 8.1) on the premise part of the if–then rules to obtain the membership values of each linguistic fuzzy set. These membership values from the premise part are combined

226


through a min operator to determine the firing strength (weight) of each rule, in order to generate a qualified consequent (either fuzzy or crisp) of each rule depending on this firing strength. Second, the qualified consequents are aggregated to produce a crisp output as the defuzzification operation. Several types of fuzzy reasoning have been proposed (Lee 1990) that depend on the types of fuzzy reasoning and fuzzy if–then rules employed. Takagi and Sugeno (1983) proposed the most commonly used model. In this model, the output of each rule is a linear combination of input variables plus a constant term, with the final model output being the weighted average of each rule’s output. We will illustrate the process by constructing a fuzzy logic model for fish stock recruitment. Only Ev will be input in the fuzzification process where it will be transformed from crisp inputs into two fuzzy sets, corresponding to two separate regimes.

8.4.1

Fuzzy Knowledge Base

In general form the two fuzzy rules used in this knowledge base, according to the choice of fuzzification of Ev as two fuzzy sets, are: Rt = a1 − b1 St (8.6) Rule 1: If Evt is in regime 1, then yt = log St Rt Rule 2: If Evt is in regime 2, then yt = log = a2 − b2 St (8.7) St where a1 , a2 , b1 and b2 are Sr parameters to be estimated. It can be seen from (8.6) and (8.7) that if there is no significant difference in Sr relationships between regimes 1 and 2, the fuzzy model in (8.6) and (8.7) can be combined and simplified into one model as described in (8.3). In this case, the fuzzy logic model is an extension of the traditional Ricker Sr model. With the if–then rules defined in (8.6) and (8.7), the membership function for the two fuzzy sets in the “premise” if–part is illustrated in Fig. 8.1 (the solid line). The “consequent” parts (i.e. the “then–clause”) of the two fuzzy rules are defined by the traditional Ricker Sr relationship, which is the fuzzy model proposed by Takagi and Sugeno (1983).

8.4.2

The Fuzzy Reasoning

For any observed Evt and corresponding St at year t, the fuzzy reasoning process involves the following steps: Step 1: Compare the Ev with the Fmf (fuzzy membership function) (Fig. 8.1, solid line) to obtain the membership values of each fuzzy set (i.e. fuzzification): Rule 1: wt1 = Fmfregime 1 (Ev) (8.8) Rule 2: wt2 = Fmfregime 2 (Ev) ;

(8.9)


227

Step 2: Generate the consequent of each rule depending on the membership values. For each Rule i, the output from the “consequent” part is calculated by the function defined in (8.6) and (8.7) as: yˆt1 = a1 − b1 St (for regime 1)

(8.10)

yˆt2 = a2 − b2 St (for regime 2)

(8.11)

Step 3: The final output of the fuzzy logic model is inferred from the two rules by a defuzzification process to compute the weighted average as: yˆt =

wt1 yˆt1 + wt2 yˆt2 = wt1 yˆt1 + wt2 yˆt2 wt1 + wt2

(8.12)

since wt1 + wt2 = 1 from (8.8) and (8.9).

8.4.3

Parameter Estimation

The Sr parameters a1 , a2 , b1 and b2 are estimated by minimizing the mean squared error (Mse): n

Mse (a1 , a2 , b1 , b2 ) = = =

1X (yt − yˆt )2 n 1 n 1 n

t=1 n X t=1 n X t=1

(8.13)

[yt − (wt1 yˆt1 + wt2 yˆt2 )]2 [yt − wt1 (a1 − b1 St ) − wt2 (a2 − b2 St )]2

(8.14)

These parameters are estimated by minimizing (8.14), which is equivalent to the least square estimation (Lse): Y = XB (8.15) where Y = (y1 , . . . ,yn )′is an n × 1 vector of observed fish stock productivity indices ′ t defined as yt = log R St , B = (a1 , b1 , a2 , b2 ) is a 4 × 1 parameter vector, and 

w11  .. X= .

−w11 S1 .. .

w12 .. .

 −w12 S1  ..  .

wn1 −wn1 Sn wn2 −wn2 Sn

is an n × 4 matrix constituted by observed fish spawner biomass and the Ev index. The parameter vector can be estimated by: ˆ = X ′ X −1 X ′ Y. B

228

8.5 8.5.1


Data Analyses and Model Comparison Data Description

To illustrate our approach, we chose Sr data sets that varied in duration and contrast. We wanted our data sets to encompass a variety of situations and did not want to only present data that showed the utility of the fuzzy logic approach. Data were from one stock for each of 5 species of Pacific salmon (Oncorhynchus spp.), plus Pacific herring (Clupea harengus pallasi) and Pacific halibut (Hippoglossus stenolepis) (Tab. 8.1 and Fig. 8.2). To represent environmental variables for the salmon stocks, we computed sea surface temperature anomalies (Ssta) from the lighthouse station nearest to individual salmon streams (http://www.pac.dfo-mpo.gc.ca/sci/Pages/lighthouse data.htm). We used the mean of April–July measurements during the year salmon went to sea since Mueter et al. (2002) found that measurements during this period were associated with the survival of BC pink, sockeye, and northern chum salmon. For herring, we used the annual mean Sst (Chen and Ware 1999), and for halibut, the Pacific Decadal Oscillation Index (Pdo) (Mantua et al. 1997; Chen 2001).

8.5.2

Model Comparison

Three criteria measured model fit: (i) root mean squared error (Rmse), where smaller Rmse values indicate better fit; (ii) squared correlation coefficients (R2 ), where higher R2 values indicate better fit; and (iii) Akaike information criteria (Aic) (Sakamoto et al. 1987), used to penalize the Mse according to the number of model parameters, where smaller Aic values indicate better fit. Sr data from the seven populations (Tab. 8.1) were fitted to four models: Ricker model (8.3), Ricker climatic model (8.4), crisp regime Sr model in Section 8.3, and the fuzzy logic Sr model in Section 8.4. Residuals were examined for homogeneity, normality and time series autocorrelation. Residuals for the herring and halibut populations were temporally correlated. After testing a series of auto–regressive and moving–average models using the maximum likelihood method on these data sets, we used the first order auto–regressive model that we found provided satisfactory fit. The density–dependent parameter in the Ricker Sr model (8.3) was significant for all stocks except chinook (Tab. 8.2, column 1-b), which means that there was a significant stock–recruitment relationship for the remaining stocks. In almost all cases, including Ev (Ricker climatic model (8.4)) improved the model fit as shown by decreasing Rmse, increasing R2 and decreasing Aic (Tab. 8.1, columns labeled as “1” and “2”). These improvements were statistically significant for sockeye salmon, Pacific herring, and Pacific halibut (Tab. 8.2, column 2-Ev). We conclude that the recruitment of these populations was affected by the environmental index (Ev). To test whether environmental regimes were significant, we fit the crisp

Rmse

Aic

1

2

3

4

1

2

3

4

1

2

3

4

Chinook (17) Chum (29) Coho (52) Pink (16) Sockeye (31) Herring (42) Halibut (60)

1.07 0.62 0.58 0.53 0.69 1.01 0.23

1.01 0.63 0.57 0.52 0.64 0.98 0.21

1.06 0.58 0.58 0.47 0.67 0.98 0.22

1.03 0.57 0.58 0.54 0.64 0.95 0.19

0.15 0.44 0.44 0.41 0.21 0.36 0.81

0.29 0.45 0.46 0.45 0.34 0.42 0.85

0.56 0.69 0.81 0.89 0.88 0.46 0.99

0.58 0.71 0.81 0.86 0.89 0.51 0.99

52.5 56.8 92.5 26.6 67.1 123.3 -3.4

51.3 58.6 92.5 27.4 63.3 120.9 -14.1

53.6 54.1 95.1 30.2 66.6 122.1 -0.5

52.8 53.7 94.4 29.1 64.5 119.9 - 17.6

Table 8.1: Summary statistics for model 1 (Ricker model (8.3)), Model 2 (Ricker climatic model (8.4)), model 3 (crisp regime Sr model in Section 8.3), and model 4 (fuzzy logic model in Section 8.4). The duration of each time series in years is given in parentheses. Populations included in the analysis are West coast Vancouver Island chinook salmon, Fraser River chum salmon, Babine River (Skeena watershed) coho salmon, Fraser River pink salmon, Fraser River sockeye salmon, West coast Vancouver Island Pacific herring, and Pacific halibut.


R2

Criteria Models

229

230

D.G. Chen and J.R. Irvine Models

1-b

2-Ev

4-a

4-b

Chinook Chum Coho Pink Sockeye Herring Halibut

NS S S S S S S

NS NS NS NS S S S

NS NS NS NS S S NS

NS NS NS NS NS S S

Table 8.2: Results of statistical tests (S = significant, NS = non–significant). Model (1b) is the Ricker model (8.3) to test the significance of the parameter b. Model (2-Ev) is the Ricker climatic model (8.4) to test the significance of including the Ev into the Ricker model. Model (4-a) is the fuzzy logic model testing the significance of the stock productivity parameter H0 : a1 = a2 and Model (4-b) is the fuzzy logic model testing the significance of density–dependent parameter H0 : b1 = b2 .

regime Sr model in Section 8.3 (Fig. 8.2) using long–term time series means to classify each regime, and the fuzzy logic model in Section 8.4 (Fig. 8.3). With the possible exception of pink salmon, the fuzzy logic Sr model outperformed the crisp regime Sr model (Tab. 8.1, columns labeled “3” and “4”). Therefore, the fuzzy logic model is used to test whether: 1. there was a significant difference in stock productivity between regimes 1 and 2, i.e. H0 : a1 = a2 (Tab. 8.2, column 4-a); and 2. density–dependent effects were significant between regimes, i.e. H0 : b1 = b2 (Tab. 8.2, column 4-b). If neither were significant, that would indicate that the stock was not affected by the environmental variable (Ev) and, therefore, the simple Ricker Sr model (8.3) would be appropriate for stock recruitment analysis (Fig. 8.3). Differences were not significant for chinook, chum, coho, and pink salmon, which was consistent with our earlier finding that Ssta did not improve the Ricker model. For data sets like these, the simple Ricker Sr model (8.3) would be appropriate for stock recruitment analysis. However, the fuzzy logic model demonstrated that there was a significant difference in productivity between regimes for sockeye salmon, there were significant differences in density–dependence for halibut, and both parameters varied between regimes for herring (Fig. 8.3).

8.6

Conclusion

Fuzzy logic models consistently outperformed traditional stock–recruitment approaches as measured by several diagnostic criteria. Data sets were highly variable in


231

Figure 8.3: Stock–recruitment data with the radius of the bullets (•) proportional to the magnitude of fuzzy membership values of the Ev in Fig. 8.1: the higher the Ev, the larger the radius. The lines are the resultant Sr models. For chinook, chum, coho and pink salmon, there is only one combined Ricker model since there were no significant changes between the two regimes. For sockeye, there are different Ricker productivity parameters in the two regimes, for herring, both a and b are different, and for halibut, the b’s are different.

terms of length and contrast, as is often the case in this type of work. The standard Ricker model was significant for datasets from six of seven populations (species) examined. Including an environmental variable without applying fuzzy reasoning improved the fit of the Ricker model in all cases, and further improvements were made when the environmental data were assigned a membership function to particular regimes using a fuzzy logic approach. For three of the populations (species), a fuzzy logic approach identified significant differences in productivity and/or density dependent effects between regimes. The application of fuzzy logic is not restricted to the simplistic examples of two regimes presented here. For instance, there appear to have been 11 regime shifts in the North Pacific since 1650 (Gedalof and Smith 2001). The fuzzy logic approach in Section 8.4 can be easily adopted and extended to multiple regimes by using multiple if–then rules. We suggest a reasonable approach often is to start with testing for

232


effects of two possible regimes as we did in this Chapter. If there is no detectable significance between the two regimes, as we found for most of our Pacific salmon data sets, there is no need to develop a procedure for more regimes. If differences between regimes are significant, as we found with sockeye salmon, herring and halibut, then the possibility of multiple regimes should be considered. Fuzzy logic can also be used to investigate the possibility of climate change occurring without regime shifts by setting fuzzy membership values between 0 and 1. For instance, Mackinson et al. (1999) suggested that rules be combined in a fuzzy system using a simple weighting factor to account for a systematic long–term influence of Evs on recruitment success. Peterman et al. (2003) recently demonstrated the value of using a Kalman filter when evaluating temporal patterns in productivity. Bristol Bay sockeye salmon productivities increased rapidly, corresponding to a regime shift in the mid–1970’s, but trends were more complex than indicated by the simple step functions used by many earlier workers. However, Peterman et al. (2003) did not investigate mechanisms for the productivity changes they found. Fuzzy reasoning could be a powerful next step used to evaluate the correlation between environmental variables and productivity in analyses such as these. Major changes to marine ecosystems resulting from global climate variation pose enormous challenges to those responsible for the assessment and management of fishery and other resources. The successful implementation of policies that will enable sustainable fisheries, especially during period of climate change, will require a good understanding of relationships between numbers of spawners and the subsequent recruitment, and how these relationships are affected by climate change. In this Chapter we showed the usefulness of fuzzy systems in evaluating simple examples of climate change. Chen and Ware (1999) used a neural network model with a fuzzy logic based decision–making procedure to select all–possible neural network models. Saila (1996) reviewed possible applications of Ai to fishery research including the neural network and fuzzy logic models with genetic algorithms to natural resource management and ecological modeling. Other Ai methodologies described in this book can also provide advantages over traditional techniques in the analyses of Sr relationships (for other examples of applications in fisheries, cf. Chapter 3). Science is at an exciting stage in terms of the development and application of Ai methodologies and we encourage scientists to investigate the possibility of applying the various Ai methods described throughout this book.

References Beamish, R. J. and D. J. Noakes (2002). The role of climate in the past, present, and future of Pacific salmon fisheries off the west coast of Canada. American Fisheries Society Symposium 32, 231–244. Chen, D. G. (2001). Detecting environmental regimes in fish stock-recruitment relationships by fuzzy logic. Can. J. Fish. Aquat. Sci. 58, 2139–2148.


233

Chen, D. G. (2002). A fuzzy logic view on classifying stock-recruitment relationships in different environmental regimes. In F. Recknagel (Ed.), Understanding Ecology by Biologically-Inspired Computation, Berlin, pp. 329–352. Springer-Verlag. Chen, D. G., B. Hargreaves, D. M. Ware, and Y. Liu (2000). A fuzzy logic model with genetic algorithms for analyzing fish stock-recruitment relationships. Can. J. Fish. Aquat. Sci. 57, 1878–1887. Chen, D. G. and J. R. Irvine (2001). A new semiparametric model to examine stockrecruitment relationships incorporating environmental data. Can. J. Fish. Aquat. Sci. 58, 1178–1186. Chen, D. G. and D. W. Ware (1999). A neural network model for forecasting fish stock recruitment. Can. J. Fish. Aquat. Sci. 56, 2385–2396. Gedalof, Z. and D. J. Smith (2001). Interdecadal climate variability and regime-scale shifts in Pacific North America. Geoph. Res. Let. 28, 1515–1518. Hilborn, R. and C. J. Walters (1992). Quantitative fisheries stock assessment: choice, dynamics and uncertainty. New York: Chapman & Hall. Kandel, A. (1992). Fuzzy expert systems. Boca Raton: CRC Press. Koslow, J. A., K. R. Thompson, and W. Silvert (1986). Recruitment to northwest Atlantic cod (Gadus morhua) and haddock (Melanogrammus aeglefinus) stocks: influence of stock size and climate. Can. J. Fish. Aquat. Sci. 44, 26–39. Lee, C. C. (1990). Fuzzy logic in control systems: fuzzy logic controller. IEEE Trans. On Systems, Man, and Cybernetics 20 (2), 419–435. Lek, S., A. Belaud, I. Dimopoulos, J. Lauga, and J. Moreau (1995). Improved estimation, using neural networks, of the food consumption of fish populations. Mar. Freshwater Res. 46, 1229–1236. Mackinson, S., M. Vasconcellos, and N. Newlands (1999). A new approach to the analysis of stock-recruitment relationships: ‘model-free estimation’ using fuzzy logic. Can. J. Fish. Aquat. Sci. 56, 686–699. Mantua, N. J., S. R. Hare, Y. Zhang, J. M. Wallace, and R. C. Francis (1997). A Pacific interdecadal climate oscillation with impacts on salmon production. Bull. Amer. Meteor. Soc. 78, 1069–1079. McGinn, N. A. (2002). Fisheries in a changing climate. Symposium 32, American Fisheries Society, Bethesda. McNeil, D. and P. Freiberger (1993). Fuzzy logic. New York: Simon & Schuster. Mueter, F. J., D. M. Ware, and R. M. Peterman (2002). Spatial correlation patterns in coastal environmental variables and survival rates of salmon in the north-east Pacific Ocean. Fish. Oceanogr. 11, 205–218. Peterman, R. M., B. J. Pyper, and B. W. MacGregor (2003). Use of the Kalman filter to reconstruct historical trends in productivity of Bristol Bay sockeye salmon (Oncorhynchus nerka). Can. J. Fish. Aquat. Sci. 60, 809–824. Quinn II, T. J. and R. B. Deriso (1999). Quantitative fish dynamics. Oxford: Oxford University Press.

234


Ricker, W. E. (1975). Computation and interpretation of biological statistics of fish population. Bulletin No. 191, Fisheries Research Board of Canada. Saila, S. B. (1996). Guide to some computerised artificial intelligence methods. In B. Megrey and E. Moksness (Eds.), Computers in Fisheries Research, London, pp. 8–40. Chapman & Hall. Sakamoto, Y., M. Ishiguro, and G. Kitagawa (1987). Akaike Information Criterion Statistics. Dordrecht: Kluwer Academic Publishers. Takagi, T. and M. Sugeno (1983). Derivation of fuzzy control rules from human operator’s control actions. In Proc. Of the IFAC Symp. On Fuzzy Information, Knowledge Representation and Decision Analysis, pp. 55–60. Zadeh, L. A. (1965). Fuzzy sets. Information and Control 8 (3), 338–353.

Part IV

Practical Decision Support

CHAPTER 9

A Hybrid Case-Based Reasoning System for Supporting the Modeling of Estuaries Sara Passone, Paul W.H. Chung, and Vahid Nassehi

9.1

Introduction

The physical behavior of estuaries is intrinsically complex. Influenced by tides and the mixing of fresh and sea waters, estuaries are characterized by highly variable environmental conditions. In addition, high concentration of human activities in these areas since the beginning of civilization has also led to the alteration of estuaries’ topography and their large-scale pollution. Numerical modeling is considered a very effective tool for studying the complex behavior of natural water systems such as estuaries and estimating the effects of man-made or natural changes on them. On the other hand, the scope and scale of water related problems require the involvement of a large number of various experts with different backgrounds. Thus, communication between modelers and policy makers is crucial in devising water resources management schemes. However, the differences in expertise and approach adopted by groups of professionals often complicate useful and productive cooperation and, consequently, hinder the selection of a right and realistic management strategy. The current models are in fact not easily accessible. In depth knowledge of modeling theory and extensive experience are required for the selection of a correct modeling approach and interpretation of the obtained simulation results. Therefore, the usability of computational models appears lower than their actual capability. The aim of this Chapter is to present a computing system for modeling estuaries that can be utilized by users from different backgrounds in estuarine science. The system aims to provide a general user with the necessary guidance for selecting the model that best matches his/her goals and the nature of the problem to be solved. It is based on the cooperative action of three modules: a case-based reasoning scheme, a genetic algorithm (Ga) and a library of numerical estuarine models. The main idea is for the system to record and adopt solutions from past experience to new problems. With respect to the possible correlation between the features of the estuary and the physical phenomenon to be modeled, the case-based module returns possible solutions from the system’s memory. The selected model is then adapted by the genetic algorithm component, which, implemented by combining the classical evolutionary approach with problem-specific knowledge, estimates a valid set of hundreds of model parameters to suit the particular estuarine environment. A

238

S. Passone et al.

case study is also given to demonstrate the system’s ability to provide the user with an appropriate estimation of the available model strategies and the robustness of the designed routine for adjustment of the model parameters. This system has the ability to extend the application of computer-based numerical estuarine models to a broader class of users through an effective organization and re-use of the knowledge necessary for the design of modeling strategies. Thus, it facilitates understanding of complex concepts and communication between the users so that time and effort needed in a multidisciplinary work of this nature is significantly reduced.

9.1.1

Background: Status and Threats to Estuaries and their Environment

The number of estuaries around the world is enormous. In United States alone there are about 850 of them. An estuary is a unique environment where the land meets the sea and where fresh and salt waters mix. Under the influence of tides, weather, seasonal river flows and climate, the estuarine water level, salinity, temperature, and sediment load are characterized by high temporal and spatial variability that make these habitats exceptional in terms of flora and fauna. These favorable environmental conditions have always attracted high density of human settlements - seventy percent of the largest cities in the world are located in estuarine areas. Unfortunately, the intensive use of estuaries for transportation, food production, waste-disposal and stresses induced by urbanization, land reclamation and flood protection have dramatically modified the morphology and the ecosystem of these water courses (Kennish 1986). Among the disturbances with an impact on the fragile and complex equilibrium of these water systems pollution remains a major threat. Nutrient enrichment, organic carbon loading, oil spills and toxic chemical contaminants (Gesamp 1990; Kennish 2002) affect these coastal systems with deleterious consequences for water quality, living resources and, therefore, human health. In addition, other anthropogenic activities (i.e. overfishing, agriculture and aquaculture, energy production, marine transport, land reclamation, tourism, waste-disposal and water diversion) have altered the estuarine environment and the life of its organisms either by changing the existing environmental conditions (temperature, water discharge, current directions) or reducing the existing resources beyond any possible sustainability. Furthermore, these disturbances have, directly and indirectly, more exposed estuaries to the impetus of hurricanes, storms and coastal floods. These priority problems are common to a number of regions such as Australia, United States, the Mediterranean and the North Sea. But they have also recently extended to developing countries whose industrialization can be considered relatively recent. In general, many studies regarding estuaries have been conducted and several remedial programs are now underway (Commission of the European Communities, 1999). However, despite the effort being carried out, the pressure on the environmental quality of such coastal systems is expected to remain high (Thomann

CBR System for Supporting the Modeling of Estuaries

239

1998). The evidence is that the human occupancy of coastal areas will increase worldwide during the 21st century, accelerating the degradation of estuaries and surrounding areas and deterioration of estuarine waters (Kennish 2002). In addition, while several of these problems have been studied and serious actions against them have been taken in developed countries, they become more acute in developing countries day by day. Recovery from eutrophication, chemical contamination and fishery exploitation has been demonstrated possible for estuaries in Europe and United States. Nevertheless, their impact is expected to rise in many less advanced countries which still do not have strict and effective regulations regarding fisheries management, abatement of sewage waste disposal in coastal waters and the employment of toxic pesticides. There are also estuarine problems common to developed as well as developing countries. As the demand of clean water arises sharply around the world, freshwater inflows are diverted to satisfy the domestic needs and the industrial and agricultural requirements of a growing population with severe repercussions on estuarine water quality and biological reproduction of estuarine organisms. It is also predicted that coastal subsidence, due to extraction of groundwater, oil and gas, and global warming, responsible for the possible rise of the sea level and extreme weather conditions, will severely modify estuaries’ hydrological regime, morphology and ecology.

9.1.2

Estuarine Management

Estuaries are studied by numerous disciplines and estuarine research is in continuous expansion. Oceanographers, engineers and natural scientists such as biologists and geophysicists, have been considering the estuarine systems from different points of view. A wide range of data and information concerning estuaries have been collected in the past decades, which have allowed to improve our understanding of estuarine processes. The complexity of physical, chemical and biological mechanisms that govern these water systems makes estuarine management a challenging and problematic task. The need is to integrate in an interdisciplinary framework various aspects of estuarine science to assess the impact and effectiveness of a number of possible strategies (Grigg 1989). This requires planners, regulators and scientists to work together to solve, mitigate and manage estuarine problems. Unfortunately, estuarine management continues to be characterized by poor communication between scientists and policy-makers. The differences in expertise and approach adopted by various groups of professionals often complicate useful and productive co-operation in selecting a right strategy for realistic situations. In particular, it is often impossible for policy makers to access and easily interpret results obtained from modeling, monitoring and data management (Bridgewater 2003). While the current research continues working on reducing the uncertainty that still surrounds estuarine processes, it also focuses on developing effective instruments to support and advise managers and policy makers by integrating different

240

S. Passone et al.

work perspectives as well as combining classical working methods (field campaigns and modeling) with information and communication technologies. The scope is to have multi-function computer tools that can analyze the problem under different perspectives (i.e. water quality, ecology, socio-economics, emergency), propose alternative solution schemes and outline the possible consequences arising from each action suggested.

