Discussions on Interpretabilit of Fuzz Sstems using Simple Eamples Hisao Ishibuchi, Yusuke Nojima Department of Computer Science and Intelligent Sstems, Osaka Prefecture Universit - Gakuen-cho, Naka-ku, Sakai, Osaka, 599-853 Japan Email: hisaoi@cs.osakafu-u.ac.jp, nojima@cs.osakafu-u.ac.jp Abstract Two conflicting goals are often involved in the design of fuzz rule-based sstems: Accurac maimization and interpretabilit maimization. A number of approaches have been proposed for finding a fuzz rule-based sstem with a good accurac-interpretabilit tradeoff. Formulation of the accurac maimization is usuall straightforward in each application area of fuzz rule-based sstems such as classification, regression and forecasting. Formulation of the interpretabilit maimization, however, is not so eas. This is because various aspects of fuzz rule-based sstems are related to their interpretabilit. Moreover, user s preference should be taken into account when a single fuzz rule-based sstem is to be chosen from several alternatives with different accurac-interpretabilit tradeoffs. In this paper, we discuss the difficult in measuring the interpretabilit of fuzz rule-based sstems using ver simple eamples. We do not intend to propose an new interpretabilit measure. Our intention is to help to activate discussions on how to measure the interpretabilit of fuzz rule-based sstems. Kewords Fuzz sstems, fuzz rules, accurac-interpretabilit tradeoff, multiobjective design of fuzz sstems. Introduction Handling of the tradeoff between the accurac maimization and the interpretabilit maimization has been a hot issue in the design of fuzz rule-based sstems since the mid-990s []-[3]. A number of approaches have alread been proposed for improving the accurac of fuzz rule-based sstems while maintaining their interpretabilit [], [2], [4]-[2]. Genetic algorithms have been frequentl used in those approaches to search for an accurate and interpretable fuzz rule-based sstem. This is because genetic algorithms can perform not onl continuous optimization for parameter tuning but also discrete optimization for structure determination. Studies on fuzz genetics-based machine learning are called genetic fuzz sstems [22]-[24]. In some recent studies [3], [25]-[36], multi-objective genetic algorithms have been used to search for multiple Pareto-optimal fuzz rule-based sstems along the accurac-interpretabilit tradeoff surface. Those studies are often referred to as multi-objective genetic fuzz sstems [37]. Recentl multi-objective genetic algorithms have also been used for machine learning [38] and data mining [39]. Let us denote a fuzz rule-based sstem b S. We can also view S as a set of fuzz if-then rules. In each application area of fuzz rule-based sstems such as classification, regression and forecasting, the specification of the accurac of S for the given training data is not difficult (e.g., the number of correctl classified training patterns b S). Let us denote the accurac measure of S as Accurac(S). A design problem of fuzz rule-based sstems can be formulated as follows: Maimize Accurac(S). () Due to the accurac-interpretabilit tradeoff relation, the accurac maimization in () often leads to the deterioration in the interpretabilit of fuzz rule-based sstems. This means that we often obtain from () an accurate and complicated fuzz rule-based sstem with poor interpretabilit. In some application areas, not onl the accurac but also the interpretabilit is ver important. Thus we often want to maimize the accurac of fuzz rule-based sstems without degrading their interpretabilit. This maimization problem can be formulated as follows: Maimize Accurac(S) subject to Interpretabilit(S), where Interpretabilit(S) is the interpretabilit measure of the fuzz rule-based sstem S and is the required minimum level of the interpretabilit. Of course, we can formulate the maimization problem of the interpretabilit under the given minimum accurac level as follows: Maimize Interpretabilit(S) subject to Accurac(S). (3) One ma want to maimize