Analysis and guidelines to obtain a good uniform fuzzy partition granularity for fuzzy rule-based systems using simulated annealing

In this contribution, we will analyse the importance of the fuzzy partition granularity for the linguistic variables in the design of fuzzy rule-based systems (FRBSs). In order to put this into e€ect, we will study the FRBS behaviour considering uniform fuzzy partitions with the same number of labels for all the linguistic variables, and considering uniform fuzzy partitions with any number of labels for each linguistic variable. We will present a method based on Simulated Annealing (SA) in order to obtain a good uniform fuzzy partition granularity that improves the FRBS behaviour. It is an ecient granularity search method for ®nding a good number of labels per variable. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Fuzzy rule-based systems; Data base; Fuzzy partition; Granularity; Simulated annealing International Journal of Approximate Reasoning 25 (2000) 187±215 www.elsevier.com/locate/ijar q This research has been supported by CICYT projects TIC96-0778 and PB98-1319. * Corresponding author. Tel.: +34-58-24-40-19; fax: +34-58-24-33-17. E-mail addresses: [email protected] (O. CordoÂn), [email protected] (F. Herrera), [email protected] (P. Villar). 0888-613X/00/$ see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 8 8 8 6 1 3 X ( 0 0 ) 0 0 0 5 2 9 1. Introduction Fuzzy rule-based systems (FRBSs) represent one of the most important areas for the application of fuzzy set theory. These systems constitute an extension of classical rule-based systems, because they deal with fuzzy rules instead of classical logic rules. They have been successfully applied to a wide range of problems presenting uncertainty and vagueness in di€erent ways [2,15,24]. An FRBS presents two main components: (1) the inference system, which implements the fuzzy inference process needed to obtain an output from the FRBS when an input is speci®ed, and (2) the knowledge base (KB), which represents the knowledge about the problem being solved. The KB is composed by the rule base (RB) containing the collection of fuzzy rules, and by the data base (DB) containing the membership functions of the fuzzy partitions associated to the linguistic variables. Two main tasks need to be performed to design an FRBS for a speci®c problem: to select the fuzzy operators involved in the inference system, i.e., to de®ne the way in which the fuzzy inference process will be performed, and to derive an appropriate KB about the problem under solving. The accuracy of the FRBS in the solving of this problem will depend directly on both components. Focusing on the second design task, many approaches have been presented to automatically learn the RB from numerical information (input±output data pairs representing the system behaviour) when there is no knowledge provided by an human expert. However, there is not a similar e€ort for deriving the DB, although its design is a critical task since most of the RB learning methods assume the existence of a previously de®ned DB, and thus it will signi®cantly condition the behaviour of the ®nal FRBS. A very common way to proceed involves considering uniform fuzzy partitions with the same number of terms for all the linguistic variables of the problem, that is, the same granularity. The aim of this article is to analyse the in ̄uence of the granularity of the fuzzy partitions in the FRBS performance. To be precise, we will deal with this problem from a double perspective: · We will try to give an answer to the question: is it a good operation mode to consider uniform fuzzy partitions with the same number of labels for all the linguistic variables? · We will also develop an ecient method for obtaining good uniform fuzzy partitions ®nding a good granularity per linguistic variable. To do so, we will work with di€erent RB automatic learning methods and we will compare their behaviour when considering DBs with a di€erent number of linguistic terms for each linguistic variable. The membership functions considered will always be triangular-shaped, symmetrical and uniformly distributed, thus making the granularity of the fuzzy partitions the unique 188 O. Cord on et al. / Internat. J. Approx. Reason. 25 (2000) 187±215 parameter of the DB having in ̄uence on the learning method and, consequently, on the ®nal FRBS behaviour. Moreover, we propose simulated annealing (SA) as the method to search for a good uniform fuzzy partition granularity, i.e., a granularity that produces an FRBS with good accuracy, and in some cases, the one with the best behaviour. This paper is organized as follows. In Section 2, we present the role of the DB in the FRBS design process. In Section 3, we analyse the in ̄uence of the uniform fuzzy partition granularity on the FRBS behaviour taking three realworld applications as a base. First, we study the FRBS behaviour considering the same number of labels in each linguistic variable (Section 3.1), and later, considering any number of labels in each linguistic variable (Section 3.2). Finally, the conclusions of the study are presented. In Section 4, we present a SA method for obtaining a uniform fuzzy partition granularity with good behaviour and validate it on the said problems. In Section 5, some concluding remarks are provided. A short description of the RB learning methods used in the paper is given in Appendix A, while the characteristics of the problems considered as benchmarks can be found in Appendix B. Finally, the SA procedure is brie ̄y described in Appendix C. 2. The role of the data base in the design of FRBSs The composition of the KB of an FRBS directly depends on the problem being solved. The best situation is when there is a human expert able to express his/her knowledge in the form of fuzzy rules, thus providing the de®nitions for the DB (the relevant input and output linguistic variables for the system, the term sets for all of them and the membership functions of the fuzzy sets de®ning their meaning) and for the RB (the fuzzy rules themselves). Unfortunately, this situation is not very common: usually the expert is not able to provide all this information or there is no expert information about the problem under solving. In the last few years, many approaches have been proposed to solve this problem. These approaches try to automatically learn the RB from numerical information (input±output data pairs representing the system behaviour), using di€erent techniques such as ad hoc data-driven algorithms [2,4,18,31], least square methods [2], gradient descent algorithms [22], hybrid methods between the latter two ones [20], clustering algorithms [32], neural networks [29] and genetic algorithms (GAs) [7]. As we have mentioned, there is not a similar e€ort for deriving the DB. However, the DB has a signi®cant in ̄uence on the FRBS performance. In fact, studies such as the ones developed in [3,33] show, for the case of Fuzzy PI controllers, that the system performance is much more sensitive to the choice of the semantics in the DB than to the composition of the RB. Considering a O. Cord on et al. / Internat. J. Approx. Reason. 