9.1.3

Estuarine Modeling

Computer-based mathematical modeling is essential for designing effective estuarine management plans. Numerical estuarine models consist of solving estuarine problems using mathematical and numerical techniques. Their use allows investigating the status of estuaries, identifying and estimating alternative management strategies, making sure that all the phases of the adopted management program work properly. In the past thirty years numerical modeling for water engineering and environmental problems has become very sophisticated. Modeling systems are now powerful software packages with user-friendly front-ends and a menu of facilities to support pre-processing and post-processing operations (Chau and Chen 2001). However, selecting the model that is suitable to the specific estuary considered and the particular aspects of it to be simulated is still not straightforward. It continues to be a very difficult task for which the user needs to have detailed knowledge about the application and possible limitations of the available models. The choice of a model depends on many factors such as the topographical and hydrodynamic characteristics of an estuary and the chemical-biological processes that need to be simulated. But it is the nature of the problem that characterizes the model so that selection among competing models depends on the aim of the investigation and the degree of performance required. The use and purpose of a model often drive the user’s preference. A particular model must be selected taking into account not only the characteristics of the problem, but also the accuracy of their application and aim of the investigation. User’s requirements can influence the degree of the accuracy required and, consequently, many assumptions and approximations may be made to obtain the desired balance between accuracy and costs. In order to have a realistic and sensible investigation the model selection must be done without an overestimation of the available capability with a careful consideration of the resources on hand and inevitable consequences deriving from the approximations introduced (Hinwood and Wallis 1975). For instance, averaging over one or more spatial dimensions produces a simpler and more economical model. However, this decision imply many other assumptions and approximations which easily condition the model performance. Furthermore, the user must be able to evaluate and interpret simulation outputs so that, in case of unrealistic predictions, the user can detect where and why a mistake has been made. Thus, it is evident that, although the current generation of models are


241

very sophisticated, the use of models is still limited by the extensive mathematical background and interdisciplinary knowledge required for their correct application (Dyke 1996). Therefore, reservations have been expressed about the ability of a nonexpert user in modeling to correctly apply these tools. To exploit the full potential of estuarine modeling, it is necessary that the user is assisted in the formulation of a model.

9.1.4

Water Modeling and Artificial Intelligence

While the current models only responds to main modeling requirements of effectiveness, precision and low cost, future models need to be implemented with a useroriented approach, where the modelers are not the final users of the results but, instead, those who need to base their decisions on the simulation are the consumers. Models must not only provide technically sound solutions but they should also support the decision making process (Cunge and Erlich 1999). There is a consensus that only through the development of intelligent computer systems it is possible to increase the understanding of estuarine water quality problems and build effective water quality management programs (Thomann 1998). In particular, there is the requirement for advanced computer environments where numerical models are integrated with intelligent reasoning (Comas et al. 2002). According to this approach, modeling software must be considered as a component of more extended knowledge management systems, where data are collected, transformed, explained and communicated. Thus, other information tools need to be developed which, integrated as parts of the same units, co-operate in solving problems. The co-ordination of different knowledge facilities, each supervising a particular task, permits sharing expertise, working in parallel and profiting from multi-faceted knowledge bases and different sources of information (Cortés et al. 2001). For this purpose artificial intelligence technologies have been recently introduced in the field of water resources management, giving rise to a new discipline called hydro-informatics (Abbott 1991). In order to build effective decision support systems for integrated water resource management, hydroinformatic research has been focused on four main topics: data capture, storage, processing and analysis (Savic and Walters 1999); real-time diagnosis and prediction (Cunge and Erlich 1999); uncertainty and risk management (Mpimpas et al. 2001; Hall 2002); open modeling (Gijsbers et al. 2002).

9.1.5

Case-Based Reasoning for Estuarine Modeling

The decision support systems recently proposed for providing assistance in the field of water resources management, have been developed using the expert system methodology as general framework to integrate existing knowledge and information obtained from the collected data and the predictions given via modeling (Chau and Chen 2001; Comas et al. 2002). However, it is general opinion (Watson and Marir 1994) that the logic of expert system methodology is limited when flexibility and

242

S. Passone et al.

adaptability are the priorities. Despite the number of rules that can be implemented, the system’s answer is limited to the predetermined scenarios included into the system and the addition of any rule may require careful revision of the entire system’s logic (Holt and Benwell 1999). The case-based reasoning system for estuarine modeling (Cbem) presented in this Chapter is based on the utilization of the case-based reasoning (Cbr) technique in combination with numerical modeling and genetic algorithms (further aspects of Gas are discussed in Chapter 7). Cbr methodology consists of deriving the solution of a new problem from similar situations encountered in the past. By classifying the past events according to a set of common problem descriptors (Kolodner 1993), for any new problem encountered a suitable solution is retrieved from the memory of the Cbr system and, then, adapted to suit the nature of the new case examined. The logic of the Cbr solving procedure is particularly appropriate to a complicated discipline such as water resources management and, in particular, estuarine modeling. This is mainly due to the variety of expertises required. For instance, the study of any estuarine phenomenon via numerical modeling relies on an example-byexample based knowledge. A purely theoretical approach in estuarine modeling is not possible and the effort should often be supported by practical experience. The complexity of estuaries and the actual interaction of many problem factors must be estimated through the codification of previous studies and correct assessment of numerous assumptions for different cases and problems. Cbem, which is organized in three modules (i.e. a case-based reasoning scheme and a Ga and a library of numerical estuarine models), enables the potential users, who may not have in depth knowledge of modeling, to define the type of problem and the characteristics of the estuary and obtain meaningful numerical simulations. One case study is provided with the purpose of illustrating Cbem’s design issues and effectiveness. This is the Upper Milford Haven estuary, Wales, Uk. In particular, the specific problem of salt intrusion has been studied.

9.2

The CBEM Architecture

The system has three main software components: a case-based module where the case description and the case retrieval take place, a genetic algorithm module, responsible for the case adaptation and a library of numerical simulation models, which contains the computer codes of the numerical models described in the casebased module and called by the genetic algorithm during the adaptation phase (Fig. 9.1). Based on the features of the estuary under investigation and the physical phenomenon to be modeled, the case-based module retrieves a suitable solution among the past cases contained in the system’s memory. The selected model is then returned from the model library and specifically adapted by the genetic algorithm unit, which estimates a valid set of model parameters to suit the new estuarine physical conditions.


243

Figure 9.1: Structure of the Cbem system.

9.3

The CB Module

The case-based (Cb) module allows the user to describe new and past cases (case description). It is also responsible for the retrieval process (case retrieval).

9.3.1

Description Component

In Cbem a case is divided into two parts: the estuary, which is the object of the investigation, and the related models, each of which is employed to simulate a specific physical phenomenon for that estuary. The estuary description (Fig. 9.2) contains indices representing the features of the estuary domain, while the model description (Fig. 9.3) includes information about the model characteristics and the estuarine problem simulated. This distinction is due to several practical reasons. A given estuary may have been studied and modeled for different purposes, or a specific estuarine process may have been repeatedly simulated for the same estuary but using different model strategies to satisfy various quality requirements of the results

244

S. Passone et al.

Figure 9.2: Estuary Description – Upper Milford Haven estuary, Wales, Uk.

and simulation purposes. This distinction also makes the case representation more accessible and readable to the user, who has to supply the necessary information. In addition, it facilitates and speeds up the retrieval process. Dividing the cases into parts enables the search mechanism to identify easily similarity between correspondent aspects of the cases and make rapid conclusions about the cases themselves (Kolodner 1993). An estuary is described in terms of physical and hydrographic features, while the model description consists of indices outlining technical features and model performance. Some of these attributes are expressed in qualitative terms (e.g. width-depth ratio, meandering, required simulation time) to facilitate the retrieval process, and others are numeric data (e.g. intertidal area, total tidal area). In addition there are also features that, as object symbols (Chung and Inder 1992), are defined according to existing classification schemes for estuaries and models (Dyer 1997; Hinwood and Wallis 1975) (e.g. geomorphologic type, tidal range, salinity stratification, model dimension). The knowledge in the description component is organized using the rule-based approach and fuzzy-set theory (cf. also Chapter 8 for further applications of fuzzy set theory). Thus, some indices (i.e. the average width to average depth ratio, the degree of meandering, the salinity stratification, the Coriolis and wind forces) have been constructed in such a way that their value is suggested by the system as the


245

Figure 9.3: Model Description.

estuary description gradually progresses. For instance, by using the fuzzy set theory, the width to the depth ratio is derived from the average width and average depth given in the estuary description with the numerical value of their ratio expressed according to a fuzzy qualitative scale. According to this indexing the average width to the average depth ratio is classified over the classes of low, moderate, high and very high with different membership grades. The class with the highest membership value is displayed on the case description screen. For instance, if the estuary is represented by (low/0.44, moderate/0.97, high/0.23, very high/0.21) the ratio is defined in the case description as moderate. In addition, by including in the system’s knowledge base a set of properly designed rules, the salinity stratification of any estuary is easily estimated with respect to other indices (i.e. tidal range and width-to-depth ratio) previously given in the estuary description. Knowledge is also employed in form of dimensional analysis through which estimates of the magnitude of the wind and Coriolis forces, whose effects are represented in the model equations with specific mathematical terms, are provided. Based on the dimensional analysis’s outcome, the system suggests if it is appropriate either to take into account or discard their effects in the modeling procedure. The user can also select the value of such indices if he/she does not agree with the system suggestion.

9.3.2

Retrieval Component

As the user decides to investigate a problem using Cbem, he/she enters the feature values of the estuary to be modeled using the estuary description scheme. He/she then defines the type of problem and the purpose of the investigation (Fig. 9.4). In order to illustrate the retrieval process implemented in Cbem the physic phe-

246

S. Passone et al.

Figure 9.4: Model Selection Screen. The salt intrusion problem is examined for management purpose.

nomenon of salt intrusion investigated for management purpose is initially considered. In addition, we will present the model solutions proposed by Cbem when the same estuarine problem (i.e. salt intrusion) is examined for a different application purpose. The indices for the purpose and the problem definition are defined in the model selection screen using the same classes employed for the homonymous indices within the model description. The model approximations are also contained. These are selected conforming to what was previously declared in the estuary description about the presence of inlets, the bed shape and the inclusion of wind and Coriolis forces. In the model selection screen the user can change the previous choice and define different governing assumptions. Furthermore, a lower-bound limit for the degree of similarity (i.e. cut-off) must also be indicated. Only cases with similarity greater than or equal to the cut-off value will be retrieved. The value of the cut-off must be between 0 and 1 as the grade of similarity is calculated on a normalized scale (Kolodner 1993). Once the indices of the model selection screen have been chosen, the retrieval process is activated. The search engine selects from the system’s case-base only those cases for which the current problem has been previously modeled.


247

Figure 9.5: Set of cases retrieved from the system’s case-base ranked with respect to their similarity to the Upper Milford Haven estuary.

9.3.2.1

Similarity Score

The user is provided with two similarity scores. The first similarity rating between a new and a past case is defined based on the values of the physical and hydrographic characteristics contained in their estuary descriptions (Fig. 9.5). For each retrieved estuary Cbem gives the value of the similarity and the number of models previously employed for the specific estuary to simulate the considered estuarine problem (i.e. salinity intrusion). The user is then required to select, based on this similarity rating, the retrieved estuaries for further examination in the second phase of the retrieval process. This computes the second similarity degree of the cases, which is estimated with respect to the type of investigation to be conducted, the accuracy and simulation time required. Cbem returns to the user the chosen models, described by the value of their similarity, accuracy, dimension and the similarity of the related estuary (Fig. 9.6). Based on the information provided, the user, aware of the degree of similarity estimated for both the estuary and the model, is responsible for the final selection of the model. The two degrees of similarity are computed using the nearest neighbor matching procedure (Kolodner 1993). The necessary steps for the computation of the similarity between cases to study the salinity intrusion within an estuary for management purpose are given in Table 9.1 and 9.2. It must be noticed that the retrieval procedure implemented in the present scheme uses different sets of matching and importance criteria according to the type of estuarine problem considered. The

248

S. Passone et al.

Figure 9.6: Cases retrieved after the second phase of the retrieval process. The retrieval process is carried out to find a suitable model to investigate the salt intrusion in the Upper Milford Haven estuary for management purpose.

nearest neighbor matching is also combined with the heuristic criteria of exclusion and preference expressed with respect to the model dimension. The model dimension, which depends on the type of problem as well as the estuary’s physics, needs to be chosen so that the physical phenomenon under investigation is well-represented without underestimating or oversizing the problem domain. Cases are not retrieved from the case-base if the dimension of the related models is inappropriate for the type of problem considered (criterion of exclusion). Some cases are also preferred over others if their model dimension as evaluated by the system is more suited to simulate the current problem (criterion of preference). For instance, in case of the Upper Milford Haven estuary, because of the criterion of exclusion only 5 models of the 7 retrieved in the first stage of the process are finally presented to the user (Fig. 9.6). Furthermore, preference is expressed for the 1-D network model as more appropriate for simulating the hydrodynamics of a branching estuary like the Upper Milford Haven. As evident from Fig. 9.6, the case ‘FAL, 1-D network, estuary sim. = 0.940, model sim. = 0.82’) has the highest first and second similarity score. The model dimension, 1-D network, matches quite well the geometry and the geomorphology of Upper Milford Haven (Bikangaga and Nassehi 1995), while its model characteristics (i.e. ‘low’ for simulation time and ‘moderate’ for accuracy) meet the requirements of the purpose of the investigation (i.e. management tool). In order to illustrate the ability of Cbem to return adequate model solutions based on the specific nature of the problem analyzed, the case of Upper Milford Haven estuary and the phenomenon of salt intrusion are also here considered for a different application purpose. Figure 9.7 presents the correspondent model selection


249

1. select a set of 4 features F from the estuary description: the ratio of the average width to the average depth (a), the geomorphological estuary type (b), the tidal range (c), and the meandering rate (d). 2. assign the degree of relevance W= 1 to (a),(b) and (c) and 0.75 to (d). 3. determine the similarity values S = sim(FIk , FR k ), with sim as similarity function and, I and R referring to the input and retrieved cases, respectively. 4 P

W k Sk

4. normalize the match aggregate score = k=1 4 P

. Wk

k=1

Table 9.1: First Similarity Score.

screen in which ‘research’ is entered for purpose. Figure 9.8 illustrates the results obtained in the first and second stage of the retrieval process, respectively. While the similarity score computed with respect to the estuary description is the same as that one given in Fig. 9.5, different results are generated by Cbem for the model similarity degree. The second case in the list (‘Tay, 2-D moving, estuary sim. = 0.59, model sim. = 0.94’) has the highest value of model similarity. The 2-D moving model, which is capable of coping with changes in the flow channel boundaries (Nassehi and Kafai 1999), provides the necessary accuracy for a research exercise. Unfortunately the high level of accuracy provided comes with a significant simulation time due to the complex computational scheme that the model employs. 9.3.2.2

Similarity Measurements

The similarity with respect to the indices given in the estuary description is measured differently if the index is expressed either on quantitative or qualitative scale or it is defined as an object symbol. For those indices that are represented by numerical values, the similarity is measured using a similarity function. For instance, the similarity with respect to any quantitative index F is given as:  I ! −1 F − F R 2  √ (9.1) SIM F I , F R = 1 + FR with I and R referring to the input and the retrieved case, respectively. In addition, for those qualitative indices such as the average width to the average depth ratio and grade of meandering represented using the fuzzy set theory, the

250

S. Passone et al.

1. exclude case-models with respect to the model dimension. 2. select the following features from the model description: purpose (h), accuracy (i), simulation time (j). 3. apply a set of pre-determined rules to establish the functional role Mk of each feature with respect to the purpose of the current investigation. 4. assign a grade of relevance P=0.75 to (h), 1 to (i) and 0.75 (j). 5. apply the criterion of preference with respect to the model dimension. If a model is ‘preferred’, assign the value 0.5 to the match value Mk and to the grade of relevance P. 6. normalize the match aggregate score=

N P

Pk Mk

k=1 N P

with N equal to 4 if the Pk

k=1

criterion of preference is valid, otherwise N is equal to 3. Table 9.2: Second Similarity Score.

similarity measure is based on ‘the geometric model distance’ (Liao et al. 1998), which computes the degree of similarity as function of the multiple class memberships defined using fuzzy sets. For instance, if the multiple membership values of the average width to the average depth ratio for a new case I and a past case R are represented as I(low/a1, moderate/a2, high/a3, very high/a4) and R(low/b1, moderate/b2, high/b3, very high/b4), the degree of similarity is estimated as follows:

I

SIM F , F

R

=1−

4 P

k=1

|ak − bk | 4

(9.2)

Differently from the previous two types of indices considered, the degree of similarity along object symbols (i.e. geomorphological type, tidal range, purpose) cannot be computed employing any standard function. It is up to the system’s designer to establish the similarity between classes of each object. Except for the event when two cases are represented by the same value and the associated similarity is clearly 1, in any other situation it is necessary to assign to each pair of values a pre-determined similarity value. The similarity function must be, then, defined by giving each pair of symbols a weighted distance that takes into account the possible relationship between the two classes, and their characterization. The adequacy of each retrieved model is then evaluated through a pre-determined


251

Figure 9.7: Model selection screen. The problem of salt intrusion is investigated for a application purpose specified as research.

set of match values that rank the accuracy, simulation time consumed and purpose of the model based on the investigation aim of the new problem.

9.4

The GA Module

The retrieved model is selected from the model library, which contains the computer codes of the models described in the case-base unit, and given to the genetic algorithm module (Ga) for being adapted to suit the new case (for a general introduction of genetic algorithms, see also Chapter 7 in this book). The Ga module, which is responsible for the case adaptation phase of the case-based reasoning process, carries out the optimization of the model by combining the classical evolutionary theory with problem-specific information. The Ga module incorporates knowledge from the practice of estuarine model calibration by using modified genetic operators. Furthermore, the present scheme benefits from the co-operation with the Cb module by including in the initial population the parameter values from the most similar cases. As it will be illustrated later, the use of knowledge augmented operators and case-based initialization improves the search performance. It finds a better set of parameters for the new problem and requires less time than the classical genetic approach. The Cbem procedure terminates and returns to the user a model scheme retrieved among the past cases as the best match, and a new set of parameters

252

S. Passone et al.

Figure 9.8: Results generated by Cbem in case the problem of salt intrusion is investigated for research purpose.

that provides satisfactory performance. The model can now be applied to simulate different scenarios. The implemented genetic algorithm focuses on the optimization of the Manning’s coefficient, which represents in the model equations the resistance provided by the estuary bed to the water flow. A value of the Manning’s coefficient must be given for each of the sections into which the estuarine domain is divided with respect to the numerical scheme adopted (Fig. 9.9) (i.e. based on the numerical method considered the model equations are discretized for each domain section). Thus, the genetic algorithm is designed to find the appropriate set of Manning’s coefficient values that better suit the representation of the estuarine processes given by the model and the numerical solution scheme chosen.

9.4.1

Chromosome Representation

The decimal base is used for the chromosome representation (Fig. 9.10). As the Manning’s coefficients differ from one value to another only in the last two digits, a chromosome is represented as a sequence of integer numbers each of which corresponds to the second and third decimal places of each Manning’s coefficient value in the set. Thus, the decimal base provides the necessary correspondence between the sets of Manning’s coefficients and the chromosomes. As the number of elements in a discretized domain can be considerably high, this representation also facilitates the passage to and from the phenotypical representation during the selection process and, the transformation of the chromosomes by genetic operators.


253

Figure 9.9: Finite element discretization of the Upper Milford Haven estuary (a Manning’s coefficient is assigned to each element in the domain).

254

S. Passone et al.

Figure 9.10: Example of a set of Manning’s coefficients and representation of the corresponding chromosome.

9.4.2

Fitness Function

The fitness function consists of estimating the discrepancy ρ between the water surface elevations (Hm ) measured at specific locations within the estuary, and their corresponding simulated values (Hs ). For each sampling station a set of experimental data is given which corresponds to the water surface elevations observed at different n time levels. By representing the total number of sampling stations as J and the total number of samples collected at each station during a tidal period as N , the series of all measured water surface elevations can be expressed as Hm ={(hm )jn , j=1,. . . , J; n=1,. . . ,N} and the set of all simulated values as Hs ={(hs )jn , j=1,. . . , J; n=1,. . . , N}. Hence, the discrepancy between Hm and Hs is given as (Babovic and Abbott 1997): 1/2  N J X 2 X (hs )nj − (hm )nj  (9.3) ρ (Hs , Hm ) =  j=1 n=1

In order to find which chromosome gives a minimum for equation (9.3), water surface elevations for all chromosomes in each generation must be simulated. Hence, hs at each station j for the time levels n is calculated. Thus, for the purpose of calculating the fitness function each chromosome is transformed in the corresponding set of Manning’s coefficients and included in the model input data required. The model is then made run for a number of tidal cycles until the convergence is achieved. Finally, the simulated results are stored in an output datafile and given to the genetic algorithm for calculating the fitness of the chromosome.


9.4.3

255

Initial Population

Ga scheme uses three different modes for initializing the population of chromosomes: the division of the chromosome in segments each of which has the same allele value (zonation option); the arrangement of the values of the genes in descending order (scaling option); and the inclusion, in the initial population, of chromosomes, of sets of Manning’s coefficients inherited from similar cases (use of case information). The initial population is based on the estuary divided into zones of influence each having a specific physical behavior. This is due to the fact that the resistance to the flow changes with respect to the variation of the estuarine physical characteristics. Therefore, it is expected adjacent elements to have similar values for the Manning’s coefficient. The division of the estuary is reflected in each chromosome by partitioning it into a number of segments corresponding to the estuarine zones (Fig. 9.11). Based on this chromosome’s structure the genes in the same segment are initially assigned with the same value, which is randomly generated.

Figure 9.11: Example of chromosome initialized using the zonation and scaling options.

The observation that the flow resistance generally decreases towards the estuary mouth is also taken into account during the initialization. Based on this evidence, the Ga program sorts the alleles of chromosomes in descending order, with lower values for genes that correspond to elements of the domain allocated towards the estuary mouth (scaling option). In the example provided in Fig. 9.11, the values of the genes gradually decrease from zone A towards zone C, which include the estuary’s head and mouth, respectively. The last feature implemented for generating the initial population consists of seeding the cluster with appropriate Manning’s coefficient series selected from the system’s case-base. Based on the principle that similar problems should have similar solutions (Louis and Johnson 1997), estuaries that do not significantly differ from one another should have similar sets of Manning’s coefficients. The sets are preventively adapted to suit the discretization scheme employed for the estuary under investigation.

256

S. Passone et al.

9.4.4

GA Operators

The Ga operators of selection, crossover and mutation are also designed to incorporate concepts from the theory of estuarine calibration for the purpose of Manning’s coefficient optimization.

9.4.5

Selection

Starting from the initial population the subsequent generations are formed by selecting the chromosomes according to their fitness. The selection procedure implemented consists of : • keeping 10% of the best chromosomes (i.e. with highest fitness values) in the next generation (i.e. elitist approach) • and having the other 90% of the next generation randomly reproduced according to their fitness values (i.e. roulette wheel) and then transformed by crossover and mutation in order to introduce diversity in the population. The application of the elitist logic stops the search to converge too quickly.

9.4.6

Crossover and Mutation

The present scheme also contains different forms of the more common random mutation and crossover. The crossover and mutation operators are devised to guide the search towards chromosomes with a real physical meaning for estuarine calibration. Therefore, the traditional genetic operators are modified according to the previously made observations of adjacent genes representing adjacent elements and chromosome’s segments corresponding to specific estuary’s zones. The crossover operator swaps between chromosomes segments which correspond to specific estuary’s zones. The number of cut-points in a chromosome is randomly chosen each time the crossover operator is applied (Fig. 9.12). The mutation operator implemented here is based on the concept that close elements are generally characterized by similar Manning’s coefficients. Thus, the chromosomes are mutated by changing the value of a randomly chosen gene and its closest neighbors (Fig. 9.13).

9.4.7

Model Results with GA Calibration - Upper Milford Haven Estuary

In order to illustrate the effectiveness of the Ga module, we consider the 1-D network model used for Fal estuary (Uk). This has been retrieved by Cbem as the most appropriate for modeling salt intrusion in the Upper Milford Haven estuary for management purpose (Fig. 9.6). The water surface elevations obtained from the


257

Figure 9.12: Chromosomes before and after the crossover operation.

model will be sufficient for demonstrating the superiority of the Ga module when compared to the classical manual calibration. The Ga model optimization operated is executed with the population set to 30 individuals and the rate of crossover and mutation equal to 0.5 and 0.01, respectively. The Ga based calibration is carried out for 15 generations. The estuary is divided into three main zones corresponding to the two branches (i.e. Western Cleddau and Eastern Cleddau rivers) and the main channel (the Decleddau reach) (Fig. 9.9). Based on this partition of the estuary, the chromosome population is initialized using the zonation and the scaling options. These chromosomes are then transformed by the modified mutation and crossover operators. Only one set of Manning’s coefficients from the case-library is included in the initial population. This is the set of parameters employed for this particular model to simulate the salt intrusion in the Fal estuary. The simulated water surface elevations, generated using the set of Manning’s coefficients selected by the Ga module, are presented for the stations of Port Lion,

Figure 9.13: Chromosome before and after the mutation operation.

258

S. Passone et al.

Picton Point and Easthook in Fig. 9.14. The simulations at these locations show that the tidal wave has a regular shape within the main channel becoming gradually distorted as it propagates upstream from the junction into the branches. This is consistent with the geomorphological characteristics of the Upper Milford Haven estuary. The observed and simulated water surface elevations, obtained by the manual optimization of the model, at the described locations are also shown in Fig. 9.14. The comparison between the observed data and the simulated water surface elevations for the manual and the Ga based calibrations shows that the model using the Ga module yields a better performance. The superiority of the Ga based calibration over the trial and error optimization is also demonstrated by considering the time necessary to carry out these two processes. In general, manual calibration of a model requires two weeks to one month (working time) while the Ga based calibration takes 10 hours of Cpu time on a shared Sun workstation (i.e. sun-cc211).