both the accurac and the interpretabilit. In this case, a simple approach is to use a scalarizing function f (.) which combines the accurac and interpretabilit measures into a single objective function: Maimize f (Accurac(S), Interpretabilit(S)). (4) A well-known scalarizing function is the weighted sum: Maimize w Accurac(S) + w 2 Interpretabilit(S), (5) where w and w 2 are non-negative weight values. In addition to the weighted sum in (5), we can use various scalarizing functions developed in the field of multiple criteria decision making (MCDM [40]-[42]). In general, it is not eas for human users to specif an appropriate scalarizing function for multi-objective problems. Users ma want to eamine some fuzz rule-based sstems with different accurac-interpretabilit tradeoffs (instead of a single best solution with respect to a specific scalarizing function). In this case, the design of fuzz rule-based sstems can be formulated as the following multi-objective problem: Maimize { Accurac(S), Interpretabilit(S)}. (6) A large number of Pareto-optimal fuzz rule-based sstems can be obtained b multi-objective genetic algorithms such as NSGA-II [43], SPEA [44] and SPEA2 [45]. 649
In man cases, the interpretabilit maimization is handled as the compleit minimization. Thus the above-mentioned formulations in -(6) can be reformulated accordingl. For eample, the multi-objective formulation in (6) is rewritten as Maimize Accurac(S) and minimize Compleit(S), (7) where Compleit(S) is a compleit measure. The main difficult in the above-mentioned formulations in -(7) for the design of accurate and interpretable fuzz rule-based sstems is the formulation of their interpretabilit. Whereas the formulation of the accurac of fuzz rule-based sstems is usuall straightforward from their application task such as classification and regression, it is not eas for human users to appropriatel formulate the interpretabilit. This is because various aspects of fuzz rule-based sstems are related to their interpretabilit [46]-[52]. Moreover, it is not eas for human users to mathematicall formulate each of those aspects even when the close relation of each aspect to the interpretabilit of fuzz rule-based sstems is clear. In this paper, we eplain the difficult in formulating the interpretabilit of fuzz rule-based sstems using simple numerical eamples. More specificall, we demonstrate the difficult in comparing different fuzz rule-based sstems with respect to their interpretabilit even in ver simple situations. 2 Interpretabilit of Fuzz Partitions When we use the same tpe of fuzz partitions with different granularities (e.g., uniform fuzz partitions with smmetric triangular membership functions (MFs) in Fig. ), we can sa that the increase in the number of membership functions degrades the interpretabilit of fuzz partitions. For eample, the fuzz partition with two membership functions in Fig. (a) is the most interpretable among the four alternatives in Fig.. In this case, we can formulate the interpretabilit of fuzz partitions b the number of membership functions. That is, the interpretabilit maimization is realized b minimizing the number of membership functions in fuzz partitions. minimized) when the same fuzz partition is used for all input variables. For eample, we can sa that the 33 fuzz grid in Fig. 2 (a) is more interpretable than the 44 fuzz grid in Fig. 2 (b). The comparison, however, becomes difficult when we use different fuzz partitions for each input variable as shown in the following two eamples. 2 2 (a) 33 fuzz grid. (b) 44 fuzz grid. Figure 2: Comparison between the 33 and 44 fuzz grids. Eample : Let us consider the 44 and 35 fuzz grids in Fig. 3. Both fuzz grids have eight membership functions in total (i.e., 4+4 = 3+5). Thus the are evaluated as having the same interpretabilit if the are compared using the number of membership functions. The 35 fuzz grid is, however, viewed as being more interpretable than the 44 fuzz grid if we use the number of fuzz subspaces as an interpretabilit measure (i.e., 5 < 6). Since a single fuzz rule is usuall generated for each fuzz subspace, the 35 fuzz grid in Fig. 3 (b) can be viewed as being more interpretable than the 44 fuzz grid in Fig. 3 (a) if we evaluate the interpretabilit using the number of fuzz rules. Some human users, however, ma intuitivel feel that the 44 fuzz grid with the same fuzz partition for the two input variables is more interpretable than the 35 fuzz grid with the different fuzz partitions. 