Uncertainty propagation in environmental decision

Available online at www.sciencedirect.com

Procedia Environmental Sciences 2 (2010) 576–584

International Society for Environmental Information Sciences 2010 Annual Conference (ISEIS)

Uncertainty propagation in environmental decision making using random sets Kejiang Zhang a* Gopal Acharia

a

Department of Civil Engineering, University of Calgary, 2500 University Dr NW, Calgary, Alberta, T2N 1N4, Canada

Abstract Significant uncertain information is involved in environmental decision making due to complexities of natural systems, lack of sufficient data, and the interpretation of information that may be in numerical or linguistic forms. Uncertainties can be present in identification of criteria, interactions among criteria, evaluations of alternatives, eliciting weights from experts, and the choice of aggregation operators. Uncertainties arising from performance evaluations of criteria for each alternative and weights can be identified as aleatory (random) and epistemic (informal and lexical) uncertainty. These two types of uncertainty were best respectively represented as probability density function and possibility distribution. A methodology was presented in this paper to propagate these two kinds of uncertainty through aggregation operators. Random set theory is used as a uniform framework to integrate aleatory uncertainty and epistemic uncertainty. Evidence theory is utilized to approximate the probability measure when both probability density functions and possibility distributions are transformed into random sets. This methodology facilitates the incorporation of aleatory and epistemic information into the multicriteria environmental decision makings.

© 2010 Published by Elsevier Ltd. Keywords: aleatory and epistemic uncertainty; multicriteria decision making; random sets; evidence theory

1. Introduction Environmental decision making is a complex process due to the complexities of natural systems that are not well understood, lack of all relevant data, and the varying needs of different stakeholders [1]. Multiple criteria decision analysis (MCDA) is a class of decision-making methodology that assisting a decision-maker through the decision process via explicit formalized models [2] [3] [4]. Kiker et al. (2005) [5] present a review of the available literature and provide recommendations for applying different MCDA techniques, including the Analytical Hierarchy Process (AHP), ELimination and Choice Expressing the REality (ELECTRE), Multiattribute Utility Theory (MAUT), PROMETHEE, and their combinations. There are four methods in multicriteria decision making (MCDM): x The outranking approaches such as ELECTRE and PROMETHEE methods x The value and utility theory approaches such as AHP x Interactive methods such as the step method (STEM) [6] x Theinteractive multiple goal programming [7]. MCDA is widely used in environmental management [8] - [11] as it allows the measurements of criteria performances by a variety of benchmarks with different units [12]. With increasing complexity of environmental problems, more complicated MCDA models incorporating competing objectives such as economic and

* Corresponding author. Tel.: +1-403-220-2467. E-mail address: [email protected].

1878-0296 © 2010 Published by Elsevier doi:10.1016/j.proenv.2010.10.063

Kejiang Zhang and Gopal Achari / Procedia Environmental Sciences 2 (2010) 576–584