9.5

Conclusion

Cbem is a case-based reasoning system for estuarine modeling that aims to provide the necessary guidance for selecting the estuarine model that better matches to user’s requirements and the characteristics of the problem under investigation. Cbem consists of three modules: a case-based module, a genetic algorithm module and a library of numerical codes. These modules, integrated to work as a single tool, are activated to perform specific tasks of the case-based reasoning methodology. The system accumulates past experience in form of previous cases and, change and adapt them to solve new problems. The current prototype is equipped with examples of past solutions recorded according to the features of specific estuaries and the models employed. However, the system’s ability extends beyond simple retrieval of similar cases. It provides the user with direct information regarding the modification of previous schemes according to a given new problem. The implemented genetic algorithm, hybridized with specific knowledge of estuarine model calibration, is used for adjusting the values of the Manning’s friction coefficient for the selected model to suit particular estuarine environments. Cbem is designed for a user whose field is related to the management and protection of estuaries. Cbem is not required to have any specific and deep expertise in estuarine science. For instance, the indexing scheme utilized for the estuary description, which has been demonstrated to provide a satisfactory set of indices for estimating the similarity between estuaries, does not include canonical estuarine parameters (i.e. stratification parameter, circulation parameter, densimetric Froude number and estuarine Richardson number) as inexperienced users in estuarine hydrodynamics may misuse this type of data (Dyer 1997). The system is also able to distinguish between the constraints and requirements related to the estuary’s physical characteristics as well as those regarding the specific problem to be simulated.


259

Figure 9.14: Simulation of the water surface elevations at (a) Port Lion, (b) Picton Point and (c) Easthook during the spring tide of 25th April 1979.

260

S. Passone et al.

Thus, through the comparison with past cases the user is able to understand how the selection of a modeling strategy is affected by the specific estuary’s physical and hydrological characteristics, and the model requirements for accuracy, investigation purpose and simulation time consumed Therefore, although no numerical modeling expertise is necessary, in order to properly use Cbem, the user must be able to classify the model according to some general technical features and in terms of model performance. Further development is essential for overcoming the current limits of Cbem. For instance, more interaction between the user and Cbem is needed. This would permit to progressively and critically train the user and increase his/her critical judgment. The current version of Cbem is also limited by the fact that the Ga module optimizes the model with respect to only one type of parameter: the Manning’s coefficient, representing the resistance of the estuary bed to the flow. Through the implementation of a multi-objective genetic algorithm Cbem would be able to provide the calibration of other parameters such as the dispersion coefficient. In addition, the model description scheme (e.g. numerical method, time step, accuracy) also needs to be organized in more detail. This will provide the user with a more precise estimation of the overall model’s adequacy. Thus, in the logic of model composition, this will allow the retrieval mechanism to identify among the past cases those containing important information (e.g. set of specific assumptions, routines and data) that should be combined to form a model strategy for the new case. Finally, Cbem should also be extended to support the interpretation of simulation results. It would be useful if Cbem could be provided with a critical background and uncertainty analysis to guide the user through the evaluation of the predicted scenarios and the design of appropriate estuarine management policies.

References Abbott, M. (1991). Hydroinformatics – Information Technology and the Aquatic Environment. Aldershot: Avebury Technical. Babovic, V. and M. Abbott (1997). The evolution of equations from hydraulic data, 1. Theory. Journal of Hydraulic Research 3, 397–410. Bikangaga, J. and V. Nassehi (1995). Application of computer modelling techniques to the determination of optimum effluent discharge policies in tidal water systems. Water Resources 29 (10), 2367–2375. Bridgewater, P. (2003). Science for sustainability: an estuarine and coastal focus. Estuarine, Coastal and Shelf Science 56, 3–4. Chau, K. W. and W. Chen (2001). A fifth generation numerical modelling system in coastal zone. Applied Mathematical Modelling 25, 887–900. Chung, P. and R. Inder (1992). Handling uncertainty in accessing petroleum exploration data. Revue de L’Institut Francais du Petrole 47 (3), 305–314.


261

Comas, J., E. Llorens, E. Mart´ı, M. Puig, J. Riera, F. Sabater, and M. Poch (2002). Knowledge acquisition in the streames project: the key process in the expert system development. In BESAI Workshop, ECAI conference. Cortés, U., M. S` anchez-Marrè, R. Sang¨ uesa, J. Comas, J. Roda, M. Poch, and D. Ria˜ no (2001). Knowledge management in environmental decision support system. Ai Communication 14, 3–12. Cunge, J. and M. Erlich (1999). Hydroinformatics in 1999: What is to be done? Journal of Hydroinformatics 1, 21–31. Dyer, K. (1997). Estuaries: a Physical Introduction. New York: John Wiley & Sons. Dyke, P. (1996). Modelling Marine Processes. London: Prentice-Hall. Gesamp (1990). The state of the marine environment. Technical Report 39, Gesamp, Imp, London. Gijsbers, P., R. Moore, and C. Tindall (2002). HarmonIT: towards an open modelling interface and environment to harmonise european developments in water related simulation software. In I. Cluckie, D. Han, J. Davis, and S. Heslop (Eds.), Proc. 5th Int. Conf. on Hydroinformatics, Cardiff, pp. 1268–1275. Grigg, N. (1989). Chemicals In The Aquatic Environment: Advanced Hazard Assessment, Chapter Water resources modelling. Berlin: Springer-Verlag. Hall, J. (2002). Random set uncertainty handling in hydroinformatics. In I. Cluckie, D. Han, J. Davis, and S. Heslop (Eds.), Proc. 5th Int. Conf. on Hydroinformatics, Cardiff, pp. 1388–1393. Hinwood, J. B. and I. Wallis (1975). Classification of models of tidal waters. Journal of Hydraulics Division, Proc. of A.S.C.E. 101 (HY10), 1315–1351. Holt, A. and G. Benwell (1999). Applying case-based reasoning techniques in gis. The International Journal of Geographical Information Science 13 (1), 9–25. Kennish, M. (1986). Ecology of Estuaries, Vol.1: Physical and Chemical Aspects. Boca Raton: CRC Press. Kennish, M. (2002). Environmental threats and environmental future of estuaries. Environmental Conservation 29 (1), 78–107. Kolodner, J. (1993). Case-Based Reasoning. San Mateo: Morgan Kaufmann. Liao, W. T., Z. Zhang, and C. R. Mount (1998). Similarity measures for retrieval in case-based reasoning systems. Applied Artificial Intelligence 12, 267–288. Louis, S. J. and J. Johnson (1997). Solving similar problems using genetic algorithms and case-base memory. In Proceedings of the Seventh International Conference on Genetic Algorithms, San Mateo, pp. 101–127. Morgan Kaufman. Mpimpas, H., P. Anagnostopoulos, and J. Ganoulis (2001). Modelling of water pollution in the thermaikos gulf with fuzzy parameters. Ecological Modelling 142 (1-2), 91–104. Nassehi, V. and A. Kafai (1999). A moving boundary computer model for hydrodynamics and salt transport in the tay estuary. Advances in Environmental Research 3 (3), 293– 308. Savic, D. and G. Walters (1999). Hydroinformatics, data mining and maintenance of UK water networks. Anti-Corrosion Methods and Materials 46 (6), 415–425.

262

S. Passone et al.

Thomann, R. (1998). The future ”golden age” of predictive models for surface water quality and ecosystem management. Journal of Environmental EngineeringASCE 124 (2), 94–103. Watson, I. and F. Marir (1994). Case-based reasoning - a review. Knowledge Engineering Review 9 (4), 327–354.

CHAPTER 10

Credal Networks for Hazard Assessment of Debris Flows Alessandro Antonucci, Andrea Salvetti, and Marco Zaffalon

10.1

Introduction

Debris flows are among the most dangerous and destructive natural hazards that affect human life, buildings, and infrastructures. Worldwide, these phenomena claim hundreds of lives and millions of dollars in property losses every year. Starting from the 1970s significant scientific and engineering advances in the understanding of the processes and in the recognition of debris flow potential have been achieved (for a review see Costa and Wiekzorek 1987; Iverson et al. 1997). Several experiments have been conducted in many laboratories with the aim of investigating the processes of generation, motion and deposition of debris flows. In parallel, field investigations focused on geologic and topographic characteristics of the source areas and statistical analysis of the triggering variables (e.g., total rainfall depth and maximum intensity) have been carried out in order to derive indicative threshold values for debris flow initiation. Takahashi (1991) was the first to propose a comprehensive theory behind the mechanism of debris flow dynamics (Section 10.2.1). Despite these advances, many aspects of the whole process are still poorly understood, and hazard identification, as well as hazard map construction, keep on relying heavily on human expertise. There is also a further limitation in the practical use of models of debris flows, even if accurate: i.e., the fact that collecting measures of model parameters “on the field” can be difficult, as only vague observations are often possible to do. In order to close the gap between theory and practice it seems necessary to merge theoretical models with qualitative human knowledge, and to recognize that uncertainty is a pervasive component of the phenomena under study. It is also particularly important to identify a framework for credibly dealing with uncertainty that is flexible enough to cope with complex scenarios and different sources of information, such as theoretical models, historical data, and expert assessment. We identify this framework with credal networks (Cozman 2000a), an imprecise probability model that extends Bayesian networks (Pearl 1988) to manage sets of probability mass functions (see Section 10.2.2.3). Imprecise probability (Walley 1991) is a very general theory of uncertainty (regarding further applications of Walley’s theory cf. Chapter 1 in this book) that measures chance and uncertainty without sharp numerical probabilities, and that is founded on rationality criteria similar to the

264

A. Antonucci et al.

Bayesian theory (see Walley 1996b and de Cooman and Zaffalon 2004, Section 2 for introductions to the theory). This Chapter presents a credal network model developed to characterize the debris flow hazard in the Ticino canton, southern Switzerland. The model is based on the qualitative representation of causal knowledge and on imprecise probability quantifications of uncertainty. The causal structure is represented by a directed graph, connecting the triggering factors for debris flows, drawn on the basis of literature results and domain expertise (Section 10.3.1). Section 10.3.2 describes the quantification of uncertainty by probability intervals, showing how intervals are obtained from historical data, expert knowledge, and physical theories about debris flows. The advantages of using intervals here are twofold. When inference of probabilities from data is concerned, probability intervals allow us to carefully deal with the problem of modeling prior ignorance, producing credible posterior probabilities (see the description of the imprecise Dirichlet model in Section 10.2.2.2). When the problem is to turn expert’s knowledge into probabilistic assessments, probability intervals provide quite a flexible modeling tool, not forcing an expert to express knowledge by precise numbers. Overall, credal networks allow us to represent theoretical and empirical knowledge coherently as a single model of debris flow hazard. We are not aware of other models of debris flows hazard with this characteristic. The primary aim of a model like the one presented in this Chapter is to support experts in the prediction of dangerous events of debris flow. We have made preliminary tests in this sense by evaluating the predictions of the model on a set of historical cases of debris flows happened in the Ticino canton. The case studies highlight the good capabilities of the model: for all the areas the model produces meaningful probabilities of hazard. We discuss critically the results in Section 10.5, showing how the results are largely acceptable by a domain expert, and suggesting lines for further sophistication of the proposed model. The positive performance of the model will have to be confirmed by further analysis in the direction of putting the model to work in practice. However, the evidence already supports the proposed approach as a promising way to play a significant role in the modeling of complex phenomena such as debris flow initiation.

10.2

Background

10.2.1

Debris Flows: Triggering Factors and Mobilization

Debris flows are a gravity-induced mass movement intermediate between a landslide and a water flood. They are composed of a mixture of water and sediment with a characteristic mechanical behavior varying with water and soil content. Debris flow events happen because of some natural forces that destroy the structure of the soil and sustain the particles dispersed in the flow, which were originally part of a stable

Credal Networks for Hazard Assessment of Debris Flows

265

mass on a gully bed or on a basin slope. Three types of debris flow initiation are relevant. The first type is due to erosion of a channel bed. As a consequence of intense rainfall, surface runoff appears on a steep channel bed, in which a large amount of material was accumulated; the water destabilizes and entrains the available debris to form the debris flow. The second type of initiation is due to landslide: the slid soil mass transforms into debris flow. The third type is the destruction of natural dams. A previous landslide, which dams up a creek, is suddenly destroyed by the water over-topping the dam, which abruptly collapses. The mechanism to disperse the materials in flow depends on the properties of the materials (grain size, friction angle), channel slope, flow rate and water depth, particle concentration, etc., and, consequently, the behavior of flow is also various. According to Costa (1984), prerequisite conditions for most debris flows include an abundant source of unconsolidated fine-grained rock and soil debris, steep slopes, a large but intermittent source of moisture (rainfall or snow-melt), and sparse vegetation. Rainfall triggering of debris flows and other landslides in steep terrain has been intensively studied. Investigations have included the estimation of empirical relationships between debris flow occurrence and rainfall intensities and durations (e.g., Caine 1980; Cannon and Ellen 1985) and deterministic assessments of the hydrologic processes involved (e.g., Campbell 1975; Johnson and Sitar 1990; Wilson and Wieczorek 1995). Principles of soil mechanics, grain-flow mechanics and mixture mechanics provide a framework for assessing debris flow mobilization on a slope or in a river bed. Several hypotheses have been formulated to explain mobilization of debris flows. The models generally focus on failure and mobilization of an infinite slope of homogeneous, isotropic soil. Mobilization hypotheses generally start from the assumption that the soil contains sufficient water to saturate the porosity. The first hypothesis for mobilization derives from the debris flow motion model of Johnson (1984). Takahashi (1981) presented an alternative hypothesis for mobilization, which is part of his model of debris flows, as a water-saturated inertial grain flows governed by the dispersive stress concept of Bagnold (1954). Takahashi’s model best applies where debris flows mobilize from flash floods that abruptly determine surface-water surcharges in relatively gently sloping, sediment-choked channels. The key role of soil mechanics is emphasized from other hypotheses for debris flow mobilization (e.g., Anderson and Sitar 1995), which underline the role of soil liquefaction caused by increasing pore pressure. In this study we adopt Takahashi’s theory (see Section 10.3.1), as the most appropriate to describe debris flow appearance in channel bed with high availability of granular and incoherent material, corresponding to the types of events observed in Switzerland.

266

10.2.2 10.2.2.1

A. Antonucci et al.

Methods Credal Sets and Probability Intervals

In this Chapter we will restrict the attention to random variables which assume finitely many values (also called discrete or categorical variables). Denote by X the possibility space for a discrete variable X, and let x be a generic element of X . Denote by P (X) the mass function for X and by P (x) the probability of x ∈ X . Let a credal set be a closed convex set of probability mass functions (Levi 1980). PX denotes a generic credal set for X. For any event X ′ ⊆ X , let P (X ′ ) and P (X ′ ) be the lower and upper probability of X ′ , respectively. These are defined as follows: P (X ′ ) = P (X ′ ) =

min P (X ′ )

P ∈PX

max P (X ′ ).

P ∈PX

Lower and upper (conditional) expectations are defined similarly. Note that a set of mass functions, its convex hull and its set of vertices (also called extreme mass functions, i.e., mass functions that cannot be expressed as convex combination of other mass functions in the set) produce the same lower and upper expectations and probabilities. Conditioning with credal sets is done by element-wise application of Bayes rule. The posterior credal set is the union of all posterior mass functions. Denote by y PX the set of mass functions P (X|Y = y), for generic variables X and Y . As far as (conditional) independence is concerned, in this paper we adopt the concept of strong independence.1 Two variables are strongly independent when every vertex in P(X,Y ) satisfies stochastic independence of X and Y , i.e., for every extreme mass function P ∈ P(X,Y ) , P (x|y) = P (x) and P (y|x) = P (y) for all (x, y) ∈ X × Y (see also Moral and Cano (2002) for a complete account of different strong independence concepts and Cozman (2000b) for a deep analysis of strong independence, for related aspects of imprecise probabilities cf. also Chapter 1). Let IX = {Ix : Ix = [lx , ux ] , 0 ≤ lx ≤ ux ≤ 1, x ∈ X } be a set of probability P intervals for X. The credal set originated by IX is {P (X) P : P (x) ∈ Ix , x ∈ ′ + X, P P (x) = 1}. I is said reachable or coherent if u X x x∈X ,x6=x′ lx ≤ 1 ≤ x∈X lx′ + x∈X ,x6=x′ ux , for all x′ ∈ X (Tessem 1992). IX is coherent if and only if the related credal set is not empty and the intervals are tight, i.e., for each lower or upper bound in IX there is a mass function in the credal set at which the bound is attained (Campos et al. 1994; Walley 1991). 1 Here we follow the terminology introduced by Cozman. Note that other authors use different terms (Couso et al. 2000).

Credal Networks for Hazard Assessment of Debris Flows 10.2.2.2

267

The Imprecise Dirichlet Model

Consider a random sample, i.e., a set of values x ∈ X for the discrete random variable X, drawn independently with chances θx . We call chances the objective probabilities of the values x ∈ X . The chances θx , which are also the parameters of an underlying multinomial distribution, are rarely known in practice, and here we assume that all the information about them is represented by the sample. Following the Bayesian approach (Bernardo and Smith 1996), one’s beliefs about the chance of x can be summarized by the epistemic probability (or, more simply, probability) of x, which is the expectation of θx with respect to the distribution of the chances conditional on the data. Indeed, the Bayesian approach to statistical inference regards the parameters of an unknown distribution as random variables. The uncertainty about the parameters prior to the availability of the sample is modeled by a density function, called prior. The mentioned posterior distribution is obtained by updating the prior, after observing the data, by means of Bayes’ theorem. In the case of multinomial samples, the Dirichlet (s, t) density is the traditional choice for the prior. Here s and t are hyper-parameters. In particular, t is a vector of tx hyper-parameters (x ∈ X ). Denoting by θ the vector of the chances, the Dirichlet (s, t) density is given by π(θ) ∝

Y

θxstx −1

(10.1)

x∈X

P where s > 0, 0 < tx < 1 for each x ∈ X , x∈X tx = 1, and the proportionality constant is determined by the fact that the integral of π(θ) over the simplex of possible values of θ is 1. The constant s determines the weight of the prior towards the number of units in the sample: the larger s, the larger the number of units needed to smooth the effect of the prior information. The Bayesian approach is applied also when substantial prior knowledge is not available. Special, so-called, non-informative priors (e.g., Bayes-Laplace’s uniform prior), are used in this important case. Unfortunately, the outlined method has some drawbacks related to the subjectivity in the choice of the prior (a discussion of the problems with non-informative priors is reported in Walley 1991, pp. 226–235). A more objective way to model prior ignorance has been proposed by Walley (1996a) as a generalization of the Bayesian approach. According to Walley, prior ignorance should be modeled by a set of prior densities. In particular, with multinomial samples, the set is constituted by all the Dirichlet densities obtained by fixing s in (10.1) and letting the t-hyper-parameters take all the possible values in their domain of definition. The resulting model is called imprecise Dirichlet model. If we regard a precise prior as a particular state of information, the rationale behind the imprecise Dirichlet model becomes clearer: i.e., to use the set of all the possible prior states of information as a model of prior ignorance (of course, for this interpretation to hold, s must be fixed).

268

A. Antonucci et al.

By updating the set of priors to a set of posteriors conditional on the data, the imprecise Dirichlet model allows posterior inference to be realized similarly to the Bayesian case, with an important difference: the imprecise Dirichlet model leads to lower and upper posterior probabilities. These are achieved by (i) computing the expected value of a chance with respect to the posterior obtained for generic value of t and (ii) by minimizing and maximizing this quantity over t’s domain of definition. The width of the interval delimited by the lower and upper probabilities, is determined by the “weight” of our prior ignorance towards the number of units in the sample. The interval is vacuous before observing any data, i.e., the epistemic probabilities range from 0 to 1. As data are collected the interval narrows, producing a precise probability in the limit of an infinite sample. Stated differently, working with intervals rather than precise numbers is a way to produce reliable epistemic probabilities for any sample size. The posterior interval of probability produced by the imprecise Dirichlet model for a given x is # (x) # (x) + s , (10.2) N +s N +s where # (x) counts the number of units in the sample in which X = x, N is the total number of units, and s is like in (10.1). The above lower and upper epistemic probabilities can also be interpreted as lower and upper bounds on the relative frequency of x if we imagine that there are s hidden observations as well as the N revealed ones. With the imprecise Dirichlet model, s is also interpreted as a parameter of caution of the inferences and is usually chosen in the interval [1, 2]. By the larger value s = 2, inferences from the imprecise Dirichlet model encompass a wide range of those obtained with Bayesian models developed for the case of prior ignorance (see Walley 1996a, Bernard 1996, and Bernard 2001 for insights about the choice of s). Expression (10.2) permits to infer probability intervals from multinomial samples and hence, as highlighted previously, to infer the credal sets originated by these intervals. Note that sets of probability intervals obtained using (10.2) are reachable. In this chapter all the model probabilities inferred from data will be of the type (10.2). 10.2.2.3

Credal Networks

We define credal networks in an incremental way, starting from Bayesian nets. A Bayesian network is a pair composed of a directed acyclic graph and a collection of conditional mass functions. A node in the graph is identified with a random variable Xi (we use the same symbol to denote them and we also use “node” and “variable” interchangeably). Each node Xi holds a collection of conditional mass functions P (Xi |pa (Xi )), one for each possible joint state pa (Xi ) of its direct predecessor nodes (or parents) P a(Xi ).


269

Bayesian nets satisfy the Markov condition: every variable is independent of its non-descendant non-parents given its parents. From the Markov condition, it follows (Pearl 1988) that the joint mass function P (X) = P (X1 , . . . , Xt ) over all the t variables of the net is given by P (x1 , . . . , xt ) =

t Y i=1

P (xi |pa (Xi ) ) ∀(x1 , . . . , xt ) ∈ ×ti=1 Xi ,

(10.3)

where pa (Xi ) is the assignment to the parents of Xi consistent with (x1 , . . . , xt ). A credal network is a pair composed of a directed acyclic graph and a collection of conditional credal sets. The graph is intended to code strong dependencies: every variable is strongly independent of its non-descendant non-parents given its parents. pa(X ) A generic variable, or node of the graph, Xi holds the collection of credal sets PXi i , one for each possible joint state pa (Xi ) of its parents P a(Xi ). We assume that the credal sets of the net are separately specified (Ferreira da Rocha and Cozman 2002; Walley 1991): this implies that selecting a mass function from a credal set does not influence the possible choices in others. This assumption is natural within a sensitivity analysis interpretation of credal nets. Now we can define the strong extension (Cozman 2000b), i.e., the set P of joint mass functions associated with a credal net: o n pa(X ) P = CH P (X) as from (10.3) : P (Xi |pa (Xi ) ) ∈ PXi i , i = 1, . . . , t where CH denotes the convex hull operation. In other words, P is the convex hull of the set of all the joint mass functions as from (10.3), that are obtained by selecting conditional mass functions from the credal sets of the net in all the possible ways. The expression for the strong extension should make it clear that there is a natural sensitivity analysis interpretation of credal networks: a credal net can be regarded as the set of the Bayesian nets corresponding to the vertices of the strong extension.

10.3

The Credal Network for Hazard Assessment of Debris Flows

10.3.1

Network Description

The physical processes and the triggering factors responsible for debris flow initiation have been already mentioned in Section 10.2.1. Experts’ intuitive knowledge, field observations, and theoretical approaches have been logically related and fourteen nodes were finally identified to represent the model structure. This section details the construction of the proposed model for the hazard assessment of debris flows, their physical meaning and their qualitative or quantitative causal relationships, as expressed by the network in Fig. 10.1.

270

A. Antonucci et al.

Geology (G)

Permeability (T)

Land Use (L)

Local slope (N)

Watershed area (W)

Granulometry (F)

Actual available debris thickness (A)

Precipitation intensity (D)

Hydrologic soil type (H)

Maximum soil water capacity (R)

Channel width (C)

Water depth (B)

Movable debris thickness (theoretical) (M)

Movable debris thickness (Q)

Figure 10.1: The causal structure.

The leaf node Q (movable debris thickness) is the goal of the analysis: it is defined as the depth of debris likely to be transported downstream during a flood event. We assume the probability distribution associated with different depths as an integral indicator of the hazard level. Node G (geology) describes the characteristics of the bedrock in a qualitative way. Rocks are continuously weathered as a result of wind, sun, rain, or frost. Debris flows require a minimum thickness of colluvium (loose, incoherent deposits at the foot of steep slope) for initiation (Reneau et al. 1984). Debris flows occur in colluvium filled bedrock hollows, produced from a variety of bedrock although no clear dependence was found between bedrock geology and debris flow occurrence. This observation is considered in the graph with the connection to node A (actual available debris thickness) and primarily expresses the propensity of different rocks to produce sediment. Additionally, bedrock properties determine the maximum rate of infiltration and deep percolation, so influencing the generation of surface runoff and the concentration in the drainage network. This influence is accounted for by the connection of the geology to the hydrologic soil type (H), which influences the maximum soil water capacity (R). A second important variable is permeability (T), i.e., the rate at which fluid can flow through the pores of the soil. If soil has high permeability, rainwater will soak into it easily. If permeability is low, rainwater will tend to accumulate on the surface or flow across the surface if it is not horizontal. The causal relation among geology and permeability determining the different hydrologic soil types was adopted according to the work of Kuntner (2002), who recently used the Geostat informa-


271

tion system2 (Kilchenmann et al. 2001) to build a correspondence table between the hydrologic soil groups of the curve number soil conservation service (USDA 1993) infiltration method and the soil of Switzerland. The basic assumption is that soils with high permeability and extreme thickness show a high infiltration capacity, whereas shallow soils with extremely low permeability have a low infiltration capacity. Another factor that is a significant cause of debris movement is the land use (L) cover of the watershed. The land use characterizes the uppermost layer of the soil system and has a definite bearing on infiltration. It describes the watershed cover and usually considers every vegetation type (fallow as well as non-agricultural uses), such as water surfaces, roads, glacier, etc. A forest soil, for example, has a higher infiltration rate than a paved urban area. It is often difficult to precisely assess the role of land use on debris flows. Vegetation is also affected by bedrock type, slope and rainfall, but these influences are too weak in order to be considered into the graph. For each land use (L) and hydrologic soil type (H) it is possible to define a curve number value and a corresponding maximum soil water capacity (R). The infiltration capacity of the soil decreases from h1 to h4 (see descriptions in Tab. 10.1). State h1 h2 h3 h4

Description Low runoff potential Moderate infiltration rates Low infiltration rates High runoff potential

P (hi |g2 , t4 ) [0.0000, 0.0247] [0.0000, 0.0247] [0.0988, 0.1235] [0.8765, 0.9012]

Table 10.1: A conditional credal set for node H, specified via probability intervals inferred from data.