2 2 0 (a) Two MFs 0 (b) Three MFs (a) 44 fuzz grid. (b) 35 fuzz grid. Figure 3: Eample with the 44 and 35 fuzz grids. 0 0 (c) Four MFs (d) Five MFs Figure : Fuzz partitions with different granularities. We can also use the number of membership functions as an interpretabilit measure (i.e., as a compleit measure to be Eample 2: Let us consider the 55 and 38 fuzz grids in Fig. 4. The 55 fuzz grid in Fig. 4 (a) has less membership functions than the 38 fuzz grid in Fig. 4 (b): 5+5 < 3+8. Thus the 55 fuzz grid is evaluated as more interpretable than the 38 fuzz grid in Fig. 4 (b) if we use the number of membership functions as an interpretabilit measure. The 55 fuzz grid, however, has more fuzz subspaces than the 38 fuzz grid: 25 > 24. Thus the 38 fuzz grid is viewed as being more interpretable than the 55 fuzz grid if we use the number of fuzz subspaces (i.e., the number of fuzz rules) as an interpretabilit measure. 650
2 2 (a) 55 fuzz grid. (b) 38 fuzz grid. Figure 4: Eample 2 with the 55 and 38 fuzz grids. As we have alread eplained in this section using the two eamples, the comparison of different fuzz grids with respect to their interpretabilit is not so eas. This means that the choice of an interpretabilit measure is difficult. If we use the number of fuzz subspaces in Fig. 4, the 38 fuzz grid is viewed as being more interpretable than the 55 fuzz grid. When we use not onl the number of fuzz subspaces but also the number of membership functions, these two fuzz grids are viewed as being non-dominated with each other with respect to the interpretabilit. In this case, the multi-objective formulation in (7) is handled as the three-objective problem: Maimize Accurac(S), and minimize {Compleit (S), Compleit 2 (S)}, (8) where Compleit (S) and Compleit 2 (S) are different compleit measures to be minimized (e.g., the number of fuzz subspaces and the number of membership functions). Even if we use these two measures, the 35 fuzz grid is viewed as more interpretable than the 44 fuzz grid. Thus we need another measure if we want to include some bias toward fuzz grids with the same fuzz partition for all input variables. For eample, the 44 fuzz grid is evaluated as more interpretable than the 35 fuzz grid if the maimum number of membership functions for each input variable is used as an interpretabilit measure (ma{4, 4} < ma{3, 5}). 3 Interpretabilit of Fuzz Rule-Based Sstems In genetic fuzz sstems [22]-[24], almost all aspects of fuzz rule-based sstems can be optimized since genetic algorithms perform continuous, discrete and combinatorial optimization. For eample, genetic fuzz sstems can be used for choosing an appropriate tpe of fuzz rules (e.g., Takagi-Sugeno, simplified Takagi-Sugeno and Mamdani). In this section, we discuss the interpretabilit of fuzz rule-based sstems with different fuzz partitions and different tpes of fuzz rules. Let us consider a simple function approimation problem of a single-input and single-output sstem = f () in Fig. 5. Our task is to design an accurate and interpretable fuzz rule-based sstem from the given input-output data in Fig. 5. For this task, Takagi-Sugeno fuzz rules are written as Rule R : If is A then i i a b, i,2,..., N, (9) i i Figure 5: Input-output data in Eamples 3 and 4. When an input value is presented to the fuzz rule-based sstem with the N fuzz rules in (9), the output value is estimated as follows: N ( ai bi ) i ( ) N ( ) i Ai Ai ( ), (0) where () is the estimated output value for the input value, and A i () is the membership value of the antecedent fuzz set Ai for the input