Accuracy-Interpretability Trade-O�ff for Precise Fuzzy Modeling

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Accuracy-Interpretability Trade-Off for Precise Fuzzy Modeling using simple indices. Application to Industrial Plants ? M. Galende ∗ G.I. Sainz ∗,∗∗ M.J. Fuente ∗∗ ∗

CARTIF Centro Tecnol´ ogico, Parque Tecnol´ ogico de Boecillo parcela 205, 47151 Boecillo (Valladolid), Spain (e-mail: [email protected],[email protected]). ∗∗ Department of Systems Engineering and Control, School of Industrial Engineering, University of Valladolid, Paseo del Cauce s/n, 47011 Valladolid, Spain (e-mail: [email protected],[email protected]) Abstract: In general, the techniques of fuzzy modeling are oriented to obtaining rule based systems with high accuracy but rarely interpretable in accordance with the fuzzy logic principles. Both concepts are in conflict and it is necessary to achieve a good trade-off between the two aspects. Here, an index to measure interpretability in fuzzy rule based systems, which has been proposed by the authors to improve accuracy-interpretability trade-off in previous work, is used for complex industrial problems, such as biotechnological processes in a wastewater treatment plant. This interpretability index is based on the aggregation of simple indices of complexity and similarity. Using generic neuro-fuzzy systems, a post-processing rule selection based on a genetic approach is done, considering the error as an index for accuracy and the aggregate index based on similarity for interpretability. In this work, this methodology is applied to an industrial plant to model two different processes: the biomass and substrate concentrations of a wastewater treatment plant. Both are modeled using different neuro fuzzy systems, FasArt and NefProx, and improved on the basis of the accuracy-interpretability trade-off. The results show that the approach permits fuzzy models to be obtained with a better balance between accuracy and interpretability, while also improving the simplicity of the model. Keywords: Fuzzy modeling, Genetic algorithms, Accuracy, Interpretability, Complexity 1. INTRODUCTION Fuzzy logic systems have proved their usefulness in a large number of applications (Bonissoene et al. (1999)). In the scientific literature, it is possible to find methodologies, algorithms and applications based on fuzzy logic theory or in combination with other approaches: neural networks, genetic algorithms, etc. Applications for control, modeling, patterns recognition, computer vision and signal processing are very known and used. If the scientific literature is revised, it is usual to find two well known modeling approaches to generate rule based fuzzy models (Casillas et al. (2003b,a)): (1) Precise Fuzzy Modeling, these models are generated by data and, in general, the procedure is based on minimizing the error. In general, this model has a good accuracy but low level of interpretation. (2) Linguistic Fuzzy Modeling, where the knowledge of experts guides the process. The rules have a good level of interpretability but poor accuracy. ? This work was supported by the Spanish Ministry of Science and Innovation under grants no. CIT-460000-2009-46 and DPI2009144410-C02-02.

Copyright by the International Federation of Automatic Control (IFAC)

Both modeling approaches have drawbacks concerning accuracy or interpretability because both objectives are in conflict when the principles of fuzzy logic are considered. In general, precise fuzzy modeling is the most popular approach in engineering areas and a large number of algorithms of this approach can be found in technical and scientific literature. In some cases, the interpretability of the fuzzy rule system is involved in the modeling but this is not very usual in general. This paper is focused on the balance of both aspects (accuracy and interpretability) for modeling problems, taking advantage of the well-known precise fuzzy approaches and improving their interpretability through: • a set of indices based on complexity, simplicity and similarity measures. • a rule selection based on a two-objective genetic approach. In this way, this proposal improves precise fuzzy modeling to obtain models with a better accuracy-interpretability trade-off. The way in which fuzzy modeling deals with accuracy is clear: the most common way is based on error. On the