The amount of rainfall which cannot infiltrate is considered to accumulate into the drainage network (superficial runoff), increasing the water depth and eventually triggering a debris flow in the river bed. These processes are described by the deterministic part of the graph, related to runoff generation and concentration, and debris flows mobilization mechanism which take into account topographic, morphologic and climatic parameters, such as local slope (N) of the source area, watershed area (W), channel width (C), and precipitation intensity (D). The climate of the regions in which debris flows are observed is as varied as geology. In addition to the duration and intensity of a storm that ultimately produces a debris flow, the an2 The Geostat database is operated by the Swiss Federal Office of Statistics and contains a federal Gis database of geocoded, spatially relevant data sets coming from various, mostly governmental, sources. Three thematic maps are available: land use raster map, with a grid size of 100 × 100 meters; soil type raster map, containing 144 soil classification units, derived from the combination of typical landscape units and soil characteristics; and a simplified geotechnical raster map, which contains 30 classes characterizing the first geological layer under the soil.

272

A. Antonucci et al.

tecedent soil moisture conditions is recognized as an important characteristic. The significant period of antecedent rainfall varies from days to months, depending on local soil characteristics. In this study the peak runoff and the corresponding water depth (B) in the torrent channel was determined using the mentioned curve number infiltration method and the rational formula method. The corresponding theoretical equation is3 b(c, d, n, r, w) =

(

k1 w 5√ k2 c 3 tan n

0

·

r 2 ) (d− 20 r (d− 20 )+r

if d >

r 20

otherwise

√ where k1 = 0.278 is a constant and k2 = 25 3 m/s is the coefficient of bed roughness. The equation follows from Manning-Strickler’s formula and the rational formula (see Maidment 1993 as general reference). Intuitively the formula expresses the proportionality between the water depth b and the watershed area w, as well as the decrease of b for increasing slope n and channel width c. The term r/20 reflects water dispersion reducing the effect of the precipitation d. According to the rational formula, the maximum runoff in a catchment is reached when all parts of the watershed are contributing to the outflow. This happens when the time of concentration (the time after which the run-off rate equals the excess rainfall rate) is reached. In the specific case we focused on the identification of areas as likely sources of debris flows: it is well known from several observations, that debris flows often occur in different surges and the hydrological components only constitute one of the several variables involved. Furthermore it is widely recognized that the drainage network of small torrents is not univocally defined and highly variable during the flood event. In this perspective topographic variables, like the length of the main channel and the time of concentration, loose the classical meaning assumed in standard hydrological applications. For these reasons fixed rainfall duration (one hour) was defined as critical rainfall interval for further analysis. Rivers and torrents adjust their width, depth, and slope to maintain a balance between the water and sediment supplied from upstream, and that exported at the outlet. In this view there may be influences from geology and intensity of precipitation to channel width. We have neglected these associations in the graph structure as they would be quite weak; they are also still a debated research issue in geomorphology.4 The channel width is obviously decisive to determine the water depth, given the runoff generated within the watershed according to the standard hydraulic assumptions. Field experience in the study region (Ticino canton) indicates that debris flows often start in very steep and narrow creeks, with reduced accumulation area upstream. The complexity and the organization of the channel geometry is 3

By abusing notation, we will use the same letter for generic values of nodes and the corresponding variable in the theoretical formulae. 4 Furthermore, channel width is systematically measured in our application, thus blocking any possible influence mediated by channel width, on the variable to predict (Q).


273

therefore usually low and almost similar in the debris flow prone watersheds. For this reasons it was decided to adopt only three categories of channel width. Further physically based analysis of stream geometry based on topographic studies, fractal analysis and network evolution (Rinaldo 1999) are outside the aim of this study. The granulometry (F), represented by the average particle diameter of the sediment layer, is required to apply Takahashi’s theory (Takahashi 1981) which describes the debris flow mobilization mechanism. A corresponding node was therefore defined in the graph. Note that the theory needs also the friction angle, but no node was created in this case as there exists an empirical one-to-one relationship between granulometry and friction angle. According to the definition of the variables, Takahashi’s theory can finally be applied, in order to determine the theoretical thickness of debris (M) that could be destabilized by intense rainfall events. The effect of a water depth b on the movable quantity m was quantified by Takahashi as i−1 h tan f −1 −1 . (10.4) m(b, f, n) = b k3 tan n The relation is linear, with a coefficient taking into account the local slope n and the internal friction angle f (which can be obtained from the granulometry), where k3 = Cb (db − 1), with db ≃ 2.65 the relative density of the grains and Cb ≃ .7 the volumetric concentration of the sediments. The variables involved in (10.4) must satisfy the constraints tan f 1 b 1 ≤ ≤1+ · 1+ ′ . 1+ k3 tan n k3 f

If the inequality on the left-hand side is violated, shallow landslides can occur also in absence of water depth, but technically speaking these are not debris flows. If the remaining inequality fails, the movable quantity as in (10.4) is thinner than the granulometry f ′ and no flow can be observed. Equation (10.4) proposes a theoretical value for the movable quantity, which does not take into account how much material is physically available. The resulting value must be compared with the actual available debris thickness (A) in the river bed. The minimum of these two values is the leaf node of the graph (Q), which is a proxy indicator of the debris flow hazard level. So, the final relation is given by q(a, m) = min{a, m}.

10.3.2

Network Quantification

This Section reports the quantification of uncertainty for all the variables in the network. Some of the variables are categorical (Tab. 10.2), while the remaining variables have been discretized (Tab. 10.3). Quantifying uncertainty means therefore to specify the conditional mass functions P (Xi |pa(Xi )) for all the nodes Xi and all the possible instances of the parents pa(Xi ). As stated in the Introduction, we allow

274

A. Antonucci et al. Node Possible values T moderately high, normal, mod. hindered, hindered, strongly hind. L forest, pasture, lakes, unprod. vegetation, bare soil, urban area G unconsolidated soil, malrs and conglomerates, limestone, crystalline rock, porphyry, gneiss H low runoff potential, moderate infiltration rates, low infiltration rates, high runoff potential

Table 10.2: Possible values of the qualitative variables.

Node Dim. Discretization intervals C [m] [2,5),[5,7),[7,+∞) Q, A, M [cm] [0,10),[10,50),[50,+∞) D [mm/h] [0,15),[15,30),[30,50),[50,+∞) F [mm] [0,10),[10,100),[100,150),[150,250),[250,+∞) N [deg] [12,15),[15,18),[18,20),[20,22),[22,24),[24,27),[27,30),[30,90] B [cm] [0,3),[3,10),[10,20),...,[90,100),[100,125),[125,150),[150,+∞) W [km2 ] [.2,.4),[.4,.6),[.6,.8),[.8,1),[1,1.5),[1.5,2),[2,3),[3,4),[4,5),[5,+∞) R [c.n.] [30,38.2),[38.2,45.5),[45.5,55),[55,63.5),[63.5,69.1),[69.1,75.4), [75.4,82),[82,95.1),[95.1,100] Table 10.3: Dimensions and discretization intervals of the quantitative variables.

the specification to be imprecise, in the sense that each value P (xi |pa(Xi )) can lie in an interval. The intervals for P (xi |pa(Xi )) (xi ∈ Xi ) define a set of mass functions pa(X ) PXi i , according to Section 10.2.2.3, to which P (Xi |pa(Xi )) belongs. Let us now consider how the intervals have been determined. In the most informative cases, the conditional mass function expresses a functional relation and the intervals reduce to a single degenerate (i.e., 0-1 valued) mass function. We embed this theoretical background about the nodes B, M and Q and the related parents by degenerate mass functions P (B|pa(B)), P (M |pa(M )) and P (Q|pa(Q)) for all the joint states of parents (since variables are discrete and the theoretical formulae are stated for continuous values, this has also involved the choice of representative values for all the states of parents). In the other cases, probabilistic knowledge coming from data or the expert is quantified by probability intervals. We used the imprecise Dirichlet model (Section 10.2.2.2) to infer probability intervals from Geostat (we used the value 2 for the hyper-parameter s, which is the most cautious choice suggested by Walley). Note that the length of the inferred intervals depends on the number of available records, as from (10.2). In the limit of a very large amount of data they are nearly precise. This is the case of the intervals for


275

the root nodes G, T, L and N (e.g., Tab. 10.4), as there is about one million records in the database. The situation changes for the remaining nodes H and R (e.g., State

Description

P (gi )

g1

Unconsolidated soil

0.1821

g2

Malrs and Conglomerates

0.0014

g3

Limestone

0.0930

g4

Crystalline Rock

0.0043

g5

Porphyry

0.0066

g6

Gneiss

0.7126

Table 10.4: Precise mass function for the root node G, as inferred from data.

Tab. 10.1). Conditional probabilities are produced on the basis of the subsets of the database related to the conditioning states of the parents. The available records are thus usually much smaller than one million, for each probability, producing different sizes of intervals. Up to now our quantification process was accomplished under the support of quantitative information (theory and data). Nevertheless some nodes can be quantified only by expert’s knowledge. The capability to integrate in the same model quantitative and empirical knowledge is actually a remarkable feature of the credal network model. An environmental engineer, skilled in assessing the relevant processes in the area under study, expressed the mass functions for W, D, C, F and A, by a collection of intervals for each mass function (see for example Tab. 10.5). On State

Description [mm]

P (fi )

f1

< 10

[0.01, 0.10]

f2

10–100

[0.10, 0.20]

f3

100–150

[0.25, 0.40]

f4

150–250

[0.25, 0.35]

f5

> 250

[0.12, 0.18]

Table 10.5: Expert’s estimates for granulometry.

the technical side, we verified coherence of the proposed intervals (Section 10.2.2.1), whence the non-emptiness of the corresponding credal sets.

10.4

Using the Model to Support Domain Experts

Section 10.3 has described our imprecise probability model for hazard assessment of debris flows. In this section we discuss how to make this model work as a decision support system. To this extent, we need to discuss what kind of outcome we expect

276

A. Antonucci et al.

from the credal network, and by which algorithms we can produce it. With respect to the first point, we focus on the task called updating, which is one of the most popular computations with Bayesian and credal networks. Let E be a vector of variables of the credal network, and let e denote an observed state of E. E is also called the vector of evidence variables, and e is called evidence. Consider any node X not in E. The updating corresponds to compute the posterior probability P (X|E = e) with Bayesian nets, and the lower and upper posterior probabilities P (X|E = e) and P (X|E = e), with credal networks. The updating is intended to update our prior beliefs about X to posterior beliefs given a certain evidence E = e. It is important to stress a methodological point here: the updating represents the correct way to update beliefs if and only if the unobserved network variables are missing at random (Grunwald and Halpern 2003), which roughly means that variables cannot be missing in a systematic way. This point is quite delicate as the failure of the missing-at-random assumption can lead the updating to produce highly misleading results. It is important to realize that this problem has not to do with the model of domain knowledge, rather, with the way the information is provided to the model. With respect to the application under study, this means that the way the field surveys are made in the Ticino canton must be taken into account. In the Ticino canton, the measurement process of the network variables is made in such a way that almost each of them falls in one of the following two categories: variables that are always measured, and variables that are never measured. These variables satisfy the missing-at-random condition in a trivial way. Few other variables, such as granularity, are measured unless the conditions prevent this to happen precisely, and in this case only a set-based, or vague, observation is made. We propose a treatment of vague observations in Section 10.5.2. This, together with the mentioned missing-at-random assumption, appears to provide a reasonably accurate model of the measurement process for the present application. From the computational point of view, the updating problem with credal nets is NP-hard (Ferreira da Rocha and Cozman 2002) also when the graph is a polytree (for a review of updating algorithms with credal nets see Cano and Moral 1999). A polytree is a directed graph with the characteristic that forgetting the direction of arcs, the resulting graph has no cycles. Contrast the situation with Bayesian nets, for which the updating takes linear time in the same conditions. The hardness result is related in some way to the fact that strong extensions can have a very large number of extreme mass functions. Indeed, the updating can be computed in principle by (i) exhaustively enumerating the vertices Pk of the strong extension; and by (ii) minimizing and maximizing Pk (X|E = e) over k, where Pk (X|E = e) can be computed by any updating algorithm for Bayesian networks (recall that each vertex of the strong extension is a Bayesian network). The exhaustive approach is attractive because it is conceptually very easy, but it is often not viable due to the large number of vertices of


277

the strong extension. The present work is a nice exception to the rule, as the type of network, jointly with the way evidence is collected, make the exhaustive approach viable in reasonable times. In practice, we make the above steps (i)–(ii) along the lines used to define the strong extension in Section 10.2.2.3: we consider pa(X) (for all all the possible joint ways to select vertices from the credal sets PX nodes X and all the states of their parents that are consistent with E = e); and the second step follows by executing a Bayesian network updating for each joint selection. Note that the exhaustive algorithm needs credal sets be specified via sets of vertices, rather than as probability intervals. This is not a problem as there are well-known algorithms to convert from one representation to the other. To produce extreme mass functions in the present work, we used the software tool lrs (http://cgm.cs.mcgill.ca/~avis/C/lrs.html) as a preprocessing step of the exhaustive algorithm. See also Campos et al. (1994, p. 174), for a specific algorithm to produce the vertices of a credal set defined via intervals of probability.

10.5

Case Studies

10.5.1

Description

The methodology developed in this work has been applied to some case studies concerned with hazardous areas of the Ticino canton, in Switzerland. The empirical study was intended to validate the model in a preliminary way, by evaluating its adequacy in critical setups: we chose testing areas that underwent a considerable event of debris flow. The conditions of the areas prior to the event (see Tab. 10.6) have been given as input to the credal network model. This information was extracted from Geostat for geology, land use, watershed area and slope (derived from the original digital elevation model). The precipitation intensity, the channel width, the granulometry of the material and the actual available debris thickness were available in form of an event report. The credal network was expected to predict the hazard on this basis, by indicating that substantial detachment of sediment should be expected with significant probability.

10.5.2

Results and Discussion

The results of the analysis are listed in Tab. 10.7. The hazard level is identified with the probability of a defined debris thickness to be transported downstream. In the first three cases the results indicate a high debris flow hazard level in the source area (which corresponds to an instable debris thickness greater than 50 cm, see Tab. 10.3). High probabilities of state q3 are in good agreement with historical observations at the Buffaga Creek (case 1) and at the Cumaval Creek (case 2) were channel bed erosion of more than 50 centimeters were extensively observed after the debris flow event. In case 3 (S. Defendente Creek), a non-negligible probability of

278

A. Antonucci et al.

Case Geology 1 2 3 4 5 6

Land Granulometry Precip. Watershed Slope Channel Avail. use [mm] [mm/h] [km2 ] [deg] [m] [m] Gneiss Forest 10–100 100 0.26 20.8 4 1.0–1.5 Porphyry Forest < 10 70 0.32 19.3 6 1.0–1.5 Limestone Forest < 10 70 0.06 19.3 4 0.7–1.0 Gneiss Unprod. veg. 100–150 30 0.11 21.8 8 > 1.5 Gneiss Forest < 10 50 0.38 16.7 4 0.7–1.0 Gneiss Bare soil 150–250 30 2.80 16.7 8 1.0–1.5

Table 10.6: Details about the case studies. Case 1 happened at Buffaga Creek in the municipality of Ronco sopra Ascona, 28 Aug. 1997. Case 2 at Cumaval Creek in Riva S. Vitale, 2 Jun. 1992. Case 3 at S. Defendente Creek in Capolago, 2 Jun. 1992. Case 4 at Val Creek in Ghirone, 1 Aug. 1998. Case 5 at Cassinello Creek in Claro, 4 Jul. 2000. Case 6 at Foioi Creek in Cavergno, 31 Aug. 1992.

medium movable debris thickness is also related to the small granulometry, which characterized the watershed. Significant probabilities (about 0.90) of the intermediate state q2 (instable debris thickness between 10 and 50 cm) were obtained for case 5 (Cassinello Creek), due to the gentler bed slope (30◦ ) as compared with the other cases. This result is confirmed by data collected during a field survey after the event: a relatively regular and shallow erosion depth was observed along the channel bed, with only few deeper bed erosion. According to antecedent hazard mapping, geologists did not assign a high debris flow hazard level to the Cassinello Creek and the event was mainly triggered by very intense and concentrate precipitation. In case 4 (Val Creek) the low hazard prediction can plausibly be explained through some of the assumptions adopted to design the network: (i) in that case the watershed area is very small and the hydrological hypotheses assumed (rational formula and critical duration of one hour, see Section 10.3.1) in the network are probably insufficient to explain the processes that triggered the debris flow. Additionally (ii), the observed hourly precipitation was not extreme (30 mm/h); the

State q1 q2 q3

1 0.0000 0.0055 0.9945

2 [0.0018, 0.0969] [0.0048, 0.0636] [0.8385, 0.9933]

Cases 3 0.0076 0.2764 0.7160

4 0.5628 0.4372 0.0000

5 0.0062 0.9034 0.0904

6 0.5926 0.3677 0.0397

Table 10.7: Posterior probabilities of the states of node Q, i.e., of the movable debris thicknesses, in the six cases considered. The second case presents significantly imprecise probabilities which are written explicitly as intervals.


279

antecedent soil moisture due to the rainfall of hours and days before, not considered in the graph as a node, played probably a key role in this case and suggests possible further improvements in the graph structure (Antonucci et al. 2004b). In case 6 (Foioi Creek) the probability is decreasing with increasing debris thickness. The evidence is the most extreme out of the six cases: a relatively high upstream area (2.8 km2 ), large channel depth and high medium grain size. These conditions partially explain the results. As a more general comment, it is interesting to observe that in almost all cases posterior probabilities are nearly precise. The conclusions are strong basically because of the deterministic nature of a substantial portion of the network and the strong evidences collected about the cases. This is not the case of Cumaval Creek, for which the probabilities are significantly imprecise. Note, however, that the results are strong in this case, too, as we can still obtain a single more probable state, namely q3 . Of course, obtaining strong conclusions from weakly stated probabilistic assumptions increases our confidence about the plausibility of the conclusions. Now we consider a different type of analysis. So far we have assumed that the evidence was precisely available each time, but this is usually an approximation for the grain size of debris material and the channel depth on the source area. This is not related to our empirical study only; it is a general problem that affects the application of models of hazard of debris flows, and that limits the real application of physical theories, such as Takahashi’s theory for example. We focus on grain size. We wonder whether we can draw credible conclusions in case 6 for which the observation of grain size is vague. From the historical event report, we can exclude that node F was in state f1 or f2 . There are chances that f4 was the actual state (f4 is the evidence used in the preceding experiments for F ), but this cannot definitely be established. In order to study the sensitivity of the results with respect to grain size, we take the most conservative position of letting the states f3 , f4 and f5 be all plausible evidences. Table 10.8 shows the posterior mass functions for node Q obtained by separately considering the values f3 , f4 and f5 for node F . An important conclusion, robust to the vague observation of grain size, is that the most dangerous event of debris flow (q3 ) is very improbable. On the other hand, we cannot establish whether or not some debris flow should be expected; depending on the grain size, q1 is much more probable than q2 and vice versa. As expected, the hazard is very sensitive to the actual grain size. These results are of major importance as they allow the user to indirectly assess the importance of observations. The sketched method could be used, depending of the required level of accuracy, to decide if additionally field surveys are needed to collect other data or if the obtained results are satisfactory for hazard estimation.

280

A. Antonucci et al. State q1 q2 q3

f3 0.0122 0.9481 0.0397

f4 0.5926 0.3677 0.0397

f5 0.9603 0.0000 0.0397

Table 10.8: Posterior probabilities of node Q in case 6 with respect to different values for node F. The observations of the remaining nodes are set according to Tab. 10.6.

10.6

Conclusion

We have presented a model for determining the hazard of debris flows based on credal networks. Credal networks present several advantages for this application. Besides providing a graphical language for modeling cause-effect relationships at the physical level, they make it possible to credibly deal with qualitative uncertainty and, through the imprecise Dirichlet model, with robust inference from data. These characteristics, in turn, favor obtaining credible conclusions, which are a primary objective for this type of application. The presented model has been developed for the Ticino canton, in Switzerland. Extension to other areas is possible by re-evaluating the mass functions obtained from data and expert’s knowledge, which have local nature. In fact, the graphical structure represents causal knowledge that can reasonably be regarded as independent of the geographical area, and the same can be said about the physical theories incorporated in the network (but recall that Takahashi’s theory is best suited to describe debris flows that appear on steep channel beds, so different theories might be worth considering in other cases). Another point that should seriously be taken into account moving to a different context is the assumption that the network variables are missing at random. In this were not the case, further considerations should be made (de Cooman and Zaffalon 2004). Computational issues might also be relevant when moving to another environment. In the present case, the updating was computed in less than one hour, in the worst case, on a 2.8 GHz Pentium 4 machine. Such easy computation depends on the nice interplay of the network structure and the way evidence is typically collected in the Ticino canton. In other geographical regions where the evidence is collected differently, it could be necessary to consider solution methods other than the exhaustive approach (Section 10.4). Among those, the branch and bound method of Ferreira da Rocha et al. (2003) is promising and worth considering. There are many possible improvements of the presented model that could be pursued in the future. These include the sophistication of the model to include further triggering factors as highlighted by the case studies; the extension of the possibility to make vague observations to nodes other than granulometry, as in Section 10.5.2, in the direction of making the model more realistic and easily applicable in prac-


281

tice; the study of different methods of inference in the network to determine the environmental conditions that should be avoided in order to prevent debris flows; and, perhaps most importantly, an extensive empirical study to validate the model under a variety of conditions, which is a prerequisite to put the model to work in practice. Some of these steps have already been taken in Antonucci et al. 2004a and Antonucci et al. 2004b. Debris flows are a serious problem, and developing formal models can greatly help us avoiding their serious consequences. The encouraging evidence provided in this chapter makes credal networks to be models worthy of further investigation in this respect. Acknowledgments. The authors would like to thank the Swiss Federal Office of Topography for providing the digital elevation model, and the Swiss Federal Statistical Office for the land-use, soil suitability, and geotechnical maps. Bayesian network updating has been computed by the software Smile, developed at the Decision Systems Laboratory of the University of Pittsburgh. Extreme mass functions have been obtained by D. Avis’ vertex enumeration software lrs. The authors of these public software tools are gratefully acknowledged. This research was partially supported by the Swiss Nsf grant 2100-067961.

References Anderson, S. A. and N. Sitar (1995). Shear strength and slope stability in a shallow clayey soil regolith. Reviews in Engineering Geology 10, 1–11. Antonucci, A., A. Salvetti, and M. Zaffalon (2004a). Assessing debris flow hazard by credal nets. In M. L´ opez-D´ıaz, M. Gil, P. Grzegorzewski, O. Hryniewicz, and J. Lawry (Eds.), Soft Methodology and Random Information Systems, Advances in Soft Computing, pp. 125–132. Berlin: Springer-Verlag. Antonucci, A., A. Salvetti, and M. Zaffalon (2004b). Hazard assessment of debris flows by credal networks. In C. Pahl, S. Schmidt, and T. Jakeman (Eds.), iEMSs 2004 International Congress: Complexity and Integrated Resources Management, Osnabr¨ uck, Germany. International Environmental Modelling and Software Society. Bagnold, R. A. (1954). The physics of blown sand and desert dunes. New York: William Morrow and Co. Bernard, J. M. (1996). Bayesian interpretation of frequentist procedures for a bernoulli process. The American Statistician 50(1), 7–13. Bernard, J. M. (2001). Non-parametric inference about an unknown mean using the imprecise Dirichlet model. In G. de Cooman, T. L. Fine, and T. Seidenfeld (Eds.), Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications (ISIPTA ’01), pp. 40–50. Maastricht: Shaker Publ.. Bernardo, J. M. and A. F. M. Smith (1996). Bayesian Theory. New York: Wiley & Sons.