value. We can use the same fuzz reasoning mechanism for the simplified version of Takagi-Sugeno fuzz rules: Rule R i : If is A i then is h i, i,2,..., N, () where h i is a consequent real number. Eample 3: From the input-output data in Fig. 5, one ma think that the can be approimated b a fuzz rule-based sstem with three Takagi-Sugeno fuzz rules. An eample of such a fuzz rule-based sstem is shown in Fig. 6 where each of the three lines (), and (3) is the consequent linear function of each of the three fuzz rules with the trapezoidal antecedent fuzz sets A, A 2 and A 3. The same input-output data can be also approimated b a fuzz rule-based sstem with four simplified Takagi-Sugeno fuzz rules as shown in Fig. 7. Each fuzz rule in Fig. 7 has a triangular membership function A i and a consequent real number h i. The question is which is more interpretable between the two fuzz rule-based sstems in Fig. 6 and Fig. 7. (3) () where i is a rule inde, A i is an antecedent fuzz set, a i and b i are real number coefficients of a consequent linear function of each fuzz rule, and N is the total number of fuzz rules. A A 2 A 3 Figure 6: Three Takagi-Sugeno fuzz rules. 65
h h 2 h 3 h 4 A A 2 A 3 A 4 Figure 7: Four simplified Takagi-Sugeno fuzz rules. In Fig. 8 and Fig. 9, we show the fuzz reasoning results b these two fuzz rule-based sstems in Fig. 6 and Fig. 7, respectivel. We can see that similar results were obtained from the two fuzz rule-based sstems. Since there is no large difference in the approimation accurac between Fig. 8 and Fig. 9, the interpretabilit will pla an important role in the selection between the two fuzz models in Fig. 6 and Fig. 7. If we use the number of fuzz rules as an interpretabilit measure, the Takagi-Sugeno model in Fig. 6 is viewed as being more interpretable than the simplified Takagi-Sugeno model in Fig. 7. On the other hand, if we use the total number of parameters (i.e., a i, b i and h i ) in the consequent part of the fuzz rules as an interpretabilit measure, the simplified Takagi-Sugeno model is evaluated as more interpretable than the Takagi-Sugeno model (i.e., 4 < 6). Figure 8: Results b the three Takagi-Sugeno fuzz rules. Figure 9: Results b the four simplified fuzz rules. Eample 4: The given input-output data in Fig. 5 can be also approimated b the two Takagi-Sugeno fuzz rules in Fig. 0. Whereas data points around = 0.5 are far from the two consequent linear functions in Fig. 0, the can be approimated through the interpolation mechanism of the fuzz reasoning in Eq. (0). Fig. is the fuzz reasoning result b the two fuzz rules in Fig. 0. We can see from Fig. that good approimation was realized b the two fuzz rules in Fig. 0. Actuall, the fuzz reasoning result in Fig. b the two Takagi-Sugeno fuzz rules is similar to Fig. 8 and Fig. 9. The question is which is more interpretable between Fig. 6 with the three rules and Fig. 0 with the two rules. It is clear that Fig. 0 is simpler than Fig. 6 with respect to various aspects of fuzz rule-based sstems (e.g., the number of fuzz rules, the number of membership functions, and the number of parameters). However, one ma think that Fig. 6 is more intuitive than Fig. 0. If we use the local accurac of each liner function [5] as an interpretabilit measure, the three Takagi-Sugeno fuzz rules in Fig. 6 are evaluated as being more interpretable than the two rules in Fig. 0. () A A 2 Figure 0: Two Takagi-Sugeno fuzz rules. Figure : Results b the two Takagi-Sugeno fuzz rules. Eample 5: We can use different tpes of fuzz rules in a single fuzz rule-based sstem. We show an eample of such a fuzz rule-based sstem in Fig. 2 where the second fuzz rule with the antecedent fuzz set A 2 has a consequent linear function (line ). Each of the other two fuzz rules with the antecedent fuzz sets A and A 3 has a consequent real number. As we can epect, good approimation was realized b these three fuzz rules (due to the page limitation, we can not show 652