environmental objectives, have to be developed. As various types of information obtained from different sources have to be used, information uncertainty and its effects on final decision must be considered. Most methodologies of MCDA require an aggregation step to integrate all information. Aggregation operators such as Ordered Weighted Average (OWA) and PROMETHEE algorithm are used to integrate a set of performance criteria to get a so-called mean or typical value for each alternative. These values are then used to establish a priority, ranking or recommendation. A technique is required to properly represent and transparently propagate different types of uncertainty through the aggregation operators such that it instils a degree of confidence to the decision-makers [12]. The objectives of this paper are (1) to investigate the sources and types of uncertainty inherent in decision making (2) to identify the proper representations for different types of uncertainty, (3) to develop a methodology for propagating different kinds of uncertainty through aggregation operators. 2. Uncertainty in environmental decision making [Klauer et al. (2006) [12] present that there are three types of uncertainty in MCDA: development uncertainty, model uncertainty, and weight uncertainty. Development uncertainties are caused if the decision depends on the predictions in the future. For example, let say one has a site with known contamination of many kinds. If one has to develop a budget for such a site, he/she has to make many assumptions which relate to levels and extent of remediation and sequence to be followed. These lead to development uncertainty. Model uncertainty occurs when the input data of a model used to evaluate the performances of criteria are uncertain. Weight uncertainty is introduced when different people provide the relative importance of different criteria [13]. Maier and Ascough II (2006) [1] mentioned that this type of uncertainty is caused by human inputs, which include knowledge and expertise of modeller; knowledge, and values of stakeholders; values and attitudes of managers/decision makers, and the current political ‘climate’. When sufficient information is not available, the evaluation of the performance for a specified criterion will induce uncertainty such as the evaluation of the hazard of a new chemical. CCME (2008) [14] uses linguistic terms such as “high”, “medium”, and “low” to represent the potential hazard based on a number of factors including potential human and ecological heath effects. That means the human factors not only affect the determination of weights but also the evaluation of criteria performances. Hyde et al. (2003) [15] and Maier and Ascough II (2006) [1] point out that uncertainties identifying all criteria plays a crucial role in ranking alternatives and should be jointly considered with weight uncertainty and uncertainty of criteria evaluations. For example, a political criterion will significantly affect the ranking of contaminated sites for remediation. Zhang et al (2009) [16] present a comparative approach for ranking contaminated sites by using fuzzy PROMETHEE. The criteria were identified based on the factors that potentially influence human health and ecological risk posed by contaminated sites. Risk-based and/or sustainability-based criteria [17] identification can better reduce uncertainties induced by identification of criteria. Conventionally, the criteria identified in MCDA are assumed to be independent. When there are conflicting objectives in MCDA process and interactions exist in some objectives [18] [19], uncertainty will be induced. This is another uncertainty that is induced due to the choice of aggregation operator. For example, given the criteria for selected contaminate sites, different outranking methods such as ELECTRE and PROMETHEE can be used to get the priority of contaminates for remediation. Uncertainty will then be induced in different choices. So far is not easy to evaluate the uncertainty caused due to different methods. Salminen et al. (1998) [20] compare the methods, PROMETHEE I and II, ELECTRE III, and SMART (Simple Multi-attribute Rating Techniques) based on a set of randomly generated problems. They found that in a particular problem the “best alternatives” obtained with these methods may be significantly different given the same criteria evaluations. Three types of uncertainty inherent in weights and the performances of criteria evaluations can be identified [21]) based on the nature of uncertainty, i.e., random uncertainty which is inherently random in nature, informal uncertainty arising due to the limited availability of site information, and lexical uncertainty which occurs when linguistic variables such as “low”, medium”, and “high” are used. Linguistic variables which are expert opinions are often selected by regulatory agencies in the United States and Canada to represent the priorities for alternatives based on the predetermined rules [14] [22] [23]. The random uncertainty is often referred to as aleatory uncertainty and lexical and informal uncertainty are called epistemic uncertainty. 3. Uncertainty representation Subjective probability is used to represent uncertainty with single-events or one-time occurrences and reflects a person’s belief in the occurrence of an event. It extends the use of probability to circumstances which are not