282

A. Antonucci et al.

Caine, N. (1980). The rainfall intensity-duration control of shallow landslides and debris flows. Geogr. Ann. A62, 23–27. Campbell, R. H. (1975). Soil slips, debris flows, and rainstorms in the Santa Monica Mountains and vicinity, Southern California. US Geol. Surv. Prof. Pap. 851. Campos, L., J. Huete, and S. Moral (1994). Probability intervals: a tool for uncertain reasoning. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2 (2), 167–196. Cannon, S. H. and S. Ellen (1985). Rainfall conditions for abundant debris avalanches, San Francisco Bay Region, California. Calif. Geol. 38, 267–272. Cano, A. and S. Moral (1999). A review of propagation algorithms for imprecise probabilities. In G. de Cooman, F. G. Cozman, S. Moral, and P. Walley (Eds.), Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications (ISIPTA ’99), Universiteit Gent, Belgium, pp. 51–60. The Imprecise Probability Project. Costa, J. E. (1984). Physical geomorphology of debris flows, Chapter 9, pp. 268–317. Berlin: Springer-Verlag. Costa, J. H. and G. F. Wiekzorek (1987). Debris Flows/Avalanches: Process, Recognition and Mitigation, Volume 7. Boulder, CO: Geol. Soc. Am. Reviews in Engineering Geology. Couso, I., S. Moral, and P. Walley (2000). A survery of concepts of independence for imprecise probability. Risk, Decision and Policy 5, 165–181. Cozman, F. G. (2000a). Credal networks. Artificial Intelligence 120, 199–233. Cozman, F. G. (2000b). Separation properties of sets of probabilities. In C. Boutilier and M. Goldszmidt (Eds.), Proceedings of the 16th Annual Conference on Uncertainty in Artificial Intelligence (UAI-2000), pp. 107–114. San Francisco: Morgan Kaufmann. de Cooman, G. and M. Zaffalon (2004). Updating beliefs with incomplete observations. Artificial Intelligence 159(1-2), 75–125. Ferreira da Rocha, J. C. and F. G. Cozman (2002). Inference with separately specified sets of probabilities in credal networks. In A. Darwiche and N. Friedman (Eds.), Proceedings of the 18th Conference on Uncertainty in Artificial Intelligence (UAI2002), pp. 430–437. San Francisco: Morgan Kaufmann. Ferreira da Rocha, J. C., F. G. Cozman, and C. P. de Campos (2003). Inference in polytrees with sets of probabilities. In U. Kjærulff and C. Meek (Eds.), Proceedings of the 19th Conference on Uncertainty in Artificial Intelligence (UAI-2003), pp. 217– 224. San Francisco: Morgan Kaufmann. Grunwald, P. and J. Halpern (2003). Updating probabilities. Journal of Artificial Intelligence Research 19, 243–278. Iverson, R. M., M. E. Reid, and R. G. LaHusen (1997). Debris-flow mobilization from landslides. Annual Review of Earth and Planetary Sciences 25, 85–138. Johnson, A. M. (1984). Debris Flow, Chapter 8, pp. 257–361. New York: Wiley & Sons. Johnson, K. A. and N. Sitar (1990). Hydrologic conditions leading to debris flow initiation. Can. Geotech. J. 27, 789–801.


283

Kilchenmann, U., G. Kyburz, and S. Winter (2001). GEOSTAT user handbook. Neuchˆ atel: Swiss Federal Statistical Office. Kuntner, R. (2002). A methodological framework towards the formulation of flood runoff generation models suitable in alpine and prealpine regions. Ph. D. thesis, Swiss Federal Institute of Technology, Z¨ urich. Levi, I. (1980). The Enterprise of Knowledge. London: MIT Press. Maidment, D. R. (1993). Handbook of Hydrology. New York: McGraw-Hill. Moral, S. and A. Cano (2002). Strong conditional independence for credal sets. Annals of Mathematics and Artificial Intelligence 35 (1–4), 295–321. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo: Morgan Kaufmann. Reneau, S. L., W. E. Dietrich, C. J. Wilson, and J. D. Rogers (1984). Colluvial deposits and associated landslides in the northern San Francisco Bay area, California, USA. In Proceedings of the IV International Symposium on Landslides, Toronto, pp. 425–430. Canadian Geotechnical Society. Rinaldo, A. (1999). Hydraulic networks in nature. Journal of Hydraulic Research 37, 861–859. Takahashi, T. (1981). Debris Flow. Ann. Rev. Fluid. Mech. 13, 57–77. Takahashi, T. (1991). Debris Flow. Rotterdam: A.A. Balkama. IAHR Monograph. Tessem, B. (1992). Interval probability propagation. International Journal of Approximate Reasoning 7 (3), 95–120. USDA (1993). Soil conservation service. Washington D.C.: United States Department of Agriculture. Hydrology, National Engeneering Handbook, Supplement A. Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. New York: Chapman and Hall. Walley, P. (1996a). Inferences from multinomial data: learning about a bag of marbles. J. R. Statist. Soc. B 58 (1), 3–57. Walley, P. (1996b). Measures of uncertainty in expert systems. Artificial Intelligence 83, 1–58. Wilson, R. C. and G. F. Wieczorek (1995). Rainfall threshold for the initiation of debris flows at La Honda, California. Environ. Eng. Geosci. 1, 11–27.

CHAPTER 11

Multi-Criteria Decision Support for Integrated Technique Assessment Jutta Geldermann and Otto Rentz

11.1

Introduction

Recent years have seen a world-wide change in the environmental policy towards integrated pollution prevention, taking into account all environmental media (air, water, land) and energy consumption. In the European Union, this tendency is confirmed in the Integrated Pollution Prevention and Control Directive (Ippc-Directive 96/61/EC), which obliges the member states to take an integrated approach to the protection of the environment in the licensing of environmentally relevant installations. The “best available techniques” (Bat) play an essential role in the material transformation of the Ippc-Directive. Bat will serve inter alia as a basis for the determination of reference values for emission limits and for the granting of permits for installations (Geldermann et al. 1999; Geldermann and Rentz 2004). At the present time, no specific method for supporting these important decisions is being applied, taking into account economic, technical, and especially ecological criteria. Here, the assessment of cross-media aspects remains a difficult task. Life cycle assessment (Lca) might be a suitable instrument, but is still at an early stage of development (Heijungs et al. 1992; Guinée et al. 2002). Moreover, it is very time consuming and thus expensive to collect detailed information for the characterization of the investigated techniques and their potential environmental impacts. Therefore, an appropriate decision support tool should give guidance in this multi-criteria decision situation. The Chapter aims to put up for discussion the further need for multi-criteria decision making (Mcdm) methods for integrated technique assessment. Accordingly, this Chapter is structured in three parts: Firstly, the definition of Bat and their use in the setting of environmental standards are outlined (Section 11.2). Then, the case study from the sector of industrial coating is presented, and the multi-criteria approaches Maut (multi-attribute utility theory) and Promethee (outranking approach) are applied exemplarily (Section 11.3). Finally, the experiences gained are discussed and conclusions are derived.

286

J. Geldermann and O. Rentz

11.2

Determination of BAT

In the European Union, the Ippc-Directive lays down the framework for a formal procedure for the licensing of installations. Furthermore, it pursues the protection of the environment which is not limited to single media. The aim is . . . to achieve a high level of protection of the environment taken as a whole (Article 1), by means of integrated prevention and control of pollution arising from the (mainly) industrial activities listed in Annex I. The term pollution is defined in Article 2: pollution shall mean the direct or indirect introduction as a result of human activity, of substances, vibrations, heat or noise into the air, water or land which may be harmful to human health or the quality of the environment, result in damage to material property, or impair or interfere with amenities and other legitimate uses of the environment. The member states are obliged to take an “integrated approach to the protection of the environment” when granting permits for environmentally relevant industrial installations, based on the catalogue of best available techniques. They are defined in Article 2, § 11 as the most effective and advanced stage in the development of activities and their methods of operation which indicate the practical suitability of particular techniques for providing in principle the basis for emission limit values designed to prevent and, where that is not practicable, generally to reduce emissions and the impact on the environment as a whole:

• “techniques” shall include both the technology used and the way in which the installation is designed, built, maintained, operated and decommissioned, • “available” techniques shall mean those developed on a scale which allows implementation in the relevant industrial sector under economically and technically viable conditions, taking into consideration the costs and advantages, whether or not the techniques are used or produced inside the Member State in question, as long as they are reasonably accessible to the operator, • “best” shall mean most-effective in achieving a high general level of protection of the environment as a whole. In determining the best available techniques, special consideration should be given to the items listed in Annex IV. The definition of Bat gives, inter alia, clear indications that - as a rule - a number of techniques are meant, rather than one single technique, which are suitable for application at the local sector level. Annex IV lists twelve items altogether, which are “...to be taken into account generally or in specific case.” According to Article 2 the evaluation is to be done explicitly by “...taking into consideration the costs and advantages...”, and furthermore, the consideration of the economic and technical tenability for the concerned industrial sector is required. According to Annex IV

Multi-Criteria Decision Support for Integrated Technique Assessment

287

(“...bearing in mind the likely costs and benefits of a measure...”), economic aspects are also to be taken into account when determining Bat. The assessment of ecological aspects within the determination of best available techniques is a difficult task, especially because of the required evaluation of cross-media aspects. Here, trans-medial problem shifting from one environmental medium to another (like from air into water) must be taken into account. Life cycle assessment has been developed as an instrument for the environmental assessment of products (Guinée et al. 2002). The different approaches for Lca have been harmonized to some extent in the Iso 14040 document, which explains the four steps of Lca: 1. Determination of objectives, scope, system boundaries and functional unit 2. Inventory analysis 3. Impact assessment (Classification, characterization and valuation of the emissions and consumptions) 4. Interpretation (Weighting by specific contribution as an indication of the quantitative relevance of the substances concerned and by environmental importance and “verbal-argumentative” final valuation) The Lca methods and approaches, which have been developed so far, differ mainly in the impact assessment step. The best developed methods are based on impact categories (Guinée et al. 2002), which are at present still incomplete and disputable in several aspects, but have the best scientific research background for pointing out the relation between emissions and their potential ecological impacts. Therefore, at least indications of the most important environmental aspects for the identification of Bat can be expected (Geldermann et al. 1998). The resulting ecological profile comprises several impact potentials which are measured in different units (e.g. kg CO2 -equivalent, or kg Cfc11-equivalent/m3 air), which is a typical decision situation for Mcdm. The interpretation, as the last step of a Lca, should provide the basis for a final judgement. The objects of both environmental protection and sustainability, however, can seldom be defined unambiguously. Thus, subjective assumptions play a major role. However, the inventory analysis already causes difficulties in the assessment of production techniques and industrial installations: Due to varying measurements, the differences in the input parameters and numerous interdependencies in the production process, comparable exact data are rarely available.1 Therefore, in the published Bref (best available technique reference) documents, results of Lca studies can rarely be found. Instead, data on mass and energy flows are considered without further aggregation. In addition to 1

Therefore, a description of the techniques with the help of fuzzy numbers might seem to be more realistic than with crisp numbers. Also, for the weighting of the ecological impact potentials, fuzzy numbers might be appropriate (Geldermann et al. 2000).

288


Goal and

Formulate Problem

Scope Definition

Inventory Analysis

Interpretation

Impact Assessment

Phases of LCA according to ISO 14040

Evaluate Options

Review Decision Structure

Phases of MCA (Multi−Criteria−Analysis)

Figure 11.1: Mapping the decision analysis cycle onto the phases of Lca (Iso 14040) (French and Geldermann 2005).

the consideration of the conflicting environmental criteria, also economic aspects have to be taken into account, by weighting them against technical and ecological aspects. Figure 11.1 maps the typical steps of decision analysis against the phases of Lca (French and Geldermann 2005).

11.3

Case Study in the Sector of Industrial Paint Application

A case study in the sector of industrial paint application is presented for illustration of the multi-criteria decision problem for integrated technique assessment. The collection of data on the investigated processes can be supported by mass and energy flow management models (Rentz et al. 2000). In this way, several cost and emission reduction potentials have been identified for selected paint application processes with the help of the developed mass and energy flow model. For the processes “coating of mobile phones” and “coating of Pvc-parts” destined for the automobile industry, all emission and cost relevant process steps and their interdependencies have been investigated. For both coating processes, the process steps (pre-treatment, coating, drying, etc.) and respective work sequences (e.g. paint application techniques) are identified and the employed materials (paint, detergents, etc.) and technologies (e.g. spray guns) are analysed in detail. For this purpose an extensive collection of data on consumptions and emissions was performed within the company. This served as a basis for computer-based process simulations. During the investigation within the


289

company, it soon became obvious, that an Lca impact assessment would not yield additional information or useful data aggregation. These investigations allow the identification of critical points as well as the analysis of the influences when implementing alternative paints and/or processes (including end-of-pipe measures) in the processes of industrial paint application, based on mass and energy flows and an economic assessment. The definition and modeling of different scenarios represent a basis for supporting decision processes in the light of reducing environmental problems, taking into consideration technical, economic and environmental aspects. In this way the substitution of paints and organic solvents (introduction of water-based paints or use of recovered cleaning solvents) or process modifications (e.g. introduction of high-volume low-pressure spray guns) as well as their interdependencies with further process steps can be evaluated, following an integrated approach. Besides different measures for avoiding or at the very least reducing volatile organic compound (Voc) emissions, measures reducing energy consumption, waste generation or the use of harmful substances are also taken into account. In direct combination with results obtained from the model with regard to mass and energy flows, an economic analysis has finally been performed in order to examine potentials for cost reduction. Table 11.1 summarizes the investigated scenarios for paint application on Pvcparts for the automobile sector (for the full description and discussion of technical, economic and environmental implications of the mentioned scenarios, cf. Rentz et al. (2000)). Because of the need to maintain trade secrets, the calculated values are given anonymously, but they should still allow the illustration of the basic procedure of simultaneous evaluation of the considered alternatives with regard to the various criteria. Mass and energy flow models are conceived for the investigation of mass and energy flows within a company, but they do not allow for a formal comparison of the investigated alternatives. For this purpose, methods and tools for multi-criteria decision support can be used in combination with simulation programmes (Spengler et al. 1998; Geldermann 1999).

11.3.1

Approaches for Multi-Criteria Decision Making based on the Ranking of Discrete Alternatives

The key philosophical departure point for Mcdm as a formal approach for problem solving, as distinct from the classical approaches of operations research and management science, lies in the representation of several conflicting criteria (Stewart 1992; Belton and Stewart 2002). Mcdm has been one of the fastest growing areas of operational research during the last two decades, as it is often realized that many concrete problems contain several criteria (Vincke 1986). The theory of Mcdm can be divided into multi-objective decision making (Modm) and multi-attribute decision making (Madm). Modm analyses a subset of a continuous vector space, usually restricted by constraints, by locating all efficient solutions, before determining the optimum


No.

0 1 2A 2B 2C 2D 2E 3A 3B

Scenario Short description (Manual coating) Robot Thermal Incineration on drier Adsorption on spraying cabin Thermal Incineration on spraying cabin Biofilter on spraying cabin Scenario 2D+1 Water based coat Scenario 3A+1

Resources Waste Emissions Primary measures Waste gas cleaning Paint Diluting Paint Operating Solvent Operating InvestOperating costs Paint solvent sludges costs emission costs ment [e/yr] [e/yr] [e] [e/yr] 0 -45% 0%

0 -45% 0%

0 -60% 0%

0 0% 0%

0 -42% -3%

0 -43% 0%

0 200 350

0 0 255

0%

0%

0%

0%

-60%

0%

500

50

0%

0%

0%

0%

-60%

0%

750

20

0%

0%

0%

0%

-72%

0%

350

40

-45% 6% -41%

-45% 6% -41%

-60% 0% -60%

0% -75% -75%

-81% -78% -87%

-43% 2% -44%

550 25 225

40 0 0

290

Table 11.1: Selected results of the investigated scenarios for paint application on Pvc-parts.


291

dependent on the user’s preferences. Therefore, Modm offers feasible methods for operational planning, e.g. goal programming. For the comparison of several particular alternatives or in strategic planning, when a certain number of alternatives (e.g. techniques) are to be pre-selected for further investigation, the approaches of Madm should be used, because Madm investigates a finite set of alternatives. In the literature on decision theory, the Madm methods are classified according to the given information (Hwang and Masud 1979; Zimmermann and Gutsche 1991). Basically, Madm comprises two mathematical steps (Zimmermann 1987): 1. the aggregation of the judgements with regard to each criterion and each alternative, and 2. the ranking of the alternatives according to the aggregation rules. To introduce the basic notation for Madm, consider the set A of T alternatives that has to be ranked, and K criteria that have to be optimized, A := a1 , . . . , aT : set of discrete alternatives or techniques at (t = 1...T ) F := f1 , . . . , fK : set of criteria relevant for the decision fk (k = 1...K),

(11.1)

then the resulting multiple criteria decision problem can be concisely expressed in a matrix format. The goal achievement matrix or decision matrix D := (xtk ) with t = 1, ..., T and k = 1, ..., K is a (T × K) matrix whose elements xtk = fk (at ) indicate the evaluation or value of alternative at , with respect to criterion fk :     x11 · · · x1K f1 (a1 ) ··· fK (a1 )   ..  :=  .. .. D =  ... (11.2)  . xtk .  . fk (at ) . xT 1 · · · xT K f1 (at ) ··· fK (aT )

The complex process of decision aiding, of course, goes further behind pure mathematics. It aims rather at a comprehensive support in order to reveal the subjective preferences, which differentiate the importance of the investigated criteria (also called “weights”) and of the difference between distinct criteria scores. This underlying subjectivity, especially for the setting of weights, is naturally beyond a strictly logical or mathematical analysis (cf. e.g. Weber and Borcherding 1993; Watson and Buede 1994). During the last three decades, basically two philosophies are being distinguished for Madm (cf., e.g., Bana e Costa 1990; Roy 1996; Gal et al. 1999; Belton and Stewart 2002): • The American school assumes that the decision maker has an exact conception about the utility of the scores of the alternative and the weights of the different criteria, which are to be discovered and to be interpreted by the means of decision support. Well known approaches are the multi-attribute

292

J. Geldermann and O. Rentz value/utility theory (Mavt/Maut), simple additive weighting (Saw) or the analytic hierarchy process (Ahp). • The European (or French) school supposes that the preferences are not apparent to the decision maker, and therefore the decision support is necessary for structuring the decision situation and for giving insight into the consequences of different weightings in the decision problem. The emphasis is on the recognition of the limits of objectivity. Thus, the researcher should help to build a value judgement model by seeking working hypotheses for making recommendations (Roy 1996). The methods Electre, Oreste, Topsis (cf. Glossary, page 305ff.) and Promethee are among the most prominent examples for this philosophy.

The formation and the existence of these two philosophical schools is often subject for discussions, mainly because no shared view about a psychologically “correct” modeling of the human value judgement exists (Hämäläinen 2004). It has been stated that without a generally accepted paradigm, there are only competing schools and sub-schools, where each author feels obliged to build up his work from anew from the foundation (Lootsma 1996). Also cross-cultural differences in management and decision styles are given as a reason for the formation of the different schools of multi-criteria decision analysis (Geldermann et al. 2003). In the following, the algorithms for Sar, Saw and Promethee are briefly presented, before they are applied in a multi-criteria decision support system for the investigated case study is shown.

11.3.2

Simple Additive Ranking

As a very intuitive multi-criteria approach, the simple additive ranking (Sar) is based on the ranking of the alternatives with regard to each criterion, the subsequent aggregation of the weighted ranks and finally of the normalization of the obtained values. The resulting value function indicates the goodness of the alternatives: the higher the rank, the higher the value of the alternative in comparison with the other regarded alternatives (French 1986; Eisenf¨ uhr and Weber 1994): v (at ) =

K 1X wk · Rk (fk (at )) T k=1

with

wk ≥ 0 and

K X

wk = 1 ;

(11.3)

k=1

with: wk : Weighting factor of the criterion k, T : Total number of alternatives, Rk : Ordinal rank of an alternative with regard to the considered criterion k, if fk (at ) → max : Rk (max {fk (at )} = T and Rk (min {fk (at )}) = 1 if fk (at ) → min : Rk (max {fk (at )} = 1 and Rk (min {fk (at )}) = T .


293

Figure 11.2: Graphical representation of the ordinal ranking of the investigated scenarios with regard to the considered criteria.

The normalization is not necessary for obtaining a rank order of the regarded alternatives, but allows a graphical representation which is more intuitive. Both descending and ascending rank orderings can be defined and lead to the same resulting total rank order. It has to be defined which ordinal score is to be given, if two or more alternatives score even on a certain criterion. In spite of the rough differentiation between the alternatives, the simple additive ranking is a widely used approach, but prone to rank reversals (cf. the so-called “Borda Count” as described by e.g. French 1986, p. 23). Therefore, just the rank ordering without any further aggregation should be used for deriving a graphical representation of the considered alternatives, in order to check, if there are dominating or dominated alternatives. Moreover, the received rank order allows to explain the decision makers the multi-criteria character of the decision problem and the possible goal conflicts in a very illustrative manner. If two or more alternatives score even, the respective ordinal rank is given to each of these alternatives. From the graphical representation in Fig. 11.2 can be seen, that scenario 3B shows the best values with regard to three criteria. None of the nine examined scenarios, however, dominates another or is dominated.

11.3.3

Application of Simple Additive Weighting (SAW)

The multi-attribute value/utility theory (Mavt/Maut) is based on the establishment of a value function, representing goal achievement according to each criterion, multiplied by the particular weights of the criterion (French 1986). The decision maker declares a one-dimensional value-function vk (fk (at )) that is – in the most simplistic case – normalized to the interval [0,1], where the best score on each cri-

294


Figure 11.3: Graphical representation of the simple additive weighting.

terion gets the utility value vk = 1 , and the worst one gets vk = 0:

v (at ) =

K X k=1

wk · vk (fk (at ))

with

wk ≥ 0 and

K X

wk = 1 ;

(11.4)

k=1

with: wk : Weighting factor of the criterion k, K: Total number of alternatives

vk :

( if fk (at ) → max : vk (fk (a′t )) = if fk (at ) → min : vk (fk (a′t )) =

(fk (a′t )−min(fk (at )) max(fk (at ))−min(fk (at )) max(fk (at ))−(fk (a′t )) max(fk (at ))−min(fk (at )) .

(11.5)

The higher the weighted sum of the utility values, the better the alternative. This concept is rather intuitive to the decision maker. Since the Maut is based on the concept of trade-off between the scores on different criteria, a complete “compensation” between attributes is possible, such that a sufficiently large gain in a lesser attribute will eventually compensate for a small loss in a more important attribute, no matter how important one attribute is (Bouyssou 1990; Stewart 1992). In the application for environmental assessment however, good results concerning releases into the water might, for example, counterbalance worse emissions into the air, but this mathematical representation does not match the real effects caused by the emissions in the environment (Spengler et al. 1998). Figure 11.3 shows the graphical representation of exemplary results of the case study, when equal weighting of all criteria (0.125%).


11.3.4

295

Application of PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation)

In order to overcome the assumptions of complete compensation and existence of a “true” ranking of the alternatives which only need to be discovered, the outranking methods have been developed within the so-called European or French School of Madm.2 Outranking takes into account that preferences are not constant in time, are not unambiguous, and are not independent of the process of analysis (Roy and Bouyssou 1993). Therefore, “outranking” could be thus defined: alternative at outranks at′ , if there is a “sufficiently strong argument in favour of the assertion that is at least as good as from the decision maker’s point of view ” (Brans et al. 1986). Accordingly, the outranking-relation is the result of pair-wise comparisons between the alternatives with regard to each criterion (Stewart 1992; Roy and Bouyssou 1993). The outranking method Promethee (Brans et al. 1986) derives the preference values by generalized criteria, which can be defined by the decision maker specifically for each considered criterion fk : pk (fk (at ) − fk (at′ )) = pk (d) ∈ [0, 1] .

(11.6)

The degree of preference of an alternative at in comparison to at′ can vary from pk (d) = 0, which means indifference, over a zone of weak preference to pk (d) = 1 indicating strict preference. The six types of generalized criteria presented in Fig. 11.4 have been suggested which might cover most of the decision problems, but the decision maker may model his preferences with the help of specifically shaped preference functions as well. The algorithm for Promethee can be outlined as follows (Brans et al. 1986): 1. Specify for each criterion fk a generalized preference function pk (d) (see Fig. 11.4). 2. Define a vector containing the weights, which are a measure for the relative importance of each criterion, wT = [w1 , . . . , wk ]. If all the criteria are of the same importance in the opinion of the decision maker, weights can be Pall K taken as being equal. The normalization of the weights k=1 wk = 1 is not necessarily required, but facilitates an uniform representation and comparison of different evaluations. No specific approach for the setting of weights is proposed, because the aim of Promethee (and outranking in general) is seen in the explanation of the weighting factors spontaneously expressed by the decision maker. 2

Following the work of Ackoff (1981) outranking aims to be more than a strict, “clinical” problem resolving or as problem solving in the classical sense of operational research, but also foster the approach of problem dissolving, giving hints for changes of the constraints given with the decision problem.

296

J. Geldermann and O. Rentz P(d)

P(d)

1

P(d) 1

1

d

Type 2 q Quasi-criterion

Type 1 Usual criterion

P(d)

d

P(d)

1

p

Type 4 Level criterion

d

d

P(d)

1

q

p

Type 3 Criterion with linear preference

1

q

p

d

Type 5 Criterion with linear preference and indifference area

d

Type 6 Gaussian criterion

Figure 11.4: Generalized criteria for the use in Promethee (Brans et al. 1986).

3. Define for all the alternatives at , at′ ∈ A the outranking-relation π: ( A × A → [0, 1] π: P π(at , at′ ) = K k=1 wk · pk (fk (at ) − fk (at′ )) .

(11.7)

The preference index π(at , at′ ) is a measure for the intensity of preference of the decision maker for an alternative at in comparison with an alternative at′ for the simultaneous consideration of all criteria. It is basically a weighted average of the preference functions pk (d) and can be represented as a valued outranking graph (cf. Fig. 11.5).