the fuzz reasoning result). The given input-output data in Fig. 2 can be also approimated with a similar accurac b the four simplified Takagi-Sugeno fuzz rules in Fig. 3. Since the first two fuzz rules with the antecedent fuzz sets A and A 2 in Fig. 3 have the same consequent real number, the can be merged into a single rule. The last two fuzz rules with A 3 and A 4 in Fig. 3 can be also merged into a single rule. As a result, we have a fuzz rule-based sstem with the two simplified Takagi-Sugeno fuzz rules in Fig. 4. (3) () A A 2 A 3 Figure 2: A Takagi-Sugeno rule and two simplified rules. h h 2 h 3 h 4 A A 2 A 3 A 4 Figure 3: Four simplified Takagi-Sugeno rules. () A A 2 Figure 4: Two simplified Takagi-Sugeno rules. It is clear that the fuzz rule-based sstem with onl the two fuzz rules in Fig. 4 is the simplest one in Figs. 2-4. However, we have no definite answer to the question: Which is the most interpretable in the three models in Figs. 2-4? 4 Conclusions In this paper, we demonstrated the difficult in evaluating the interpretabilit of fuzz rule-based sstems. As shown in this paper, the evaluation of the interpretabilit is difficult even in ver simple situations. Different fuzz rule-based sstems are viewed as being more interpretable according to different interpretabilit measures. This means that the choice of an appropriate interpretabilit measure is important in the design of fuzz rule-based sstems. At the same time, such a choice is difficult as shown in this paper. We hope that this paper will activate discussions on the interpretabilit and help to develop new approaches to fuzz modelling based on the accurac-interpretabilit tradeoff analsis. References [] H. Ishibuchi, K. Nozaki, N. Yamamoto, and H. Tanaka, Construction of fuzz classification sstems with rectangular fuzz rules using genetic algorithms, Fuzz Sets and Sstems 65 (2/3) 237-253, 994. [2] H. Ishibuchi, K. Nozaki, N. Yamamoto, and H. Tanaka, Selecting fuzz if-then rules for classification problems using genetic algorithms, IEEE Trans. on Fuzz Sstems 3 (3) 260-270, 995. [3] H. Ishibuchi, T. Murata, and I. B. Turksen, Selecting linguistic classification rules b two-objective genetic algorithms, Proc. of 995 IEEE International Conference on Sstems, Man and Cbernetics, 40-45, 995. [4] M. Setnes, R. Babuska, and B. Verbruggen, Rule-based modeling: Precision and transparenc, IEEE Trans. on Sstems, Man, and Cbernetics - Part C 28 () 65-69, 998. [5] J. Yen, L. Wang, and G. W. Gillespie, Improving the interpretabilit of TSK fuzz models b combining global learning and local learning, IEEE Trans. on Fuzz Sstems 6 (4) 530-537, 998. [6] Y. Jin, W. von Seelen, and B. Sendhoff, On generating FC 3 fuzz rule sstems from data using evolution strategies, IEEE Trans. on Sstems, Man, and Cbernetics - Part B 29 (6) 829-845, 999. [7] D. Nauck and R. Kruse, Obtaining interpretable fuzz classification rules from medical data, Artificial Intelligence in Medicine 6 49-69, 999. [8] Y. Jin, Fuzz modeling of high-dimensional sstems: Compleit reduction and interpretabilit improvement, IEEE Trans. on Fuzz Sstems 8 22-22, 2000. [9] M. Setnes and H. Roubos, GA-based modeling and classification: Compleit and performance, IEEE Trans. on Fuzz Sstems 8 (5) 509-522, 2000. [0] L. Castillo, A. Gonzalez, and R. Perez, Including a simplicit criterion in the selection of the best rule in a genetic fuzz learning algorithm, Fuzz Sets and Sstems 20 309-32, 200. [] H. Roubos and M. Setnes, Compact and transparent fuzz models and classifiers through iterative compleit reduction, IEEE Trans. on Fuzz Sstems 9 (4) 56-524, 200. [2] J. Aboni, J. A. Roubos, and F. Szeifert, Data-driven generation of compact, accurate, and linguisticall sound fuzz classifiers based on a decision-tree initialization, International Journal of Approimate Reasoning 32 () -2, 2003. [3] R. Alcala, J. R. Cano, O. Cordon, F. Herrera, P. Villar, and I. Zwir, Linguistic modeling with hierarchical sstems of weighted linguistic rules, International Journal of Approimate Reasoning 32 (2-3) 87-25, 2003. 653
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