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amenable to relative frequency interpretations [24]. Different people may assign different degrees of belief to a particular event. When sufficient information is not available, fuzzy set theory [25] and its extension possibility theory [26] [27] are more suitable to represent uncertainty. It is recognized that aleatory uncertainty can be better represented using probability theory, while epistemic uncertainty can be best represented by fuzzy set theory, possibility theory, evidence theory [28], and random sets [29] - [31]. When one uncertainty theory better represent one type of uncertainty and another uncertainty theory is more suitable for another one, both aleatory and epistemic uncertainty may exist simultaneously. A technique to propagate both of these uncertainties through the MCDA process is required. 4. Uncertainty propagation When some uncertainties are represented as probability distributions (pdfs), while other uncertainties as possibility distributions (or fuzzy numbers), the propagation of both types of uncertainties is needed. As the probability distribution and the possibility distribution represent different kinds of uncertainties, they cannot be propagated directly. An intuitive method is to transform a probability distribution to the corresponding possibility distribution or vice versa. If all probability density functions are transformed to the corresponding possibility distributions, fuzzy techniques can be used; if all the possibility distributions are transformed to the corresponding probability distributions (pdfs), stochastic methods such as Monte Carlo simulations can be used. Many methods are available for these transformations [32] - [35]. The limitation of transformation methods is that the transformation of a possibility distribution to the corresponding pdf will add external information and that from a probability distribution to a possibility distribution will lose information [33]. As the loss and gain of information cannot be quantified, their influence on final decision cannot be justified. Thus, random set theory is selected as a uniform framework to integrate aleatory and epistemic uncertainties. Both these two kinds of uncertainty represented simultaneously as the pdfs and possibility distributions can be transformed into a set of random sets. The computational algorithm is illustrated in Figure 1. The following sections explain each step illustrated in Figure 1. 4.1. Implication of random sets Random sets are sets defined on some probability space where values are represented as sets rather than points [29]-[31]. A random set is a multi-valued mapping and it can be considered as the imprecise observation of a random variable. It is a generalization of a random variable [36]. The comparison of a random set and a random variable is illustrated in Figure 2. Random set theory has close relation with probability theory and possibility theory [27]. Mathematically, these relations can be expressed as: Let U be a universal non-empty set and

( U the power set of U. In random set

theory, available evidence for a variable u is expressed by a basic probability assignment on ( U , i.e., by a set function m:

( U o > 0,1@ such that m 0 and ¦ A( U m A 1 . This set function is considered as a

probability measure defined in a universal set Z (which can be considered as the set of observations) related to U through a multi-valued mapping

* : Z o ( U [37].

This multi-valued mapping is caused by the existence of imprecise information. For each set m(A) express the probability of U is a pair

*, m , where

z

A ( U , the

* 1 A If m (A) > 0, the set A is called focal element. A finite random set on

m is a basic probability assignment (bpa) on

( U and * is the family of focal

elements induced by m. For simplicity, random sets used in this research refer to finite random sets as there is another type of random set, i.e. random closed set [31].


Random sets

Determination of the output intervals of the aggregation operators and the corresponding basic probability assignments Determination of the lower bound belief function and the upper bound plausibility function for each alternative Calculate the mathematical expected value for each alternative

P

ªN « ¦ mi inf Ai , ¬i 1

º

N

¦ m sup A »¼ i

i

i 1

Comparison of intervals Fig. 1. Computational algorithm of propagation of aleatory and epistemic uncertainty in multicriteria decision making X (e) = A

0

:

U

e

(a) X (e)

0

:

U

e

(b)

Fig. 2. Comparison of (a) random set and (b) a random variable, where is the sample space and U is the universe of discourse denoted as 2U

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4.2. Correlations of random sets theory with fuzzy set, probability and possibility theory A fuzzy number (or a possibility distribution) can be discretized into a set of random sets by using Į-cuts. If 1= Į1 > Į2 >…> Įn = 0, the bpa for ith interval is mi = Įi - Įi+1. A probability density function (pdf) can also be decomposed into a set of random sets. The bpa for each interval is the integration of the pdf over the corresponding interval. Molchanov (2005) [30] and Nguyen (2006) [31] concluded that random set theory closely related to evidence theory [28], i.e., the distribution function of a random finite set is a belief function and the distributional functional of a random closed set (a random closed set on d-dimensional Euclidean space (Rd) is a map X : : o , where is the closed subsets of Rd) is a plausibility function. These correlations facilitate the application of random set theory as a uniform framework to propagate hybrid uncertainties. Due to imprecision, the probability measure of a general variable u U cannot be derived and only its range can be determined by calculating its lower and upper bound based on evidence theory [28]. The lower and upper bound probabilities are called belief function and plausibility function, respectively [28] which are given as

Bel E

¦

z B E

mB

(1)

The belief of E is the sum total of the probability assignments to all sets which are subsets of E. The plausibility of E is the sum total of all the probability assignments of sets which have at least one element common, i.e. E B z . This is defined as [28]:

Pl E Obviously,

¦

B E z

mB

(2)

Bel E d Pl E . The range of probability of set E denoted as Pr E lies between Bel(E) and

Pl(E):

Bel E d Pr E d Pl E

(3)

4.3. Propagation of uncertainty through aggregation operators Initially the probability and possibility distributions, which are the representations of uncertain criteria performance evaluations, are discretized into random sets (see Figure 1), i.e., a collection of intervals and the corresponding basic probability assignments. Output intervals and the corresponding basic probability assignments are calculated for the aggregation operator. For each combination of intervals obtained from the discretization of random and fuzzy criteria performances, (e.g., if there are three uncertain criteria performances, each criterion being discretized into 3 intervals and there are 3 basic probability assignments, then there will be 3u3u3 = 27 combinations of intervals and basic probability assignments), interval techniques are used to determine the output intervals of the aggregation operator. The criteria are assumed to be independent in this research and the basic probability assignment for each output interval can be calculated directly. For example, given interval A = [1, 5], mA = 0.3, and interval B = [2, 8], mB = 0.5, obtained from the discretization of two random or fuzzy criteria evaluations. The output interval A + B is [3, 13] and the corresponding basic probability assignment for the interval A + B is the product of mA multiplying mB, i.e., 0.15. This process is repeated for each combination, finally, the distributions of belief function and plausibility function as the lower and upper bound of the cumulative probability distributions, respectively are obtained. The cumulative probability distribution lies in between of these two bounds. The discrepancy between the belief and plausibility functions is caused by the presence of epistemic uncertainty. If there are no epistemic uncertainties, i.e., only aleatory uncertainties are available, the belief function and plausibility function will converge to the cumulative probability distribution function.


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Due to the existence of aleatory and epistemic uncertainty, the output of the aggregation operator for each alternative (see Figure 1) gives a lower bound (belief function) and an upper bound (plausibility function). Comparison of values of alternatives as ranges is not easy. This can be done by determining the mathematical expected values for upper and lower bound cumulative probability distribution functions, respectively, i.e. find the mathematical expected values for belief and plausibility functions. When U is the real line, the lower and upper bound cumulative probability distributions can be determined by [38]

FL u

Bel ^uc U : uc d u`

Fu u

Pl ^uc U : uc d u`

¦

m Ai

(4)

¦

m Ai

(5)

Ai :u tinf Ai

Ai :u tsup Ai

The range of the expected value ( P ) of the random variable u can be estimated based on Eqs. (4) and (5) ([38] [39]) which is given by

P

ªN « ¦ mi inf Ai , ¬i 1

º

N

¦ m sup A »¼ i

(6)

i

i 1

Thus, we obtain an interval of the expected value of a random variable. This is induced due to the presence of imprecise information. 4.4. Ranking interval numbers As an interval of the expected values is obtained for each alternative, these interval values have to be ranked to complete the MCDA process. Teno and Mareschal (1998) [38] considered an interval as a fuzzy set, defuzzyfication techniques are then used to translate a fizzy set into a real number. Final decision is then based on the translated values. Shakhnov (2008) [40] presented three models of ranking interval-defined objects based on pairwise relations of domination with respect to probability, mathematical expectation, and utility. Xu and Da (2003) [41] presented a methodology for comparison of two interval numbers based on possibility degree formula. Zhang and Su (2007) [42] reviewed the existing methods for ranking two interval numbers. An enhanced ranking approach for interval numbers based on the possibility degree (Li, 2004) which represents the degree of one interval being greater than

a

b , then I a

> a, a @ and

ª¬ b , b º¼ be two interval numbers. If a b and I b ; if the possibility degree of I a ! I b denoted as PI a ! Ib is greater than 0, then I a ! I b ; if

another interval is used in this paper. Let

Ia

Ib

PI a ! Ib 0 , then I a I b . The possibility degree of Ia > Ib is defined as

PI a ! Ib

If I b

°1, if a t b ° °° a b b a , if a d b b d a ® a a ° °a b § b a · § a b · ° ¨ ¸¨ ¸ , if b d a b d a ¯° a a © a a ¹ © b b ¹