4. As a measure for the strength of the alternatives at ∈ A, the leaving flow is calculated: T X 1 + (11.8) π(at ; at′ ) . Φ (at ) = · T ′ ′ t =1,t 6=t

The leaving flow is the sum of the values of the arcs which leave node at and therefore yield a measure of the “outranking character” of at . The normalization using the total number of considered alternatives T is not a necessary precondition, but both for the leaving and entering flow, the same approach has to be chosen. As through the normalisation of the weights, a comparison of different evaluations is made easier. Mareschal (1998), p. 55 denotes the


297

π (t, t’) a t’

at

π (t’, t) Figure 11.5: Outranking graph.

Promethee algorithm without any normalisation, while Brans and Mareschal 1 . (1994), p. 301 propose a normalization factor of T −1 5. As a measure for the weakness of the alternatives at ∈ A, the entering flow is calculated, measuring the “outranked character” of at (analogously to the leaving flow): T 1 X − (11.9) π(at′ ; at ) . Φ (at ) = · T ′ t =1

6. Graphical evaluation of the outranking relation Basically, the higher the leaving flow and the lower the entering flow, the better the alternative. The Promethee I partial pre-order is determined by a comparison of the leaving and entering flows by a set intersection in a manner that also allows the representation of incomparability between alternatives. In the partial pre-order, an arc leads from alternative at to at′ , if at is preferred to at′ . In case a complete pre-order is requested, Promethee II yields the so-called net-flows as the difference of the leaving and entering flows avoiding any incomparabilities: Φnet(at ) = Φ+ (at ) − Φ− (at ) . (11.10) However, the partial pre-order derived by Promethee I may contain more realistic information through the indication of incomparabilities. With the help of the graphical representation, clusters of alternatives can be derived, so that a group of best alternatives might be identified. Moreover, the documentation of the incomparabilities is helpful for the identification of any further information demand (Spengler et al. 1998). In the case study, the preference function of type 3 (criterion with linear preference) is deployed (cf. Fig. 11.4). Here, the difference between the best and the

298


worst score on each criterion is taken as the parameter p for the preference function of type 3, indicating the transition from weak to strict preference between two considered alternatives with regard to the respective criterion. Figure 11.6 depicts the graphical representation of the obtained. It should be noted that the calculated entering flow is multiplied by (-1), in order to achieve an intuitively understandable graphical representation within the multi-criteria decision support system. Also the partial pre-order is given.

Figure 11.6: Graphical representation of the results of Promethee (preference function type 3, p = xmax − xmin ).

From practical experiences already gained in the application of Promethee in the context of environmental assessment and determination of Bat, it can be said that this depiction is much appreciated (Geldermann 1999). Moreover, the concept of generalized criteria, based on parameters which have a real meaning and are therefore operational and comprehensible, as a basis for the preference functions seems to be more accepted than the derivation of aggregated utility functions. The fact that the sum of all the calculated net flows of all alternatives equals zero (cf. Tab. 11.2), underlines the relative character of the evaluation, which is based on pair only comparisons of the examined alternatives.

299

Multi-Criteria Decision Support for Integrated Technique Assessment Scen.

Φ+

Φ−

Φnet

3B 1

0.494 0.378

-0.013 -0.065

0.481 0.313

2E 3A 2D 0 2B 2C 2A

0.371 0.207 0.061 0.081 0.042 0.042 0.018

-0.080 -0.186 -0.212 -0.257 -0.240 -0.267 -0.374

0.291 0.021 -0.151 -0.177 -0.198 -0.225 -0.356

Difference to previous alternative 0.169 (= 0.481 − 0.313) 0.022 0.269 0.173 0.026 0.022 0.027 0.130

Utility value (Saw) 0.943 0.774 0.752 0.483 0.310 0.285 0.263 0.236 0.106

Difference to previous alternative 0.169 (= 0.943 − 0.774) 0.022 0.269 0.173 0.026 0.022 0.027 0.130

Table 11.2: Comparison of the net fluxes Φnet calculated by Promethee and the Saw utility values.

11.4

Discussion

On closer examination, some similarities between the results of the simple additive weighting and of the Promethee application become obvious. Firstly, the ranking of the alternatives according to the Promethee net flows Φnet (using preference functions of type 3 with the parameter p as the difference between the highest and lowest score with regard to each criterion) and to the aggregated utility values from the Saw is the same. Moreover, the differences between the preference or utility values of each pair of alternatives are almost equal, as shown in Tab. 11.2. This gives rise to the claim that the outranking concept of preference, based on pair wise comparisons, and the concept of Mavt/Maut utility functions are basically the same. Both Madm approaches discussed in this Chapter are additive methods, since their definitions are based on the weighted sum of the respective utility, value or preference functions (cf. equations (11.2), (11.3), and (11.5)). Therefore, it is sufficient to investigate the case of just one criterion further. Four alternatives will be investigated, which are considered to have scores that will be maximized. The following notation is chosen for ease of presentation: f1 (a1 ) := x1 ; f1 (a2 ) := x2 ; f1 (a3 ) := x3 ; f1 (a4 ) := x4 ; min fk (at ) := xmin ; max fk (at ) := xmax . Within the Saw, the difference of the utility values of alternatives are calculated as follows: v(a1 ) − v(a2 ) =

x2 − xmin x1 − x2 x1 − xmin − = . xmax − xmin xmax − xmin xmax − xmin

(11.11)

300


Within Promethee, the calculation of the net flow is based on the preference values of the difference of the scores, which can be presented in a matrix format:

a1 a2 a3 a4

a1 0 p1 (x2 − x1 ) p1 (x3 − x1 ) p1 (x4 − x1 )

a2 p1 (x1 − x2 ) 0 p1 (x3 − x2 ) p1 (x4 − x2 )

a3 p1 (x1 − x3 ) p1 (x2 − x3 ) 0 p1 (x4 − x3 )

a4 p1 (x1 − x4 ) p1 (x2 − x4 ) p1 (x3 − x4 ) 0

The net flow of each alternative with regard to each criterion without normalization + − + is calculated as Φnet k (at ) = Φk (at ) − Φk (at ). Within the above given matrix,Φk (at ), the measure for the strength of the alternative is calculated as the sum of the preference values in the rows for each alternative, while Φ− k (at ), the measure for the weakness, is the sum of the preference values in the columns for each alternative. Without loss of generality, it can be assumed that x1 > x3 ; x2 > x3 ; x3 > x4 . net For calculation of the difference between Φnet k (a1 ) and Φk (a2 ), two cases can be distinguished, as shown for the application of the preference function of type 3 in run 2: Case 1: x1 > x2 1 + 1 − 1 net 4Φ 1 = 4 Φ1 − 4 Φ1 = (x1 −x3 )−xmin 1 (x1 −x2 )−xmin 4 xmax −xmin + xmax −xmin (x2 −x4 )−xmin 1 (x2 −x3 )−xmin 4 xmax −xmin + xmax −xmin 2 −2xmin = 14 4x1x−4x max −xmin

+ +

(x1 −x4 )−xmin xmax −xmin − (x1 −x2 )−xmin xmax −xmin

(11.12)

Case 2: x1 < x2 : 1 net 1 + 1 − 4Φ 1 = 4 Φ1 − 4 Φ1 = (x1 −x4 )−xmin 1 (x1 −x3 )−xmin 4 xmax −xmin + xmax −xmin (x2 −x3 )−xmin 1 (x2 −x1 )−xmin 4 xmax −xmin + xmax −xmin 2 −2xmin = 14 4x1x−4x max −xmin

+ +

(x2 −x1 )−xmin xmax −xmin − (x2 −x4 )−xmin xmax −xmin

(11.13)

Consequently, it can be concluded that the difference of the net flow between each pair of alternatives in the evaluation of T alternatives is 1 net 1 1 − 1 T x1 − T x2 − 2xmin Φ1 = Φ+ Φ1 = . 1 − T T T T xmax − xmin

(11.14)

This means that with a growing number of considered alternatives, the difference between the numerical results obtained by Saw and Promethee diminishes. More-


301

over, the results vary only by the constant factor

ǫ=

K X k=1

wk

−2 min(fk (at )) . T (max(fk (at )) − min(fk (at ))

(11.15)

This factor obviously becomes rather small, as the described computations confirm. Moreover, it should be noted that the derivation of the formal preference and the utility function is encumbered with numerous reservations, so that the actual numerical values are not the most important part of the decision support, but rather a means for structuring a vital discussion. Nevertheless, with the constant factor given in eq. (11.13), the difference between Saw and Promethee can now be quantified.

11.5

Conclusion

The similarities between the utility concept within Maut and the preference concept of the outranking approach are striking, in spite of the philosophical differences between them. It might be argued that especially the Outranking approach Promethee comprises also Saw through its notion of leaving, entering and net flows. From the above discussion it becomes obvious that with the different versions of Promethee, the classical Madm approaches can be reproduced, if the preference functions are modeled accordingly. If the generalized criterion of type 1 is being used, even the Sar can be reproduced. However, Promethee prescribes no means for the setting of weights, but here, the tried and tested concepts within applications of the classical Madm approaches can also be integrated into a decision support system based on Promethee, as shown in Geldermann (1999) and Geldermann and Rentz (1999). However, the outranking approaches are still considered to have no axiomatic foundation, because of the concept of weak preferences, some preconditions of measurement theory are not fulfilled (Bouyssou 1996). Therefore, further research on the stability of the results is called for, in order to better explain the rank reversals which sometimes occur (Keyser and Peeters 1996). As with all decision support tools, sensitivity analyses are of foremost importance (Geldermann and Rentz 2001). At the very least, a reservation should always be noted: that any quantification within the decision support fakes an objectivity which does not exist, or, as Vincke (1986), p. 165 puts it: “Solving a multi-criteria decision problem is much more than aggregating given preferences on a given set.”. It is human goals that matter in the case of decision processes, and therefore a wide gap often exists between theory and practice in decision analysis. Multi-criteria decision support should thus be more than competing schools. Therefore, this Chapter suggests, based on its findings, not to emphasize the differences of the approaches, but the similarities.

302


References Ackoff, R. L. (1981). The art and science of mess management. Interfaces 11 (1), 20–26. Bana e Costa, C. (1990). Readings in Multiple Criteria Decision Aid. Heidelberg: Springer-Verlag. Belton, V. and T. Stewart (2002). Multiple Criteria Decision Analysis - An integrated approach. Boston: Kluwer Academic Publ. Bouyssou, D. (1990). Readings in multiple criteria decision aid. In C. A. Bana e Costa (Ed.), Building Criteria: A Prerequisite for MCDA, Berlin. Springer-Verlag. Bouyssou, D. (1996). Outranking relations: Do they have special properties? Journal of Multi-criteria Analysis 5, 99–111. Brans, J. P. and B. Mareschal (1994). The Promcalc & Gaia decision support system for multi-criteria decision aid. Decision Support Systems 12, 297–310. Brans, J. P., P. Vincke, and B. Marshal (1986). How to select and how to rank projects: The Promethee method. European Journal of Operational Research 24, 228–238. Eisenf¨ uhr, F. and M. Weber (1994). Rationales Entscheiden. Heidelberg: Springer-Verlag. French, S. (1986). Decision Theory - an introduction to the mathematics of rationality. Ellis Horwood Ltd. French, S. and J. Geldermann (2005). The varied contexts of environmental decision problems and their implications for decision support. Environmental Science & Policy 8 378–391. Gal, T., J. T. Stewart, and T. Hanne (1999). Multicriteria Decision Making Advances in Mcdm Models, Algorithms, Theory, and Applications. Boston: Kluwer Academic Publ. Geldermann, J. (1999). Entwicklung eines multikriteriellen Entscheidungsunterst¨ utzungssystems zur integrierten Technikbewertung. Fortschrittsberichte Vdi 16. D¨ usseldorf. Geldermann, J., C. Jahn, T. Spengler, and O. Rentz (1998). Proposal for an integrated approach for the assessment of cross-media aspects relevant for the determination of ’best available techniques’ (Bat) in the European Union. Berlin. Workshop on the assessment of cross media aspects relevant for the determination of ’Best Available Techniques’ in the frame of the implementation of Article 16(2) of the Ippc-Directive. Geldermann, J., C. Jahn, T. Spengler, and O. Rentz (1999). Proposal for an integrated approach for the assessment of cross-media aspects relevant for the determination of ”best available techniques” (Bat) in the European Union. International Journal of Life Cycle Assessment 4 (2), 94–106. Geldermann, J. and O. Rentz (1999). Integrated technique assessment with imprecise information as a support for the identification of best available techniques (bat). Braunschweig. Workshop der Gor-Arbeitsgruppe ’Or und Umwelt’. Geldermann, J. and O. Rentz (2001). Integrated technique assessment with imprecise information as a support for the identification of best available techniques (bat). OR Spectrum 23, 137–157.


303

Geldermann, J. and O. Rentz (2004). The reference installation approach for the technoeconomic assessment of emission abatement options and the determination of Bat according to the Ippc-Directive. Journal of Cleaner Production 12 389–402. Geldermann, J., T. Spengler, and O. Rentz (2000). Fuzzy outranking for environmental assessment, case study: Iron and steel making industry. Fuzzy Sets and Systems Special Issue on Soft Decision Analysis 115, 45–65. Geldermann, J., K. Zhang, and O. Rentz (2003). Further development of Madmapproaches in China and in Germany. In U. Leopold-Wildburger, F. Rendl, and G. W¨ ascher (Eds.), Operations Res. Proc. 2002, Heidelberg, pp. 525–533. SpringerVerlag. Guinée, J. B., M. Gorrée, R. Heijungs, G. Huppes, R. Kleijn, A. de Koning, L. van Oers, A. Wegener Sleeswijk, S. Suh, H. A. Udo de Haes, H. de Bruijn, R. van Duin, and M. A. J. Huijbregts (2002). Handbook on Life Cycle Assessment - Operational Guide to the ISO Standards. Dordrecht: Kluwer Academic Publ. H¨ am¨ al¨ ainen, R. P. (2004). Reversing the perspective on the applications of decision analysis. Decision Analysis 1(1), 26–31. Heijungs, R., J. B. Guinée, H. A. Udo de Haes, A. Wegener Sleeswijk, A. M. M. Ansems, P. G. Eggels, R. van Duin, and H. P. de Doede (1992). Environmental life cycle assessment of products, guide & backgrounds. Technical report, Centre of Environmental Science, Leiden. Hwang, C. L. and A. S. Masud (1979). Multi-Objective Decision Making, Methods and Applications, A State of the Art Survey. Lecture Notes in Economics and Mathematical Systems. Berlin: Springer-Verlag. Keyser, W. D. and P. Peeters (1996). A note on the use of Promethee multicriteria methods. European Journal of Operational Research 89, 457–461. Lootsma, F. (1996). Comments on Roy, B.; Vanderpooten, D.: The European school of Mcda: Emergence, basic features and current work. Journal of Multi-Criteria Decision Analysis 5, 37–38. Mareschal, B. (1998). Weight stability intervals in multicriteria decision aid. European Journal of Operational Research 33, 54–64. Rentz, O., S. Nunge, and J. Geldermann (2000). Einf¨ uhrung eines innerbetrieblichen Energie- und Stoffstrommanagements bei einem mittelst¨ andischen Industrielackierbetrieb f¨ ur Kleinteile in Baden-W¨ urttemberg. Technical report, Deutsch-Franz¨ osisches Institut f¨ ur Umweltforschung. Studie im Auftrag der Landesanstalt f¨ ur Umweltschutz. Roy, B. (1996). Multi-Criteria Modelling for Decision Aiding. Dordrecht: Kluwer Academic Publ. Roy, B. and D. Bouyssou (1993). Aide multicritère à la décision. Paris: Economica. Spengler, T., J. Geldermann, S. H¨ ahre, A. Sieverdingbeck, and O. Rentz (1998). Development of a multiple criteria based decision support system for environmental assessment of recycling measures in the iron and steel making industry. Journal of Cleaner Production 6 (1), 37–52.

304


Stewart, T. J. (1992). A critical survey on the status of multiple criteria decision making theory and praxis. Omega - International Journal of Management Science 20 (5/6), 569–586. Vincke, P. (1986). Analysis of multicriteria decision aid in Europe. European Journal of Operation Research 25, 160–168. Watson, S. R. and D. M. Buede (1994). Decision synthesis: the principles and practice of decision analysis. Cambridge: Cambridge University Press. Weber, M. and K. Borcherding (1993). Behavioural problems in weight judgements. European Journal of Operational Research 67, 1–12. Zimmermann, H. J. (1987). Fuzzy Sets, Decision Making, and Expert Systems. Boston: Kluwer Academic Publ. Zimmermann, H. J. and L. Gutsche (1991). Multi-Criteria Analyse. Heidelberg: SpringerVerlag.

Glossary A1, A2, B1 B2 AHP AIDS AIM AMCO ANN AOGCM BAT BREF CDF CFC CMB CSP CUM method ELECTRE ECC EU EURS GA GDP GHG GIS GMT GWP GTAS IIASA IGBP IPR IPAR

IPCC emission scenario storylines analytic hierarchy process Acquired Immune Deficiency Syndrome Asian-Pacific Integrated Model for scenarios analyses of GHG emissions adaptive Monte Carlo optimization artificial neural network atmosphere-ocean general circulation model best available techniques best available techniques reference document cumulative probability distribution function chlorofluorocarbon chemical mass balance constraint satisfaction problem clustering method for a variable, with a fixed number of clusters that minimizes the deviation average élimination et choix traduisant la réalité European Economic Community European Union expected utility on restricted sets genetic algorithm gross domestic product greenhouse gases geographical information system global mean temperature gross world product greenhouse gas, aerosols, tropospheric ozone, and solar activity (scenario) International Institute for Applied Systems Analysis International Geosphere Biosphere Program imprecise probability of ruin imprecise probability to avaoid ruin

306 IPCC IPPC-D LCA MADM MARIA MAUT MAVT MBR MCDM MESSAGE-MACRO MiniCAM MODM m.u. ODE OECD PDO PM10 ppm PROMETHEE PV PVC QDE SAR SAW SRES SST STO TAR TOPSIS VOC WBGU WorldSCAN WRE

Glossary International Panel of Climate Change Integrated Pollution Prevention and Control Directive life cycle assessment multi-attribute decision making multi-regional approach for resource and industry allocation integrated assessment model multi-attribute utility theory multi-attribute value theory model based reasoning multi-criteria decision making combination of an energy system model (MESSAGE) and a macroeconomic model (MACRO) the mini climate assessment model multi-objective decision making monetary unit ordinary differential equation Organisation for Economic Co-operation and Development Pacific Decadal Oscillation particulate matter less than 10 micrometers in diameter parts per million preference ranking organization method for enrichment evaluation present value polyvinyl chloride qualitative differential equation simple additive rating simple additive weighting special report on emission scenarios sea surface temperature stochastic optimization third assessment report (of the IPCC) technique for order preference by similarity to ideal solution volatile organic compounds German Advisory Council on Global Change to the Federal Government general equilibrium model of world economy (CPB Netherlands Bureau for Economic Policy Analysis Wigley-Richels-Edmonds (scenarios)

Subject Index Entries in this index are generally sorted by page number as they appear in the text. Page numbers that are marked in bold face indicate that the entry appears in a title or subheading. Page numbers that are marked in italics indicate that the entry appears in the caption of a table or figure.

Symbols E-admissibility . . . . . . . . . . . . . . 22, 45, 46 Γ-maximax . . . . . . . . . . . . . . . . . . 24, 45, 47 Γ-maximin . . . . . . . . . . . . . . . . . . . 23, 45–47 A abstraction . . . . . . . . . . . . . . . . . . . . . 92, 105 acyclicity . . . . . . . . . . . . . . . . . . . . . 171, 172 adaptation . 85, 130, 135, 138, 184, 185, 190, 203 adaptive capacity . . . . . . . . . . . . . . . . . . 184 agapasm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 aggregation . . . . . . . . . . . . . . . . . . . . . . . . 291 AHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 AIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 air pollution . . . . . . . . . . . . . . . . . . . . . . . 213 algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 backtrack-free . . . . . . . . . . . . . . . . . 172 backtrack-free search . . . . . . . . . . .175 backtracking . . . . 168, 168, 169, 170 consistency . . . . . . . . . . . . . . . . . . . . 169 CSP . . . . . . . . . . . . . . . . . . . . . . 169, 172 evolutionary . . . . . . . . . . . . . . . . . . . 210 genetic (GA) . . . . . . . . . . . . . . . . . . . . 5, 6, 205, 206, 207, 208, 209, 211,

212, 232, 237, 242, 251 linear time . . . . . . . . . . . . . . . . . . . . . 172 QSIM . . . . . . . . . . . . . . . . 104, 112, 126 TREE* . . . . . . . . . . . . . . . . . . . . . . . . 176 viability kernel . . . 70, 73, 76, 79, 92 allee effect . . . . . . . . . . . . . . . . . . . 68, 71–74 ambiguity . . . . 11, 12, 12, 13, 20, 43, 48 ambiguity aversion . . . . . . . . . . . . . . 13, 14 analysis decision . . . . . . . . . . . . . . . 18, 288, 292 dimensional . . . . . . . . . . . . . . . . . . . . 245 expert . . . . . . . . . . . . . . . . . . . . . . . . . 161 fractal . . . . . . . . . . . . . . . . . . . . . . . . . 273 inventory . . . . . . . . . . . . . . . . . . . . . . 287 policy . . . . . . . . 12, 15, 17, 28, 43, 45 qualitative . . . . . . . . . . . . . . . . . . . . . 148 sensitivity . . . . . . . . . . . . . . . . . . . . . . 19 set-valued . . . . . . . . . . . . . . . . . . . . . . . 63 stock-recruitment . . . . . . . . . . . . . . 230 anancasm . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 AOGCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 approach Bayesian . . . . . . . . . . . . . . . . . . . . . . . 267 fuzzy logic . . . . . . . . . . . . . . . . . . . . . 231

308 integrated . . . . . . . . . . . . . . . . . . . . . 184 multi-actor . . . . . . . . . . . . . . . 101, 132 outranking . . . . . . . . . . . . . . . . . . . . . 285 pattern . . . . . . . . . . . . . . . . . . . 100, 107 random set . . . . . . . . . . . . . . . . . . . . . 40 rule-based . . . . . . . . . . . . . . . . . . . . . 244 syndrome . . . . . . . . . . . . . . . . . 100, 109 tolerable windows . . . . . . . . . . . . . . . 58 archetype . . . . . . 101, 101, 103, 114, 116 artificial intelligence .148, 149, 153, 176, 209, 216, 221, 241 assessment economic . . . . . . . . . . . . . . . . . . . . . . 289 environmental . . . . . . . . . . . . .287, 298 expert . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 hazard . . . . . . . . . . . . . . . 263, 269, 275 impact . . . . . . . . . . . . . . . . . . . . 287, 289 integrated technique . . . . . . . . . . . . . 6 life cycle (LCA) . . . . . . . . . . . 285, 287 policy . . . . . . . . . . . . . . . . . . . . . . . . . 127 probabilistic . . . . . . . . . . . . . . . . . . . 264 probability . . . . . . . . . . . . . . . . . . . . . . 15 probability-based . . . . . . . . . . . . . . . 12 stakeholder . . . . . . . . . . . . . . . . . . . . . . 1 technique . . . . . . . . . . . . . . . . . 285, 288 uncertainty . . . . . . . . . . . . . . . . . . . . . 11 attractor . . . . . . . . . . . . . . . . . . . . . . . . 62, 92 autocorrelation . . . . . . . . . . . . . . . . . . . . 228 automobile sector . . . . . . . . . . . . . . . . . . 289 axiom continuity . . . . . . . . . . . . . . . . . . . . . . . 16 rationality . . . . . . . . . . . . . . . . . . . . . . 13 B backtracking . . . . . . . . . . . . . 167, 169, 170 BAT . . . . . . . . . . . . . . . . . . . . . 286, 286, 298 Bayesian statistics . . . . . . . . . . . . . . . . . . 28 beliefs posterior . . . . . . . . . . . . . . . . . . . . . . . 276 prior . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Benard, Benard . . . . . . . . . . . . . . . . . . . . . 58 best available technique (BAT) 285, 286

Subject Index best-guess . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 binary encoding . . . . . . . . . . . . . . . . . . . . 206 biodiversity . . . . . . . . . . . . . . . . . . . . . . . . 110 bivalent dominance . . . . . . . . . . 23, 45, 47 bootstrapping . . . . . . . . . . . . . . . . . . . . . . . 28 Borel algebra . . . . . . . . . . . . . . . . . . . . . . . 31 Borel field . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Borel notation . . . . . . . . . . . . . . . . . . . . . 152 bottleneck of knowledge representation 100 Boulignand, Georges . . . . . . . . . . . . . . . . 63 Brouwer, Bertus . . . . . . . . . . . . . . . . . . . . 64 Buffaga Creek . . . . . . . . . . . . . . . . . . . . . 278 burden sharing . . . . . . . . . . . . . . . . 199, 202 C Cannon, Walter . . . . . . . . . . . . . . . . . . . . . 58 Cantor structure . . . . . . . . . . . . . . . . . . . . 69 capital stock . . . . . . . . . . . . . . . . . . . . . . . 184 capturability . . . . . . . . . . . . . . . . . . . .60, 63 capture basin . . . . . . . . . . . . 59, 61, 62, 92 carrying capacity . . . . . . . 55, 56, 68, 135 case study . 99, 107, 184, 195, 238, 264, 277, 278, 280, 288 catastrophe . . . . . . . . . . . . . . . . . . . . . . . . 184 catastrophic consequence 11, 12, 15, 16 catastrophic domain . . . . . . . . . . . . . 16, 19 causal structure . . . . . . . . . . . . . . . . . . . 269 causal typology . . . . . . . . . . . . . . . . . . . . 100 cause-effect relationship . . . . . . . . . . . . 280 CDF . . . . . . . . . . . . . . . . . . . . . 29, 32, 33, 37 change climate . . . . 4, 15, 19, 221, 221, 232 environmental . . . . . . 1, 2, 11, 12, 17 global . . . . . . . . . . . . . . . . . . . . . . . . . 222 GMT . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 patterns of global . . . . . . . . . . . . . . . . 3 productivity . . . . . . . . . . . . . . . . . . . 232 channel geometry . . . . . . . . . . . . . . . . . . 272 chattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 chinook . . . . . . . . . . . . . . . . . . . . . . . . . . . .231 chromosome 205–208, 254, 254, 255, 256