25 (2000) 187±215 189 previously de®ned RB, the performance of the Fuzzy controller is sensitive to four aspects: scaling factors, peak values, width values and rules. For this reason, some approaches try to improve the preliminary DB de®nition considered once the RB has been derived. To do so, a tuning process considering the whole KB obtained (the preliminary DB and the derived RB) is used a posteriori to adjust the membership function parameters to improve the FRBS behaviour (for some examples of these kinds of methods, based on neural networks and GAs, refer to [3,6,16,20]). Nevertheless, the tuning process only adjusts the shapes of the membership functions and not the number of linguistic terms in each fuzzy partition, which remains ®xed from the beginning of the design process. Other more sophisticated approaches to learn the di€erent DB components can be found in [11,12,14,19,26,30]. Usually, the most very common way to proceed for learning the RB considers, as starting point, a DB composed of uniform fuzzy partitions with the same number of terms (usually an odd number between three and seven) for all the linguistic variables existing in the problem. Triangular or trapezoidalshaped membership functions are usually considered due to their simplicity. At ®rst sight, the selection of the granularity level in the input and output variable fuzzy partitions does not seem to be a DB design task as important as the choice of the membership function shapes for the linguistic terms. However, the granularity selection plays an important role in many characteristics of the FRBS, such as the accuracy in fuzzy modeling or the smoothness in fuzzy control. Moreover, the granularity of the input variables speci®es the maximum number of fuzzy rules that may compose the RB, thus having a strong in ̄uence on aspects such as the complexity of the rule learning, the interpretability of the FRBS obtained or its accuracy. 3. Study of the in ̄uence of the uniform fuzzy partition granularity on the FRBS behaviour Typically, the DB is de®ned by choosing an equal number of linguistic terms for all the variables and by considering uniform fuzzy partitions in the variable universe of discourse for these labels. This choice is not guided by any speci®c characteristic of the problem, nor by any general rule. In this section, we analyse the use of three learning methods to explore the problem of granularity selection. 1 First, we constrain all the variables to have the same number of labels. Later, each variable is allowed to have any number 1 See Appendix A for a description of them: Wang and Mendel [31], Cord on and Herrera [10] and Descriptive±Mogul [6] learning methods. 190 O. Cord on et al. / Internat. J. Approx. Reason. 25 (2000) 187±215 of labels. In both cases, we used the interval {3±9} as possible values for the number of linguistic terms. To compare the behaviour of the di€erent FRBSs obtained, we consider three real-world applications: Low voltage line length problem, Optimal electrical network problem and Rice taste evaluation problem. The description of the benchmark problems can be found in Appendix B. The set of data pairs of every benchmark considered has been divided into two subsets, denoted training set and test set. The former is used by the learning methods to derive the RB composition, while the latter is used to evaluate the prediction ability of the generated fuzzy models. The mean square error of the FRBS over the training and test sets (MSEtra and MSEtst) is used as a comparison measure for the di€erent FRBSs obtained 1 2jEj X el2E …eyl ÿ S…exl†† with E being the example set (training or test), S…exl† being the output value obtained from the FRBS when the input variable values are exl ˆ …ex1; . . . ; exn†, and eyl being the known desired value. 3.1. FRBSs with the same number of labels for each variable In this part of the study, the three learning methods were run with the same number of labels for all the variables. Each method was run seven times for each benchmark. The results, the MSEtra, the MSEtst and the number of rules (#R), are shown in Tables 1±3 (where the best MSEtst value found in each method appears in bold type). The analysis of these results leads us to the following conclusions: · Di€erent learning methods generate the best FRBS design using a di€erent value for the fuzzy partition granularity. · The di€erence in the FRBS accuracy is signi®cant enough to validate the importance of granularity selection as an important task that must be adequately analysed during the RB learning process. The MSEtst obtained in Tables 1±3 are also showed in Fig. 1. In this graphic, it can be seen that the general behaviour of the three learning methods is similar for each problem, but the best results are obtained using a di€erent number of linguistic terms. On the other hand, it is interesting to observe that an excessively high number of labels can cause an over-®tting problem. Particularly, considering the WM and D-Mogul methods in the low voltage line length problem (Table 1), the FRBSs with best MSEtra use nine labels, while the value of the MSEtst in both cases is signi®cantly worse than the one obtained by the FRBSs with six labels (best MSEtst). The over-®tting problem is apparent when we increase the number of labels per variable in the rice taste evaluation problem O. Cord on et al. / Internat. J. Approx. Reason. 25 (2000) 187±215 191