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

other hand, when the interpretability concept is involved then there is no concise, clear and unique idea about the meaning of interpretability or the way in which this idea can be formulated. The interpretability index InterC used in this paper was proposed and validated in Galende-Hern´ andez et al. (2011. Unpublished.). Here, this index is applied to real-world problems. The paper is organized as follows: first, a brief description of alternative points of view for fuzzy modeling, interpretability and accuracy is given. Then a very brief description of the case studies is carried out and the interpretability index is explained. Then the methodology used in this work is described and some industrial results obtained are discussed. Finally, the most interesting conclusions obtained from this work are set out. 2. FUZZY MODELING: ACCURACY VS. INTERPRETABILITY In general, the most common approach in engineering areas on techniques of fuzzy modeling have been oriented towards obtaining rule-based systems as accurate as possible, so interpretability fuzzy properties (understanding and interpretation) are the weak performances of these fuzzy systems. One interesting goal is to achieve a good balance or compromise solution between accuracy and interpretability, obtaining an accurate model with an adequate accuracy and a good level of explanation (Casillas et al. (2003b,a)). 2.1 Accuracy-Interpretability Trade-Off There are several ways to deal with the generation of fuzzy systems whose performance includes an adequate accuracy-interpretability trade-off: • Algorithms taking into account the idea of accuracyinterpretability during the generation of the fuzzy system. i.e genetic fuzzy systems (Alonso et al. (2008); Ishibuchi and Nojima (2007); Cord´ on et al. (2001)). • Interpretability of the fuzzy systems is improved in a postprocessing way. Here, two approaches are found: · Similarity measures (Jin (2000); Setnes et al. (1998)). · Orthogonal Transforms, where interpretability is improved by a complexity reduction (Destercke et al. (2007); Setnes and Babuˇska (2001)). But nowadays, the way in which a fuzzy system can be more interpretable is an open question: • Sometimes this concept appears associated with the concepts of complexity and explanation capacity (Ishibuchi et al. (2009)), which can be considered as indirect measures to evaluate the interpretability. • In Zhou and Gan (2008), a review about interpretability is presented and two levels of interpretability are considered: Low-level Interpretability for fuzzy sets and High-level Interpretability for fuzzy rules. • Another point of view about interpretability, based on the previous one with added new elements, is described in Alonso et al. (2009). Here the interpretability is focused on Description (System Struc-

ture Readability) in which the Low/High Level Interpretability concepts, and the Explanation (System Comprehension) are included. In this paper, the new interpretability index proposed in Galende-Hern´andez et al. (2011. Unpublished.) is applied to some industrial models. 2.2 Neuro-Fuzzy Systems In this work, neuro-fuzzy algorithms FasArt (Cano Izquierdo et al. (2001); Sainz Palmero et al. (2000)) and NefProx (Nauck and Kruse (1999)) have been used to generate precise fuzzy models, with different complexities and fuzzy nature, to test the proposal under several contexts of complexity. According to Herrera (2008), FasArt is an approximate neuro fuzzy approach and NefProx is a linguistic neuro fuzzy approach to model fuzzy rule based systems. If Casillas et al. (2003b) is considered, both algorithms are Mamdani fuzzy rule-based systems for precise modeling. The FasArt model is a neuro fuzzy system based on the Adaptive Resonance Theory (ART). FasArt introduces an equivalence between the activation function of each FasArt neuron and a membership function. In this way, FasArt is equivalent to a Mamdani fuzzy rule-based system with: Fuzzification by single point, Inference by product, and Defuzzification by average of fuzzy set centers. A full description of this model can be found in Cano Izquierdo et al. (2001) and Sainz Palmero et al. (2000). The FasArt system has been used in several previous works (Sainz Palmero et al. (2005); Sainz et al. (2004)) for modeling, fault detection, pattern recognition, etc. with reasonable results when its accuracy as a fuzzy model is involved; but when other aspects, such as rule interpretability, are important, then some problems appear: proliferation of rules, of fuzzy sets, etc., so this system is an adequate instance for checking this proposal, taking advantage of the knowledge learnt and stored by FasArt for each problem involved. NefProx (Nauck and Kruse (1999)) is another neurofuzzy algorithm based on supervised learning for function approximation. The user tunes the parameters and the learning algorithm generates a collection of fuzzy rules from data to minimize the error. The antecedents and consequent of these IF-THEN rules are linguistic values. In this paper NefProx is used to generate Mamdani fuzzy rules with triangular membership functions. Here, the fuzzy partitions are defined and tuned by the user, so the interpretability of these elements is guaranteed in comparison with the other algorithm described previously. 3. WASTEWATER TREATMENT PLANT: BIOTECNOLOGICAL PROCESSES In this paper, interest is focused on the modeling of the activated sludge process belonging to a real wastewater treatment plant (WWTP). The formulation and solution of such a problem is of great interest, since legal and strict specifications regulate the quality of the water discharge from this type of plant in most industrialized countries.

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Modeling may help in the management and control of those plants, not only to fulfill the law, but also to reduce the cost associated with the dynamic operations of such plants. Biomass and substrate concentrations are two of the main problems involved in wastewater treatment (details in Sainz et al. (2004)). Here both aspects are considered in order to check the proposal of accuracy-interpretability balance in fuzzy modeling.