Subject Index chromosome representation . . . . . . . . 252 citizen satisfaction . . . . . . . . . . . . 147, 148 claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 classification . . . . . . . . . . . . . 150, 210, 271 categorial . . . . . . . . . . . . . . . . . . . . . . 224 crisp . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 climate sensitivity 12, 18, 24, 28, 28, 29, 32, 33, 34, 35, 37, 48 clustering . . . . . . . . . . . . . . . . . . . . . 148, 149 coalition . . . . . . . . . . . . . . . . . 55, 59, 88–90 fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 coastal area . . . . . . . . . . . . . . . . . . . . . . . . 239 competition . . . . . . . . . . . . . . . . . . . . . . . 132 complexity . . . . . . . . . . . . . . . . . . . . . . 1, 100 connectionist . . . . . . . . . . . . . . . . . . . .87 conflicts . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 constraint . 15, 17, 46, 58, 104, 163, 165, 167, 169, 170, 173, 174, 199 affine . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 chance . . . . . . . . . . . . . . . . . . . . . . . . . 193 collective . . . . . . . . . . . . . . . . . . . . 58, 89 ecological . . . . . . . . . . . . . . . . 58, 59, 77 economic . . . . . . . . . . . . . . . . 77, 81, 83 environmental . . . . . . . . . . . . . . . . . . 57 guardrail . . . . . . . . . . . . . . . . . . . . . . . .19 insolvency . . . . . . . . . . . . . . . . . . . . . 193 meta- . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 probabilistic . . . . . . . . . . . . . . . . . . . 193 production . . . . . . . . . . . . . . . . . . . . . . 77 scarcity . . . . . . . . . . . . . . . . . . 58, 59, 90 stability . . . . . . . . . . . . . . . . . . . . . . . . 58 state-dependent . . . . . . . . . . . . . . . . . 61 viability 2, 55, 57, 59, 64, 77, 84, 85, 87, 89, 135 constraint satisfaction scheme . . . . . . 105 control meta- . . . . . . . . . . . . . . . . . . . . . . . . . . 82 open-loop . . . . . . . . . . . . . . . . . . . . . . 103 robust . . . . . . . . . . . . . . . . . . . . . . .57, 58 tychastic . . . . . . . . . . . . . . . . . . . . . . . . 58 viable . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 control unit . . . . . . . . . . . . . . . . . . . . . . . . 164

309 convergence . . . . . . . . . . . . . . . . . . . . . . . . 209 convex set of probability . . . . . . . . . . . . 21 cooperation . . . . . . . . . . . . . . 132, 132, 239 Coriolis forces . . . . . . . . . . . . . . . . . 244, 245 costs adaptation . . . . . . . . . . . . . . . . . . . . . 190 mitigation 18, 19, 40, 41, 43, 45, 46 criterion E-admissibility . . . . . . . . . . 22, 45, 46 Γ-maximax . . . . . . . . . . . . . . 24, 45–47 Γ-maximin . . . . . . . . . . . . . . . 23, 45–47 Akaike information (AIC) . . . . . . 228 bivalent dominance . . . . . . 23, 45, 47 decision . . . . . . . . . . . . . . . . . 14, 21, 22 diagnostic . . . . . . . . . . . . . . . . . . . . . 230 expected utility . . . . . . 12, 14, 16, 17 Hebbian . . . . . . . . . . . . . . . . . . . . . . . . 90 Hurwicz . . . . . . . . . . . . . . . . . 23, 45–47 interval dominance . . . . . . 21, 45, 46 maximality . . . . . . . . . . . . . . 22, 45, 46 criterion of exclusion . . . . . . . . . . . . . . . 248 criterion of preference . . . . . . . . . . . . . . 248 crossover . . . . . . . . . . . . . . . . 208, 208, 257 CSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105, 163–165, 167, 168, 168, 169–171, 171, 173, 173, 174, 176, 178 Cumaval Creek . . . . . . . . . . . . . . . . . . . . 278 cycle boom-and-bust . . 117, 118, 119, 120 decision analysis . . . . . . . . . . . . . . . 288 disaster . . . . . . . . . . . . . . . . . . . . . . . . 202 feedback . . . . . . . . . . . . . . . . . . . . . . . 114 inert hysteresis . . . . . . . . . . . . . . . . . . 76 limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 periodic heavy hysteresis . . . . . . . . 75 vicious . . . . . . . . . . . . . . . . . . . . . . . . . 113 D damage generator . . . . . . . . . . . . . . . . . . 195 debris flow . 6, 263, 263, 264, 264, 265, 269, 277, 279 decision

310 here-and-now . . . . . . . . . . . . . . . . . . 190 wait-and-see . . . . . . . . . . . . . . . . . . . 190 decision maker . . . . . . . . . . . . 5, 20, 57, 85 decision making . . . 44, 45, 48, 184, 196, 285, 289 environmental . . . . . . . . . . . . . . . . . . 11 decision support system . . 241, 292, 298 Defendente Creek . . . . . . . . . . . . . . . . . . 278 deforestation . . . . . . . . . . . . . . . . . . . . . . . 110 defuzzification . . . . . . . . . . . . . . . . .226, 227 description component . . . . . . . . . . . . .243 development path . . . . . . . . . . . . . . . . . . 112 diagram causal-loop 103, 114, 124, 124, 134, 135 cause-effect . . . . . . . . . . 124, 126, 128 cost-resource . . . . . . . . . . . . . . . . . . 139 differential inclusion . . . . . . . . . 61, 91, 92 Dirichlet density . . . . . . . . . . . . . . . . . . . 267 disaster prevention . . . . . . . . . . . . . . . . . 195 dispersive stress concept . . . . . . . . . . . 265 distribution possibility . . . . . . . 27, 33, 35, 36, 36 probability . . . . . . . . . . . . . . . . . . . . . . 13 distribution band . . . . . . . . . . . . 30, 32, 33 distribution of premiums . . . . . . . . . . . 200 domain of variable . . . . . . . . . . . . . . . . . 165 dynamics logistic . . . . . . . . . . . . . . . . . . . . . . . . . . 77 qualitative . . . . . . . . . . . . . . . . . . . . . . 65 Verhulst logistic . . . . . . . . . . . . . . . . .78 E earthquake . . . . . . . . . . . . . . . . . . . . . . . . . 210 scenario . . . . . . . . . . . . . . . . . . . . . . . 195 economic growth . . . . . . . . . . . . . . . . . . . 192 economic marginalization . . . . . . . . . . 113 economic shock . . . . . . . . . . . . . . . . . . . . 184 ecosystem . . . . . . . . . . . . . . . . . . . . . 110, 113 ecosystem service . . . . . . . . . . . . . . . . . . 110 Ellsberg experiment . . . . . . . . . . 13–15, 47 end-of-pipe . . . . . . . . . . . . . . . . . . . . . . . . 289

Subject Index envelope CDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 possibilistic . . . . . . . . . . . . . . . . . . . . . 37 environmental condition . . . . . . . . . . . 223 environmental policy . . . . . . . . . . . . . . . 285 equation differential . . . . . . . . . . . . . . . . . . . . . 205 Hamilton-Jacobi-Bellmann . . . . . . 84 logistic . . . . . . . . . . . . . . . . . . . . . . 56, 67 Lotka-Volterra . . . . . . . . . . . . . . . . . 211 ordinary differential (ODE) . . . . 101 Pearl-Verhulst . . . . . . . . . . . . . . . . . . 67 qualitative differential (QDE) . . . .3, 103, 104, 104 Verhulst logistic . . . . . . . . . . . . .68, 70 equivalence class . . . . . . . . . . . . . . . . . . . 101 erosion . . . . . . . . . . . . . . . . . . . 122, 128, 265 estuary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 EURS . . . . . . . . . . . . . . . . . . . . . . . . . . . 16, 17 eutrophication . . . . . . . . . . . . . . . . . . . . . 239 event catastrophic . . . . . . . . . . 185, 189, 191 dangerous . . . . . . . . . . . . . . . . . . . . . 264 extreme . . . . . . . . . . . . . . . . . . . 183–185 extreme weather . . . . . . . . . . . . . . . . 19 evolution agapastic . . . . . . . . . . . . . . . . . . . . . . . 57 anancastic . . . . . . . . . . . . . . . . . . . . . . 57 Darwinian . . . . . . . . . . . . . . . . . . . . . . 92 heavy 56, 66, 72, 72, 73, 74, 79, 82, 87 inert . . . . . . . . . . . . . . . . . . . . 70, 71, 76 Malthusian . . . . . . . . . . . . . . . . . . 56, 68 tychastic . . . . . . . . . . . . . . . . . . . . . . . . 57 viable . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 evolutionary programming . . . . 210, 211 expected utility . . . . . . . . . . 12, 14, 16, 17 imprecise . . . . . . . . . . . . . . . . . . . . . . . 21 lower and upper . . . . . . . . . . . . . . . . 21 expert knowledge . . . . . . . . . 109, 264, 275 explanatory power . . . . . . . . . . . . . . . . . 147 exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Subject Index

311

extension utility . . . . . . . . . . . . . . . . . . . . . . 13, 299 fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . 27 fuzzification . . . . . . . . . . . . . . . . . . . . . . . . 225 possibilistic . . . . . . . . . . . . . . . . . 36, 37 fuzzy controller . . . . . . . . . . . . . . . . . . . . 225 random set . . . . . . . . . . . . . . 26, 37, 38 fuzzy inference system . . . . . . . . . . . . . 225 fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . .221 F feature mapping . . . . . . . . . . . . . . . . . . . 154 G feature space . . . . . . . . . . . . . . . . . 151, 158 GA . . . 205, 206, 209–211, 213, 213, 215, feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 216, 237, 242 affine . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 gamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 dynamic . . . . . . . . . . . . . . . . . . . . . . . . 66 game inert . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 cooperative . . . . . . . . . . . . . . . . . . . . . 59 Malthusian . . . . . . . . . . . . . . . . . . 55, 60 differential . . . . . . . . . . . . . . . . . . 57, 59 static . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 dynamic cooperative . . . . . . . . . . . . 88 Verhulst . . . . . . . . . . . . . . . . . . . . .56, 79 dynamical . . . . . . . . . . . . . . 82, 83, 84 viable . . . . . . . . . . . . . . . . . . . . . . . . . . 66 meta- . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 finite element . . . . . . . . . . . . . . . . . . . . . . 253 geo-engineering . . . . . . . . . . . . . . . . . . . . . 16 focal element . . . . . . . . . . . . . . . . . . . . . . . . 25 geographic information system (GIS) 185 Foioi Creek . . . . . . . . . . . . . . . . . . . . . . . . 278 geomorphology . . . . . . . . . . . . . . . . . . . . . 272 formula GEOSTAT information system . . . . . 271 Manning-Strickler . . . . . . . . . . . . . .272 geotechnical raster map . . . . . . . . . . . . 271 fossil fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 GIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 frequency of insolvency . . . . . . . . . . . . 201 global change . . . 99, 100, 107, 110, 139, function 183, 189 belief . . . . . . . . . . . . . . . . 23–26, 32, 49 patterns of . . . . . . . . . . . . . . . . . . . . . 148 conditional mass . . . . . . . . . . . . . . . 273 syndrome of . . . . . . . . . . . . . .107, 108 cost . . . . . . . . . . . . . . 84, 205, 206, 215 global climate variation . . . . . . . . . . . . 232 crisis . . . . . . . . . . . . . . . . .80, 81, 82, 92 global mean temperature (GMT) . . . 34, extreme mass . . . . . . . . . . . . . . . . . . 266 40, 43 fitness . . . . . . . . . . . . . . . . . . . . . . . . . 254 global warming . . . . . . . . . . . . . 12, 39, 184 fuzzy-penalty . . . . . . . . . . . . . . . . . . 210 GMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 indicator . . . . . . . . . . . . . . . . . . . . . . . . 16 govermental bonds . . . . . . . . . . . . . . . . . 187 kernel . . . . . . . . . . . . . . . . . . . . . . . . . 151 Graham, Michael . . . . . . . . . . . . . . . . . . . 76 Lyapunov . . . . . . . . . . . . . . . . . . . . . . . 62 granulometry . . . . . . . . . . . . . . . . . . . . . . 273 mass . . . . . . . 268, 271, 275, 275, 276 graph . . . . . . . . . . . . . . . . . . . . . . . . . 105, 276 membership . . . . . . . . . . 224, 225, 226 CSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 objective . . . . . . . . . . . . . . . . . . . . . . 205 directed . . . . . . . . . . . . . . . . . . . 264, 276 plausibility . . . . . . . . . . . . . . . . . . . . . .26 directed acyclic . . . . . . . . . . . 268, 269 posterior mass . . . . . . . . . . . . . . . . . 266 outranking . . . . . . . . . . . . . . . 296, 297 probability density . . . . . . . . . . . . . . 27 state-transition . 104–106, 117, 118, probability mass . . . . . . . . . . . . . . . 266 119, 121 transfer . . . . . . . . . . . . . .24, 26, 34, 35 Great Plains . . . . . . . . . . . . . . . . . . . . . . . 122

312

Subject Index

greenhouse gas (GHG) . . . . . . . 11, 17, 19 qualitative . . . . . . . . . . . . . . . . . . . . . 155 growth rate . . . . . . . . . . . . . . . . . . . . . 56, 60 subjective . . . . . . . . . . . . . . . . . . . . . 147 guardrail . . 19, 20, 41, 41, 43, 43, 44, 46 instantiation . . . . . . . . . . . . . . . . . . 165–168 insurance business . . . . . . . . . . . . . . . . . 186 H integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 halibut . . . . . . . . . . . . . . . . . . 225, 228, 229 intense rainfall . . . . . . . . . . . . . . . . . . . . . 265 hazard level . . . . . . . . . . . . . . . . . . . . . . . . 277 interaction . . . . . . . . . . . . . . . . . . . . . . . . . 132 hazard mapping . . . . . . . . . . . . . . . . . . . .278 interval distance . . . . . . . . . . . . . . . . . . . 156 heat capacity . . . . . . . . . . . . . . . . . . . . . . . 34 interval dominance . . . . . . . . . . .21, 45, 46 heavy solution . . . . . . . . . . . . . . . . . . . . . . 90 intrinsic dynamic interaction . . . . . . . 102 herring . . . . . . . . . . . . . . . . . . 225, 228, 229 IPAR . . . . . . . . . . . . . . . . . 20, 21, 24, 43–45 heuristic criteria of exclusion and prefer- isocline . . . . . . . . . . . . . . . . . . . . . . . 138, 138 ence . . . . . . . . . . . . . . . . . . . . . . . 248 heuristics . . . . . . . . . . . . 100, 109, 167, 169 J hierarchical organization . . . . . . . . . . . . 87 Jacobian . . . . . . . . . . . . 103, 116, 119, 131 homeostasis . . . . . . . . . . . . . . . . . . . . . . . . . 58 K human health . . . . . . . . . . . . . . . . . . . . . . 238 Kalman filter . . . . . . . . . . . . . . . . . . . . . . 232 human-environment interaction . 3, 100, kernel 107 guaranteed viability . . . . . . . . . 83, 84 hydrodynamics . . . . . . . . . . . . . . . . . . . . .248 interval . . . . . . . . . . . . . . . . . . . 148, 151 hydroinformatic . . . . . . . . . . . . . . . . . . . . 241 invariance . . . . . . . . . . . . . . . . . 106, 120 hydrological regime . . . . . . . . . . . . . . . . 239 qualitative . . . . . . . . . . . . . . . . . . 3, 147 hypergraph . . . . . 170, 171, 172, 174, 174 string-type . . . . . . . . . . . . . . . . . . . . . 148 hypertree . . . . . . . . . . . . . . . . 172–174, 174 viability . 56, 57, 59, 61, 62, 62, 69, hypertree decomposition . . . . . . . . . . . 174 70, 76, 78, 80, 83, 92, 106 hysteresis . . . . . . . . . . . . . . . . . . . . . . . 74, 75 kernel distance . . . . . . . . . . . . . . . . . . . . 155 I knowledge base . . . . . . . . . . . . . . . 225, 226 idiographic tradition . . . . . . . . . . . . . . . . 99 L imprecise probability to avoid ruin (IPAR) . . . . . . . . . . . . . . . . . 20, 43 lack of information . . . . . . . . . . . . . . . . . 184 indecision . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 land use . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 land use planning . . . . . . . . . . . . . . . . . . 184 independence conditional . . . . . . . . . . . . . . . . . . . . 266 landmark . . . . . . . . . . . . . . . . . . . . . 104, 125 strong . . . . . . . . . . . . . . . . . . . . . . . . . 266 landslide . . . . . . . . . . . . . . . . . . . . . . 264, 265 law indicator adaptation . . . . . . . . . . . . . . 60, 64, 90 objective . . . . . . . . . . . . . . . . . . . . . . 147 demand . . . . . . . . . . . . . . . . . . . . . . . . . 90 risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Gutenberg-Richter . . . . . . . . 188, 195 subjective . . . . . . . . . . . . . . . . . . . . . 147 Hebbian learning . . . . . . . . . . . . . . . .90 sustainability 3, 147, 148, 155, 160, Ohm’s . . . . . . . . . . . . . . . . . . . . . . . . . 163 199 inert growth . . . . . . . . . . . . . . . . . . . . . . . . 56 regulation . . . . . . . . . . . . . . . . . . . . . . .89 information statistical . . . . . . . . . . . . . . . . . . . . . . . 58

Subject Index supply . . . . . . . . . . . . . . . . . . . . . . . . . . 90 LCA . . . . . . . . . . . .285, 287, 288, 288, 289 Lebesgue, Henri . . . . . . . . . . . . . . . . . . . . . 64 lexicographic preference . . . . . . . . . 16, 48 linguistic answer . . . . . . . . . . . . . . . . . . . 149 local governance . . . . . . . . . . . . . . . . . . . 149 location-specific exposure . . . . . . . . . . 196 location-specific risk profile . . . . . . . . 190 logistic effect . . . . . . . . . . . . . . . . . . . . 68, 71 loss catastrophic . . . . . . . . . . . . . . . . 41, 191 economic . . . . . . . . . . . . . . 19, 188, 196 GWP . . . . . . . . 41, 42, 43, 44, 45, 47 information . . . . . . . . . . . . . . . . . 34, 38 irreversible . . . . . . . . . . . . . . . . . . . . . . 16 property . . . . . . . . . . . . . . . . . . . . . . . 263 welfare . . . . . . . . . . . . . . . . . . . . . . . . . . 12 M Möbius inverse . . . . . . . . . . . . . . . . . . 25, 32 magnitude space . . . . . . . . . . . . . . . . . . 153 making decision . . . . . . 12, 15, 149, 225, 241 multi-attribute decision (MADM) 289 multi-criteria decision (MCDM) 285, 289 multi-objective decision (MODM) 289 Malthus, Thomas . . . . . . . . . . . . . . . 55, 55 man-made catastrophe . . . . . . . . . . . . . . . 4 man-nature interaction . . . . . . . . . . . . . 140 management . . . . . . . 55, 60, 99, 100, 112, 119–121, 140 earthquake risk . . . . . . . . . . . 183, 195 estuarine . . . . . . . . . . . . . . . . . 239, 239 fisheries . . . . . . . . . . . . . . 221, 232, 239 ill- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 resource . . .5, 55, 67, 132, 232, 237, 241, 242 risk . . . . . .5, 183, 183, 184, 185, 241 sustainable . . . . . . . . . . . . . . . . . . . . 110

313 management intervention . . . . . . . . . . 119 management plan . . . . . . . . . . . . . . . . . . 240 management policy . . . . . . . . . . . 112, 260 management strategy . . . . . . . . . . . . . . 237 Manning’s coefficient 252, 254, 254, 255, 256 map adaptive . . . . . . . . . . . . . . . . . . . . 64, 64 hazard . . . . . . . . . . . . . . . . . . . . . . . . . 263 propagation . . . . . . . . . . . . . . . . . . . . . 88 regulation . . . . . . . . . . . . . . . . . . . . . . .70 reset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 set-valued . . . . . . . . . . . . . . . . . . . 61, 66 tychastic . . . . . . . . . . . . . . . . . . . . . . . . 57 Markov condition . . . . . . . . . . . . . . . . . . 269 mating scheme . . . . . . . . . . . . . . . . . . . . . 208 maximality . . . . . . . . . . . . . . . . . . 22, 45, 46 MCDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 measure Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ex-ante . . . . . . . . . . . . . . . . . . . . . . . . 184 ex-post . . . . . . . . . . . . . . . . . . . . . . . . 184 mitigation . . . . . . . . . . . . . . . . 187, 191 possibility . . . . . . . . . . . . . . . 24, 27, 33 probability . . . . . . . . . . . . . . 12, 27, 45 meiosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 membership . . . . . . . . . . . . . . . . . . . . . . . .245 membership value . . . . . . . . . . . . . . . . . . 250 mental map . . . . . . . . . . . . . . . . . . . . . . . . 128 method branch and bound . . . . . . . . . . . . . 280 crossover . . . . . . . . . . . . . . . . . . . . . . 208 CUM . . . . . . . . . . . . . . . . . . . . . . . . . . 149 decomposition . . . . . . . 167, 170, 174 Gaussian elimination . . . . . . . . . . .167 heuristic . . . . . . . . . . . . . . . . . . . . . . . 171 kernel . . . . . . . . . . . . . . . . . . . . . 150, 161 LCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 maximum likelihood . . . . . . . . . . . 228 outranking . . . . . . . . . . . . . . . . . . . . . 295 possibilistic decomposition . . . . . . 40 rational formula . . . . . . . . . . . . . . . 272

314 Runge Kutta . . . . . . . . . . . . . . . . . . 211 stochastic quasi-gradient . . . . . . . 193 mitigation . . . 18, 20, 24, 40–43, 46, 184, 185, 188, 190, 192, 195, 196, 202 model atmosphere-ocean general circulation (AOGCM) . . . . . . . . . . . . . 34 auto-regressive . . . . . . . . . . . . . . . . . 228 Bayesian . . . . . . . . . . . . . . . . . . . . . . . 268 catastrophe . . . . . 185, 188, 189, 203 catastrophe management . . 188, 190 chemical mass balance . . . . . . . . . 214 credal network . . . . . . . . . . . . 264, 275 crisp . . . . . . . . . . . . . . . . . . . . . .223, 229 crisp regime . . . . . . . . . . . . . . . . . . . 228 CSP . . . . . . . . . . . . . . . . . 163, 166, 177 decision . . . . . . . . . . . . . . . . . . . . . 17, 20 digital elevation . . . . . . . . . . . . . . . .277 Dirichlet . . . . . . . . . . . . . . . . . . . . . . . 274 earthquake . . . . . . . . . . . . . . . . . . . . 188 economic-demographic . . . . . . . . . 192 endogenous growth . . . . . . . . . . . . . .43 energy balance . . . . . . . . . . 28, 34, 48 ensemble . . . . . . . . . . . . . . . . . . . . . . 101 estuarine . . . . . . . . . . . . . . . . . . 237, 238 evolutionary . . . . . . . . . . . . . . . . . . . . 56 fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . 229 fuzzy logic . . . 5, 221, 222, 224, 225, 228 GA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 hurricane . . . . . . . . . . . . . . . . . . . . . . 188 imprecise Dirichlet . . 264, 267, 267, 268, 280 imprecise probability . . . . . . . . . . .275 inverse . . . . . . . . . . . . . . .205, 211, 211 logistic . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Lotka-Volterra . . . . . . . . . . . . . . . . .212 Malthusian . . . . . . . . . . . . . . . . . . . . . 56 Markov . . . . . . . . . . . . . . . . . . . . . . . . 210 Monte Carlo catastrophe . . . . . . . 184 moving-average . . . . . . . . . . . . . . . . 228 multi-agent . . . . . . . . . . . . . . . . . . . . 188