ª¬ b , b º¼ degrades to a single value, i.e., I b

ª¬ b , b º¼

b , The possibility degree of Ia > Ib is given by

(7)

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if b a

1, ° ° a b b a , ® a a ° °¯1,

PI a ! Ib

If Ia and Ib both degrade to a single value, i.e.,

Ia

if a b d a

(8)

if b ! a

> a, a @

a and I b

¬ª b , b ¼º

b the comparison of two

intervals become ranking two real numerical values. In this scenario, the possibility degree is then defined as

1, ° ®0, if a b d a , °1, ¯

PI a ! Ib

if a ! b if a b

(9)

if a b

The comparison matrix of possibility degree for m interval numbers is determined by

§ 0 ¨ P P ¨ 21 ¨ ¨ © Pm1 where Pij is the possibility degree of m

0 Pm 2

ik

k 1

r1 , r2 , , rm

T

P1m · ¸ P2 m ¸ ¸ ¸ 0 ¹

(10)

I ai ! I a j and meets 1 d Pij d 1 and Pij Pji

¦P ,

ri One gets R

P12

0 Let

i 1, 2, , m

(11)

. The comparison of interval numbers then becomes ranking

ri , i.e., the higher the

ri , the higher the interval number. A simple example was used to illustrate the presented methodology for ranking intervals. Given four intervals: I1 = [0.188, 0.197], I2 = [0.206, 0.220], I3 = [0.198, 0.207], I4 = [0.150, 0.197]. The comparison of possibility degree for these four intervals is determined by implementing Eq. (12) and is given below,

Pij

0 0 0.809 · § 0 ¨ ¸ 0 0.993 1 ¸ ¨ 1 and ri ,i ¨ 1 0.007 0 1 ¸ ¨ ¸ 0 0 0 ¹ © 0.197

The final ranking result is

0.809, 2.993, 2.007, 0.197

T

1,,4

I 2 ! I 3 ! I1 ! I 4 .

5. Discussion Uncertainties in environmental decision making were investigated in this paper. The representation and propagation of uncertainties inherent in criteria performance evaluations and weights through aggregation operators


were discussed in this research. The effects caused by uncertainties in selection of aggregation operator, correlations amongst criteria, and the identification of criteria on final decision making were not investigated. Uncertainties in criteria performance evaluations and weights were identified as aleatory and epistemic uncertainty based on the nature of uncertain information which is respectively represented using probability theory and fuzzy set theory and its extension possibility theory. A framework was developed to propagate both aleatory and epistemic uncertainty through the aggregation operators. This methodology is suitable for any aggregation operators. The difference between propagating uncertainty through a simple aggregation operator such as a weighted average operation and complicated ones such as PROMETHEE methods lies in the techniques used to determine the output intervals. As most of the aggregation operators used in multicriteria decision making is monotonic, the output intervals can be easily determined. In this research, the criteria are assumed to be independent, thus the basic probability assignments can be easily determined. If the interactions amongst criteria are considered, and the degree of the correlation (a simple correlation matrix) is unknown, a linear optimization algorithm has to be used to calculate the basic probability assignments [43]. This will be investigated in future work. 6. Conclusions Significant uncertain information is involved in environmental decision making due to the complexities of natural systems and lack of sufficient data. Different types of uncertain information are identified as aleatory and epistemic information which are better represented using probability theory and fuzzy set theory and its extension possibility theory. A methodology was developed to propagate both aleatory and epistemic uncertainties through the aggregation operators by using the merits of random sets theory and the evidence theory. Due to the presence of epistemic information or imprecise information, the probability measure for an output of the aggregation operator is not a single value but a range which is approximated by using the lower bound (belief measure) and the upper bound (plausibility measure) based on evidence theory. The global value of each alternative is compared based on the comparison of interval numbers. This methodology facilitates the incorporation of aleatory and epistemic uncertain information into the multicriteria environmental decision making.

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