N oCoverage = ArithmeticM ean(N oCoverP artitionk ) N oCoverP oints N oCoverP artitionk = T otalP oint (5) N oCoverP oints if Activation Level βR AN D SkC (Ri , Rj ) > βR | (RuleN umber − 1)! ∀1 ≤ i < j ≤ RuleN umber (3) ∀1 ≤ kA ≤ AntecedentN umber ∀1 ≤ kC ≤ ConsequentN umber 0 < βR < 1 Redundancy =

The two-objective genetic algorithm search maximizes the accuracy and the interpretability of the fuzzy model, so the fitness functions to be minimized are: Acc = M SEtra InterC = AritmethicM ean(λj ∗ InterpretabilityIndexj )

• Consistency, avoiding incoherent rules (Eq. 4). |SkA (Ri , Rj ) > βR AN D SkC (Ri , Rj ) < βI | (RuleN umber − 1)! i < j ≤ RuleN umber (4) kA ≤ AntecedentN umber kC ≤ ConsequentN umber 1 − βR

Incoherency = ∀1 ∀1 ∀1 βI

≤ ≤ ≤ =

(6)

• Completeness or No-Coverage (Eq. 5). Complete fuzzy partition involves minimize no coverage.

One of the most usual ways to measure the accuracy of a system is the mean squared error (M SE), calculated using eq. 7 with training data. The interpretability measure is based on the index InterC explained previously. N 1 X

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M SE =

(Yi − Yi0 )2

N i=1

(7)

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Table 1. Performance of Compact Base Models

In order to use the indices proposed in Section 4, some thresholds must be tuned by the user. Considering Setnes (2003) and Casillas et al. (2003b) the thresholds used in this work are: • • • •

Model Bio-1 Bio-2 Sus-1 Sus-2

βR = 0.8 for redundancy. βI = 0.2 for incoherency. βC = 0 as the activation level of coverage. λj = 1 because all individual indices in InterC have the same importance.

(1) The most interpretable model (Best I). (2) The most accuracy (Best A). (3) The median model with an adequate interpretabilityaccuracy trade off (Median I-A).

6. EXPERIMENTAL STUDY In order to check the performance of the proposal, several processes of a WWTP have been considered (Sainz et al. (2004)): • Biomass process (WWTP.Bio): three inputs (Xi (t), qi (t), Xr (t − 1)) and one output (Xr (t)). • Substrate process (WWTP.Sus): three inputs (Si (t), qi (t), S(t − 1)) and one output (S(t)). For each case study, two base fuzzy models are generated with different complexity and fuzziness performance to check the proposal on different contexts. 6.1 Fuzzy Models Precise fuzzy models are generated based on FasArt and NefProx algorithms: one compact model and one complex model are generated for each problem or case to validate the proposal. The tuned parameters used for the FasArt and NefProx compact base fuzzy models are: WWTP.Bio-1: WWTP.Bio-2: WWTP.Sus-1: WWTP.Sus-2:

ρ = 0.9 and γ = 13. msf = 7 and maxR = −1. ρ = 0.9 and γ = 17. msf = 9 and maxR = 0.

Table 1 shows the initial performance of these models: low number of rules (RN ) and error for training and test (M SEtra/tst ), similarity (S) can be better, without redundancy (R), a low incoherency (I) and complete fuzzy partition (C). The parameters for the FasArt and NefProx complex base fuzzy models are: • WWTP.Bio-3: ρ = 1.0 and γ = 11. • WWTP.Bio-4: msf = 9 and maxR = −1.

S 0.281 0.244 0.396 0.548

R 0 0 0.002 0

I 2E-4 0 4E-4 0

C(%) 98.2 100 99.4 100

Model Bio-3 Bio-4 Sus-3 Sus-4

M SEtra/tst 0.002/0.004 0.003/0.039 0.004/0.004 0.007/0.013

RN 530 108 530 34

S 0.326 0.227 0.555 0.607

R 0.001 0 0.042 0.093

I 1E-4 0 0.001 0.011

C(%) 99.4 92.5 100 97.7

• WWTP.Sus-3: ρ = 1.0 and γ = 16. • WWTP.Sus-4: msf = 7 and maxR = −1. Table 2 shows the initial performance of these models: higher number of rules, low error, with redundant and incoherent rules and complete partitions.

6.2 Improving fuzzy models: Results

Finally, the mean values over 30 runs are calculated.