Subject Index multi-modal . . . . . . . . . . . . . . . . . . . 210 neural network . . . . . . . . . . . . . . . . .232 population . . . . . . . . . . . . . . . . . . . . . . 56 predator-prey . . . . . . . . . . . . . . . . . . 211 probability . . . . . . . . . . . . . . . . . . . . .263 qualitative . . . . . . . . . . . 100, 123, 129 Ricker . 222–224, 225, 226, 229, 230 Ricker climatic . . . . . . . . . . . . 223, 228 Ricker stock-recruitment (SR) . 222 risk management . . . . 185, 186, 188 statistical . . . . . . . . . . . . . . . . . . . . . . 222 stochastic . . . . . . . . . . . . . . . . . 184, 187 stochastic optimization . . . . 185, 195 Takahashi’s . . . . . . . . . . . . . . . . . . . . 265 Verhulst . . . . . . . . . . . . . . . . . . . . . 56, 60 Walrasian . . . . . . . . . . . . . . . . . . . . . . . 59 water quality . . . . . . . . . . . . . . . . . . 209 welfare growth . . . . . . . . . . . . . . . . . 188 model ensemble 104, 112, 114, 114, 116, 119, 120, 124, 125 modeling air pollution receptor . . . . . . . . . . 213 estuarine . . . . . . 237, 240, 241, 258 inverse . . . . . . . . . . . . . . . . . . . . . . . . 213 water . . . . . . . . . . . . . . . . . . . . . . . . . 241 module case-based . . . . . . . . . . . . . . . . 242, 243 catastrophe . . . . . . . . . . . . . . . . . . . . 188 genetic algorithm . . . . . . . . . 242, 251 multi-agent . . . . . . . . . . . . . . . . . . . . 188 vulnerability . . . . . . . . . . . . . . . . . . . 188 monotonic ensemble . . . . . . . . . . . . . . . .105 multi-actor competition . . . . . . . . . . . .139 multi-criteria decision support . . 6, 285 multiplier Lagrange . . . . . . . . . . . . . . . . . . . . . . . 86 step-size . . . . . . . . . . . . . . . . . . . . . . . 194 viability . . . . . . 57, 85, 86, 86, 87–90 mutation . . . . . 206, 208, 209, 256, 257 mutation operator . . . . . . . . . . . . . . . . . 206 mutation rate . . . . . . . . . . . . . . . . . . . . . . 213 myopic behavior . . . . . . . . . . . . . . . . . . . . 85

Subject Index N Nash equlibrium . . . . . . . . . . . . . . . . . . . 139 natural catastrophe . . . . . . . . . . . 183, 195 natural disaster . . . . . . . . . . . . . . . . . . . . 183 natural hazard . . . . . . . . . . . . . . . . . . . . . 263 natural selection . . . . . . . . . . . . . . . . . . . 206 network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Bayesian . . . . . . . . . . .6, 263, 268, 276 causal . . . . . . . . . . . . . . . . . . . . . . . . . 269 credal 263, 263, 264, 268, 269, 276, 280 drainage . . . . . . . . . . . . . . . . . . 270, 271 dynamical . . . . . . . . . . . . . . . . . . . . . . 59 evolution . . . . . . . . . . . . . . . . . . . . . . 273 neural . . . . . . . . . 59, 86, 90, 100, 210 network of interrelations . 107, 110, 111 nomothetic tradition . . . . . . . . . . . . . . . . 99 non-linear regression . . . . . . . . . . . . . . . 209 non-sustainable path . . . . . . . . . . . . . . . 110 NP-hard . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 nutrient enrichment . . . . . . . . . . . . . . . . 238 O ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 ontogeny . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 open access . . . . . . . . . . . . . . . . . . . . . . . . 112 operational research . . . . . . . . . . . . . . . .295 operators genetic algorithm . . . . . . . . . . . . . . 256 optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 84 inter-temporal . . . . . . . . . . . . . . . . . . 58 optimality criterion . . . . . . . . . . . . . . . . . 85 optimization . . 5, 86, 137, 169, 183, 256 adaptive Monte Carlo 185, 190, 193 individual . . . . . . . . . . . . . . . . . . . . . 138 inter-temporal . . . . . . . . . . . 58, 77, 85 joint . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Monte Carlo . . . . . . . . . . . . . . 183, 189 multi-objective . . . . . . . . . . . . . . . . 209 multi-objective Pareto . . . . . . . . . 210 stochastic . . . . . . . . . . . . 185, 187, 193 optimization hypothesis . . . . . . . . . . . . 129

315 optimization tool . . . . . . . . . . . . . . . . . . 205 option policy . . . . . . . . . . . . . . . . . . . . . . . . . 200 ordinal assumption . . . . . . . . . . . . 119, 120 oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 overcapitalization . . . . . . . . . . . . . . . . . . 113 overexploitation . . . . . . . . . . . . . . . 103, 112 overfertilization . . . . . . . . . . . . . . . . . . . . 122 overfishing . . . . . . . . . . . . . . . . . . . . . . . . . 238 serial . . . . . . . . . . . . . . . . . . . . . . . . . . 113 overgrazing . . . . . . . . . . . . . . . . . . . . . . . . 122 P Pacific Decadal Oscillation . . . . . . . . . 228 Pandora’s box . . . . . . . . . . . . . . . . . . . . . . 16 parameter estimation . . . . . . . . . . . . . . 227 Pareto genetic algorithm . . . . . . . . . . . 209 Pasquill stability class . . . . . . . . . . . . . 215 pattern archetypal . . . . . . . . . . . . . . . . . . . . . 100 causal . . . . . . . . . . . . . . . . . . . . . . . . . 122 cause-effect . . . . . . . . . . . . . . . . . . . . 139 core . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 dynamical . . . . . . . . . . . . . . . . . . . . . 109 dynamical degradation . . . . . . . . . 100 functional . . . . . . . . . . . . . . . . . . . . . 107 global change . . . . . . . . . . . . . . . . . . 100 syndromatic . . . . . . . . . . . . . . . . . . . 109 typical . . . . . . . . . . . . . . . . . . . . . . . . . 100 pattern of interaction . . . . . . . . . . . . . . 109 PDO . . . . . . . . . . . . . . . . . . . . . . . . . . 225, 228 Pearl, Raymond . . . . . . . . . . . . . . . . . . . . . 56 Peirce, Charles . . . . . . . . . . . . . . . . . . . . . . 57 permeability . . . . . . . . . . . . . . . . . . 270, 271 phylogeny . . . . . . . . . . . . . . . . . . . . . . . . . . 100 planning horizon . . . . . . . . . . . . . . . . . . . 191 policy admissible . . . . . . . . . . . . . . . . . . . . . . 45 best . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 climate . . . . . . . . . . . . . . . . . . . . . . 16, 48 CO2 stabilization . . . . . . . . . . . . . . . 12 mitigation . . . . . . . . . . . . . . . 16, 17, 40

316 stabilization . . . . . . . . . . . . . . . . . . . . 43 WRE . 18, 18, 24, 35, 35, 39, 43, 44 policy advice . . . . . . . . . . . . . . . . . .2, 11, 43 policy analysis environmental . . . . . . . . . . . . . . . . . . 12 policy maker . . . . . . . . . . . . . . . . . . 237, 239 policy option . . . . . . . . . . . . . . . . . . . . . . .188 policy response . . . . . . . . . . . . . . . . . . . . 183 pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 pollution prevention . . . . . . . . . . . . . . . 285 population dynamics . . . . . . . . . . . . . . . . 71 possibilistic decomposition . . . . . . 32, 33 possibility space . . . . . . . . . . . . . . . . . . . 266 posterior probability . . . . . . . . . . . . . . . 276 pre-processing . . . . . . . . . . . . . . . . . . . . . 170 precautionary principle . . . . . . . . . . 13, 58 precipitation . . . . . . . . . . . . . . . . . . . . . . . 271 predator-prey relation . . . . . . . . . . . . . 212 preference ranking organization method for enrichment evaluation (PROMETHEE) . . . . . . . . . . 295 premium rate . . . . . . . . . . . . . . . . . . . . . . 201 premiums . . . . . . . . . . . 186, 191, 199, 200 risk-based . . . . . . . . . . . . . . . . . . . . . 199 prisoner’s dilemma . . . . . . . . . . . . . . . . . 132 probability classical . . . . . . . . . . . . . . . . . . . . . . . . .14 conditional . . . . . . . . . . . . . . . . . . . . 275 cumulative . . . . . . . . . . . . . . . . . . . . . . 31 epistemic . . . . . . . . . . . . . . . . . . . . . . 267 imprecise . . . . . 6, 14, 15, 17, 24, 26, 28, 40, 41, 43, 43, 44, 44, 48, 49, 263, 264, 278 objective . . . . . . . . . . . . . . . . . . . . . . 267 probability box . . . . . . . . . . . . . . 30, 32, 38 probability interval . . 14, 264, 266, 268, 271 probability of insolvency . . . . . . . . . . . 200 probability of ruin . . . . . . 17, 20, 20, 187 probability threshold . . . . . . . . . . . . . . . 193 problem constraint satisfaction (CSP) 4, 105,

Subject Index 163, 178 control . . . . . . . . . . . . . . . . . . . . . . . . . . 92 dynamic optimization . . . . . . . . . . . 85 environment-poverty . . . . . . . . . . . 128 initial value . . . . . . . . . . . . . . . . . . . . 102 inverse . . . . . . . . . . . . . . . . . . . . 210, 213 minimization . . . . . . . . . . . . . . . . . . . .84 optimization . . . . . . . . . 187, 205, 209 salt intrusion . . . . . . . . . . . . . . . . . . 246 single actor . . . . . . . . . . . . . . . . . . . . 132 process analytic hierarchy (AHP) . . . . . . 292 coating . . . . . . . . . . . . . . . . . . . . . . . . 288 core . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 crossover . . . . . . . . . . . . . . . . . . . . . . 208 decision . . . . . . . . . . . . . . . . . . . 289, 301 defuzzification . . . . . . . . . . . . . . . . . 227 estuarine . . . . . . . . . . . . . . . . . . 239, 252 fuzzification . . . . . . . . . . . . . . . . . . . 226 learning . . . . . . . . . . . . . . . . . . . . . . . 100 mating . . . . . . . . . . . . . . . . . . . . . . . . 206 measurement . . . . . . . . . . . . . . . . . . 276 mega- . . . . . . . . . . . . . . . . . . . . . . . . . 100 non-Poissonian . . . . . . . . . . . . . . . . .196 Poisson . . . . . . . . . . . . . . . . . . . . . . . . 186 production . . . . . . . . . . . . . . . . . . . . . 287 random . . . . . . . . . . . . . . . . . . . 186, 186 real growth . . . . . . . . . . . . . . . . . . . . 187 reasoning . . . . . . . . . . . . . . . . . . . . . . 226 retrieval . . . . 243, 244, 247, 248, 249 risk management . . . . . . . . . . . . . . 195 selection . . . . . . . . . . . . . . . . . . . . . . . 252 solution . . . . . . . . . . . . . . . . . . . . . . . .163 Walrasian tâtonnement . . . . . . . . . 90 profit opportunity . . . . . . . . . . . . . . . . . .110 PROMETHEE . . . . . . 296, 297, 298, 300 public transportation . . . . . . . . . . . . . . 149 Q QDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 QSIM . . . . . . . . . . . . . . . . . . . . 104, 112, 126 qualitative addition . . . . . . . . . . . . . . . . 128

Subject Index qualitative label . . . . . . . . . . . . . . . . . . . .153 qualitative magnitude . . . . . . . . . 104, 105 qualitative modeling . . . . . . . . . . . . . . . . 99 qualitative multiplication . . . . . . . . . . 128 qualitative physics . . . . . . . . . . . . . . . . . . 92 qualitative processing . . . . . . . . . . . . . .151 qualitative qualitative . . . . . . . . . . . . . . 105 qualitative space . . . . . . . . . . . . . . . . . . . 153 qualitative state . . . . . . . . . . . . . . . 104, 105 qualitative transition . . . . . . . . . . . . . . . 105 qualitative treatment . . . . . . . . . . . . . . 149 quantity space . . . . . . . . . . . . . . . . 104, 153 questionnaire . . . . . . . . . . . . . . . . . . 147–149 R radiation balance . . . . . . . . . . . . . . . . . . . .18 radiative forcing . . . . . . . . . . . . . . . . . 18, 34 rainfall triggering . . . . . . . . . . . . . . . . . . 265 random mutation . . . . . . . . . . . . . . . . . . 205 random variable . . . . . . . . . . . . . . . . . . . 266 ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 rational behavior . . . . . . . . . . . . . . . . . . . . 13 re-insurance market . . . . . . . . . . . . . . . . 196 reasoning case-based . . 6, 237, 237, 241, 242, 251, 258 fuzzy . . . . . . . . . . . 224, 226, 226, 231 interval . . . . . . . . . . . . . . . . . . . . . . . . 148 model-based (MBR) . . . . . . . . . . . 176 qualitative . . . . . . . . . . . 147, 153, 160 reasoning unit . . . . . . . . . . . . . . . . . . . . . 225 regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 regime shift . . . . . . . . . . . . . . . . . . . . . . . . 231 regulon 56, 61, 64–67, 71, 72, 75, 77, 86, 86, 89, 90 meta- . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 state- . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 relationship spawner-recruitment . . . . . . . . . . .221 renewable resource . . . . . . . . . . . . . . . . . . 84 repeller . . . . . . . . . . . . . . . . . . . . . . . . 62, 120 reserve fund . . . . . . . . . . . . . . . . . . . . . . . 191

317 resource space . . . . . . . . . . . . . . . . . . . . . . .88 retrieval component . . . . . . . . . . . . . . . 245 risk exposure . . . . . . . . . . . . . . . . . . . . . . 189 risk reserves . . . . . . . . . . . . . . . . . . . . . . . 187 risk sharing . . . . . . . . . . . . . . . . . . . . . . . . 196 risk-prone area . . . . . . . . . . . . . . . . 185, 190 root vertex . . . . . . . . . . . . . . . . . . . . . . . . .172 rule Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . 266 fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . 226 if-then . . . . . . . . . . . . . . . . . . . . 224–226 multi-Hebbian . . . . . . . . . . . . . . . . . . 90 S salmon . . . . . . . . . . . . . . 221, 225, 228, 231 salt intrusion . . . . . . . . . . . . 248, 251, 252 satisfaction level . . . . . . . . . . . . . . 157, 158 SAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 baseline . . . . . . . . . . . . . . . . . . . . . . . . . 41 catastrophe . . . . . . . . . . . . . . . . . . . . 189 destructive . . . . . . . . . . . . . . . . . . . . .185 earthquake . . . . . . . . . . . . . . . . . . . . 185 GTAS . . . . . . . . . . . . . . . . . . . . . . . . . . 28 SRES . . . . . . . . . . . . . . . . . . . . . . . 41, 42 WRE . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 sea level rise . . . . . . . . . . . . . . . . . . . . . . . . 15 sea surface temperature (SST) 224, 228 security index . . . . . . . . . . . . . . . . . . . . . . . 46 seismic activity . . . . . . . . . . . . . . . . . . . . 191 selection . . . . . . . . . . . . . . . . . . . . . . . . . . .256 selection technique . . . . . . . . . . . . . . . . . 205 self-organization . . . . . . . . . . . . . . . . . . . . 87 set Cantor . . . . . . . . . . . . . . . . . . . . . . . . . . 69 conditional credal . . . . . . . . . . . . . 271 constrained . . 57, 60, 64, 69, 74, 80, 81–84, 91, 106 credal . . . . . . . . . . .266, 266, 269, 275 expected utility on restricted (EURS) . . . . . . . . . . . . . . . . . . . . 16 fuzzy . . . . . . . . . . . . . . . . . . . . 27, 36, 58

318 invariant . . . . . . . . . . . . . . . . . 106, 121 linguistic . . . . . . . . . . . . . . . . . . . . . . 225 no-return . . . . . . . 106, 121, 126, 127 posterior credal . . . . . . . . . . . . . . . . 266 random . . 26, 30, 30, 32, 33, 38, 39, 43 viable . . . . . . . . . . . . . . . . . . . . . 103, 106 Severi, Francesco . . . . . . . . . . . . . . . . . . . . 63 similarity . . . . . . . . . . . . . . . . . . . . . 150, 155 similarity measure . . . . . . . . . . . . .151, 161 similarity measurement . . . . . . . . . . . . 249 simple additive ranking (SAR) . . . . .292 simple additive weighting (SAW) . . 292, 293, 294 simulated annealing . . . . . . . . . . . . . . . . 210 smallholder farming . . . . . . . . . . . . . . . . 128 Smith, Adam . . . . . . . . . . . . . . . . . . . . . . . 59 soft-computing . . . . . . . . . . . . . . . . . . . . . 148 soil characteristics . . . . . . . . . . . . . . . . . 271 soil moisture . . . . . . . . . . . . . . . . . . . . . . . 272 solar-panel water-pump . . 164, 164, 166 solution bang bang . . . . . . . . . . . . . . . . . . . . . . 77 Pareto-optimal . . . . . . . . . . . . . . . . .209 set-valued . . . . . . . . . . . . . . . . . . . . . . . 91 viable . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 solution extraction . . . . . . . . . . . . . . . . . 173 spawner abundance . . . . . . . . . . . . . . . . 222 Special Report on Emission Scenarios (SRES) . . . . . . . . . . . . . . . . . . . . . 41 SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223, 224 SST . . . . . . . . . . . . . . . . . . . . . . . . . . 224, 224 stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 stabilization target . . . . . . . . . . . . . . 41, 46 stakeholder . 2, 3, 11, 184, 185, 199, 202 stakeholder dialogue . . . . . . . . . . . . . . . . . . 3 state space . . . . . . . . . . . . . . . . . . . . . . . . . . 60 state variable . . . . . . . . . . . . . . . . . . . . . . 116 statistical learning . . . . . . . . . . . . . . . . . 148 stock productivity . . . 224, 227, 230, 230 strategy adaptation . . . . . . . . . . . . . . . . . . . . . 185

Subject Index climate protection . . . . . . . . 2, 11, 12 decision . . . . . . . . . . . . . . . . . . . . . . . .189 insurance . . . . . . . . . . . . . . . . . . . . . . 195 management . . . . . . . . . . . . . . . . . . . . 99 mitigation . . . . . . . . . . . . . . . . . . . . . 185 sustainability . . . . . . . . . . . . . . . . . . . . 1, 183 sustainable fisheries . . . . . . . . . . . . . . . . 221 sustainable yield . . . . . . . . . . . . . . . . . . . . 78 symbolic coding . . . . . . . . . . . . . . . . . . . . 100 symbolic landmark . . . . . . . . . . . . . . . . . 104 symptom . . . . . . . . . . . . 107, 110, 111, 122 syndrome 3, 100, 107, 109, 111, 113, 132 Dust-Bowl . 101, 112, 121, 122, 123 Overexploitation . . . . 101, 109, 112, 114, 115, 117, 118, 120, 121 Sahel . . . . . . . . . . . 101, 112, 128, 128 syndrome core . . . . . . . . . . . . . . . . . . . . . 114 system complex . . . . . . . . . . . . . . . . . . . . . . . . 57 control . . . . . . . . . . . . . . . 64, 66, 67, 91 evolutionary . . . . . . . . . . . . . . . . .60, 61 Inert-Schaeffer meta- . . . . . . . . . . . . 80 Marchaud . . . . . . . . . . . . . . . . . . . . . . .64 meta- . . . . . . . . . . . . . . . . . . . . . . . . . . .65 Verhulst-Inert meta- . . . . . . . . . . . . 83 Verhulst-Schaeffer meta- . . . . . . . . 81 Verhulst-Scheaffer . . . . . . . . . . . . . . 82 system’s case-base . . . . . . . . 246, 247, 255 T technique stochastic optimization . . . . . . . . 184 temporal pattern . . . . . . . . . . . . . . . . . . .223 theorem Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . 267 convergence . . . . . . . . . . . . . . . . . . . . . 64 equilibrium . . . . . . . . . . . . . . . . . . . . . 64 fixed point . . . . . . . . . . . . . . . . . . . . . . 64 guaranteed coverage . . . . . . . . . . . 105 Nagumo . . . . . . . . . . . . . . . . . . . . . . . . 92 viability . . . . . . . . . . . . . . . . . 64, 66, 92 theory

Subject Index Bayesian . . . . . . . . . . . . . . . . . . . . . . . 264 control . . . . . . . . . . . . . . . 59, 85, 91, 92 decision . . . . . . . . . . . . . . . . . . . . . . . . . 14 Dempster-Shafer . . . . . . . . . . . . . . . . 26 extreme value . . . . . . . . . . . . . . . . . . 185 fuzzy set . . . . . . . . . . . . . 222, 244, 249 graph . . . . . . . . . . . . . . . . . . . . . . 59, 106 machine learning . . . . . . . . . . . . 3, 148 multi-attribute utility (MAUT) 285, 292 multi-attribute value (MAVT) . 292 non-Archimedian decision . . . . . . . 16 of choice under uncertainty (Savage) 12 possibility . . . . . . . . . . . . . . . . . . . 27, 32 probability . . . . . . . . . . . . . . . . . . 11, 15 Takahashi’s . . . . . . . . . . 273, 279, 280 uncertainty . . . . . . . . . . . . . . . . . . . . 263 viability . . . . . 55, 55, 56–58, 60, 106 Walras general equilibrium . . . . . . 90 theory of choice . . . . . . . . . . . . . . . . . . . . . 13 Ticino canton 6, 264, 272, 276, 277, 280 tolerable window approach . . . . . . . . . . 81 tractability . . . . . . . . . . . . . . . . . . . . . . . . . . 99 trajectory . . 60, 62, 76, 81, 83, 106, 127, 137 admissible . . . . . . . . . . . . . . . . . . . . . 103 emission . . . . . . . . . . . . . . . . . . . . .15, 17 forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 35 qualitative . . . . . . . . . . . 126, 126, 131 transactions cost . . . . . . . . . . . . . . . . . . . 191 tree clustering . . . . . . . . . . . . . . . . . . . . . 173 triggering factor . . . . . . . . . . 264, 269, 280 Tuscany . . . . . . . . . . . . . . . . . . 184, 185, 196 tutoring system . . . . . . . . . . . . . . . . . . . . 163 tychasm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 U uncertainty . . . . . . . . . . . . . . . . . . . . . . . 1, 2, 12, 19, 26, 27, 29, 37, 41, 47–49, 57–59, 101, 102, 106, 151, 160, 222, 239, 241, 260, 263, 264, 273 ambiguous . . . . . . . . . . . . . . . . . . . . . . 13

319 epistemic . . . . . . . . . . . . . . . . . . . . . . . 11 probabilistic . . . . . . . . . . . . . . . . . . . . 13 system . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Upper Milford Haven estuary . 242, 244, 247, 248, 248, 253, 256 upward computation . . . . . . . . . . . . . . . 173 urban area . . . . . . . . . . . . . . . . . . . . . . . . . 271 V value economic . . . . . . . . . . . . . . . . . . . . . . 184 extreme . . . . . . . . . . . . . . . . . . . . . . . 185 membership . . . . . . . . . 225, 227, 231 property . . . . . . . . . . . . . . . . . . . . . . . 191 threshold . . . . . . . . . . . . . . . . . . . . . . 263 utility . . . . . . . . . . . . . . . . . . . . . . . . . 299 variable auxiliary . . . . . . . . 150, 155, 157, 158 evidence . . . . . . . . . . . . . . . . . . . . . . . 276 independent and identically distributed . . . . . . . . . . . . . . . . . . . 186 interval . . . . . . . . . . . . . . . . . . . . . . . . 150 qualitative . . . . . . 148–150, 156, 156 quantitative . . . . . . . . . . . . . . 149, 274 vegetation type . . . . . . . . . . . . . . . . . . . . 271 Verhulst, Pierre-Fran¸cois . . . . 55, 56, 76 viability . . . . . . . .1, 60, 63, 84, 105, 135 ecological . . . . . . . . . . . . . . . . . . . . . . 135 economic . . . . . . . . . . . . . . . . . . . . . . 135 viability crisis . . . . . . . . . . . . . . . . . . . 67, 91 viability criteria . . . . . . . . . . . . . . . . . . . . 112 viability multiplier . . . . . . . . . . . . . . . . . . 88 viable-capture basin . . . . . . . . . . . . . . . . . 62 volatile organic compound (VOC) . . 289 vulnerability . . . . . . 5, 183, 190, 191, 196 W warning signal . . . . . . . . . . . . . . . . . . . . . . 56 waste disposal . . . . . . . . . . . . . . . . . . . . . 238 weather condition . . . . . . . . . . . . . 176, 177 welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Wheat Belt . . . . . . . . . . . . . . . . . . . . . . . . 121 WRE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Advanced Methods for Decision Making and Risk ...

Advanced Methods for Decision Making and Risk ...

Suggest Documents

PDF Download Advanced Methods for Decision Making ... - Google Sites

Advanced Decision Making

Criteria for evaluating group decision-making methods

MULTI-ATTRIBUTE DECISIOn-MAKInG METHODS

Introduction to Decision Making Methods

Towards Intelligent Decision Making for Risk Screening

Risk Based Decision Making Approach

Multiple Attribute Decision Making: Methods and ...

Multiple Attribute Decision Making: Methods and ...

Decision-Making Process: Concepts and Methods

entry methods and international marketing decision making

Multiple Attribute Decision Making: Methods and

Multiple Attribute Decision Making: Methods and ...

Imperfect Decision Making and Risk Taking Are

Risk, uncertainty and decision-making - UKCIP [PDF]

Risk and trust in vaccine decision making

Risk, uncertainty and decision-making - UKCIP

UNCERTAINTY AND RISK-BASED DECISION-MAKING

A Decision Support Approach for Group Decision Making under Risk ...

Methods and Models for decision Making - 1 - Lamsade

multi-criteria decision-making methods and their applications for ...

Developing a Risk Analysis and Decision Making Strategy for an ...

Multi-Criteria Decision Making Approach for Flood Risk and Sediment ...

Multi-Criteria Decision Making Methods: Comparative ...