• • • •

RN 166 80 105 9

Table 2. Performance of Complex Base Models

To run genetic algorithm NSGA-II: five random initial populations are used; individuals encoded by Gray code; genetic operator by default; number of generations to stop; and cross-validation. For a system√ with a high number of rules, each population contains RuleN umber individuals, else RuleN umber. Three representative models, or points, are selected from the Pareto front according to the objectives:

M SEtra/tst 0.003/0.005 0.004/0.008 0.007/0.010 0.011/0.021

Table 3 shows the mean values obtained over 30 runs for the different compact models and data sets considered. This table shows the accuracy (Acc = M SEtra ) and interpretability (InterC ) indices considered during the post-processing rule selection for the three characteristic points in the Pareto Front (Best I, Median I-A and Best A). The table also shows the mean variations, (∆), for each individual measure considered: mean squared error for test (M SEtst ), number of rules (RN ), similarity (S), redundancy (R), incoherency (I), and mean final percentage of completeness (C(%)). For these compact models, the interpretability rate has been improved in all cases, preserving an adequate, and even better, accuracy. The test error for WWTP.Bio1, in the worst case, is increased from 0.005 to 0.009 (108.1%), but this value is not very significant and the rest of the measures are improved: number of rules (-26%), similarity (-2.6%), null redundancy, incoherency (-72%) and complete partitions (98.2%). Figure 1 shows the real and the FasArt outputs with the best interpretability-accuracy balance from the Pareto Front in the WWTP.Bio-1 case. Table 4 shows the mean values for the complex models considered. In these cases, a better interpretability is obtained with similar accuracy, except in FasArt models. Here the error is increased from 0.004 to 0.007 (81%). This sounds very high, but it is because the initial error is extremely low. Even in this case, the accuracy loss is aceptable and the model remains accurate enough and this is balanced by the reduction reached in the number of rules (29%), similarity (-2%), redundancy(-10%), incoherency (38%) and complete partitions (98%). In NefProx complex models (WWTP.Bio-4 and WWTP.Sus-4), it is possible to obtain better interpretability-accuracy trade-off in all three models. Figure 2 is similar to Figure 1 when the complex model for WWTP.Sus-3 is considered.

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Table 4. Improved Complex Models

Table 3. Improved Compact Models WTTP.Bio-1 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C WTTP.Bio-2 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C WTTP.Sus-1 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C WTTP.Sus-2 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C

Ini 0.003 0.005 0.000 166 0.281 0 0.0002 98.2% Ini 0.004 0.008 0.000 80 0.244 0 0 100% Ini 0.007 0.010 0.000 105 0.396 0.002 0.0004 99.4% Ini 0.011 0.021 0.000 9 0.548 0 0 100%

Best I 0.004 108.1% -0.210 -26.3% -2.6% -71.9% 98.2% Best I 0.007 48.5% -0.235 -52.2% -4.4% 100% Best I 0.006 -35.4% -0.777 -28.5% -5.0% -77.7% -44.9% 99.1% Best I 0.014 -30.8% -0.596 -77.78% -31.3% 90.6%

Median I-A 0.003 57.5% -0.110 -11.7% -0.6% -35.7% 98.2% Median I-A 0.006 28.3% -0.142 -39.1% -2.6% 100% Median I-A 0.005 -23.5% -0.543 -31.7% -5.0% -69.6% -25.4% 98.4% Median I-A 0.013 -33.4% -0.312 -58.52% -8.8% 97.1%

WTTP.Bio-3 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C WTTP.Bio-4 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C WTTP.Sus-3 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C WTTP.Sus-4 Acc M SEtst (∆) InterC RN (∆) S(∆) R(∆) I(∆) C

Best A 0.003 96.5% -0.032 -11.6% 0.2% -28.0% 97.7% Best A 0.006 26.1% -0.022 -24.2% 0.9% 97.6% Best A 0.005 -19.4% -0.274 -32.5% -4.5% -59.6% -2.1% 97.5% Best A 0.013 -34.8% -0.113 -33.33% -4.6% 100%

Ini 0.002 0.004 0.000 530 0.326 0.001 1E-4 99.4% Ini 0.003 0.039 0.000 108 0.227 0 0 92.5% Ini 0.004 0.004 0.000 530 0.555 0.042 0.001 100% Ini 0.007 0.013 0.000 34 0.607 0.093 0.011 97.7%

Best I 0.003 38.1% -1.732 -32.1% -2.8% -14.0% -86.8% 98.8% Best I 0.018 -45.8% -0.330 -61.6% -5.7% 91.5% Best I 0.005 60.5% -8.451 -27.6% -1.0% -4.4% -97.5% 97.7% Best I 0.010 -23.7% -2.626 -47.0% -13.1% -92.6% 140.4% 95.1%

Median I-A 0.003 62.5% -1.037 -32.2% -3.0% -11.9% -73.3% 98.9% Median I-A 0.013 -54.3% -0.207 -48.5% -4.2% 92.3% Median I-A 0.004 73.3% -5.373 -34.1% -1.6% -8.7% -91.8% 98.3% Median I-A 0.009 -14.3% -1.804 -51.9% -12.2% -88.8% 118.1% 91.2%

Best A 0.003 81.0% -0.330 -29.2% -2.4% -10.2% -38.0% 98.3% Best A 0.012 -58.6% -0.039 -23.1% -0.04% 90.1% Best A 0.004 79.0% -0.258 -33.0% -2.0% -12.4% -20.8% 98.3% Best A 0.008 -9.3% -0.746 -57.7% -7.5% -62.5% 16.0% 86.4%

Fig. 1. Real (solid line) and FasArt (dashed line) outputs of compact models for WWTP.Bio-1

Fig. 2. Real (solid line) and FasArt (dashed line) outputs of the complex models for WWTP.Sus-3

7. CONCLUSIONS

trial problem. The proposal is based on a post-processing rule selection using a two objective approach oriented to improving accuracy and interpretability simultaneously. The interpretability index, InterC proposed in GalendeHern´andez et al. (2011. Unpublished.), is now tested with some industrial problems.

This work, from the accuracy-interpretability point of view, takes a methodology to improve models obtained by (precise) fuzzy modeling algorithms that are the common in engineering fields and applies it to an indus-

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The results obtained show that, for all case studies, it has been possible to improve the accuracy-interpretability trade-off. Simultaneous improvement of accuracy and interpretability have been possible in some cases, so the improvement of interpretability need not imply accuracy loss. In other cases, the trade-off has been improved from the interpretability point of view with a very low accuracy loss, thus this improved model remains accurate enough. In fact, it is possible to improve the accuracy and the interpretability of the (precise) fuzzy model jointly or with a low or acceptable cost from the accuracy point of view. REFERENCES Alonso, J., Magdalena, L., and Gonz´ alez-Rodr´ıguez, G. (2009). Looking for a good fuzzy system interpretability index: An experimental approach. International Journal of Approximate Reasoning, 51(1), 115 – 134. Alonso, J., Magdalena, L., and Guillaume, S. (2008). HILK: A new methodology for designing highly interpretable linguistic knowledge bases using the fuzzy logic formalism. International Journal of Intelligent Systems, 23(7), 761 – 794. Bonissoene, P., Chen, Y.T., Goebel, K., and Khedkar, P. (1999). Hybrid soft computing systems: industrial and commercial applications. Proceedings of the IEEE, 87(9), 1641–1667. Cano Izquierdo, J., Dimitriadis, Y., G´ omez S´ anchez, E., and L´ opez Coronado, J. (2001). Learnning from noisy information in FasArt and Fasback neuro-fuzzy systems. Neural Networks, 14(4-5), 407–425. Casillas, J., Cord´ on, O., F., H., and Magdalena, L. (2003a). Interpretability Issues in Fuzzy Modelling, volume 128 of Studies in Fuzziness and SoftComputing, chapter Interpretability Improvements to Find the Balance Interpretability-Accuracy in Fuzzy Modeling: An Overview, 3–22. Springer-Verlag, Berlin Heildelberg. Casillas, J., Cord´ on, O., Herrera, F., and Magdalena, L. (2003b). Accuracy Improvements in Linguistic Fuzzy Modelling, volume 129 of Studies in Fuzziness and SoftComputing, chapter Accuracy Improvements to Find the Balance Interpretability-Accuracy in Fuzzy Modeling: An Overview, 3–24. Springer-Verlag, Berlin Heildelberg. Cord´ on, O., Herrera, F., Hoffmann, F., and Magdalena, L. (2001). Genetic Fuzzy Systems: Evolutionary Tuning and Learning of Fuzzy Knowledge Bases, volume 19 of Advances in Fuzzy Systems - Applications and Theory. World Scientific, Singapore. Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transacions on Evolutionary Computation, 6(2), 182–197. Destercke, S., Guillaume, S., and Charnomordic, B. (2007). Building an interpretable fuzzy rule base from data using orthogonal least squares-application to a depollution problem. Fuzzy Sets and Systems, 158(18), 2078 – 2094. Gacto, M., Alcal´ a, R., and Herrera, F. (2009). Adaptation and application of multi-objective evolutionary algorithms for rule reduction and parameter tuning of fuzzy rule-based systems. Soft Computing - A Fusion of Foundations, Methodologies and Applications, 13(5), 419 – 436.

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