Abstract
In the last years, the numerous successful applications of fuzzy rule-based systems (FRBSs) to several different domains have produced a considerable interest in methods to generate FRBSs from data. Most of the methods proposed in the literature, however, focus on performance maximization and omit to consider FRBS comprehensibility. Only recently, the problem of finding the right trade-off between performance and comprehensibility, in spite of the original nature of fuzzy logic, has arisen a growing interest in methods which take both the aspects into account. In this paper, we propose a Pareto-based multi-objective evolutionary approach to generate a set of Mamdani fuzzy systems from numerical data. We adopt a variant of the well-known (2+2) Pareto Archived Evolutionary Strategy ((2+2)PAES), which adopts the one-point crossover and two appropriately defined mutation operators. (2+2)PAES determines an approximation of the optimal Pareto front by concurrently minimizing the root mean squared error and the complexity. Complexity is measured as sum of the conditions which compose the antecedents of the rules included in the FRBS. Thus, low values of complexity correspond to Mamdani fuzzy systems characterized by a low number of rules and a low number of input variables really used in each rule. This ensures a high comprehensibility of the systems. We tested our version of (2+2)PAES on three well-known regression benchmarks, namely the Box and Jenkins Gas Furnace, the Mackey-Glass chaotic time series and Lorenz attractor time series datasets. To show the good characteristics of our approach, we compare the Pareto fronts produced by the (2+2)PAES with the ones obtained by applying a heuristic approach based on SVD-QR decomposition and four different multi-objective evolutionary algorithms.
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References
Box G, Jenkins G (1970) Time series analysis, forecasting and control. Holden Day San Francisco
Chen SH, Tsai FM (2004) A new approach to construct membership functions and generate fuzzy rules from training instances. In: Proceedings of 2004 IEEE international conference on fuzzy systems, vol. 2, pp 831–836
Chen Y, Yang B, Dong J, Abraham A (2005) Time-series forecasting using flexible neural tree model. Inf Sci 174:219–235
Cordón O, Herrera F, Del Jesus MJ, Villar P (2001) A multiobjective genetic algorithm for feature selection and granularity learning in fuzzy-rule based classification systems. In: Proceedings of Joint 9th IFSA world congress and 20th NAFIPS international conference, vol. 3, pp 1253–1258
De Oliveira V (1999) Semantic constraints for membership function optimisation. IEEE Trans Systems Man Cybern Part A: Systems Hum 29(1):128–138
Deb K, Pratab A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evolut Comput 6(2):182–197
Fonseca CM, Fleming PJ (1995) An overview of evolutionary algorithms in multiobjective optimisation. Evolut Comput 3(1): 1–16
Horn J, Nafpliotis N, Goldberg DE (1994) A niched Pareto genetic algorithm for multiobjective optimization. In: Proceedings of the first IEEE conference on evolutionary computation, vol. 1, pp 82–87
Ishibuchi H, Nakashima T (1999) Genetic-algorithm-based approach to linguistic approximation of nonlinear functions with many input variables. In: Proceedings of 8th IEEE international conference on fuzzy systems, pp 779–784
Ishibuchi H, Yamamoto T (2003) Interpretability issues in fuzzy genetics-based machine learning for linguistic modelling. Lecture Notes Comput Sci 2873:209–228
Ishibuchi H, Yamamoto T (2004) Fuzzy rule selection by multi-objective genetic local search algorithms and rule evaluation measures in data mining. Fuzzy Sets Systems 141(1):59–88
Ishibuchi H, Nojima Y (2005) Accuracy-complexity tradeoff analysis by multiobjective rule selection. In: Proceedings of ICDM 2005 workshop on computational intelligence in data mining, Houston, pp 39–48
Ishibuchi H, Murata T, Turksen IB (1997) Single-objective and two-objective genetic algorithms for selecting linguistic rules for pattern classification problems. Fuzzy Sets Systems 89(2):135–150
Ishibuchi H, Nakashima T, Murata T (2001) Three-objective genetics-based machine learning for linguistic rule extraction. Inf Sci 136:109–133
Ishibuchi H, Takeuchi D, Nakashima T (2001) GA-based approaches to linguistic modelling of nonlinear functions. In: Proceedings of 9th IFSA world congress and 20th NAFIPS international conference, vol 2, pp 1229–1234
Jimenez F, Gomez-Skarmeta AF, Roubos H, Bab̌ska R (2001) A multi-objective evolutionary algorithm for fuzzy modeling. In: Proceedings of 9th IFSA world congress and 20th NAFIPS international conference, vol. 2, pp 1222–1228
Jin Y (2000) Fuzzy modeling of high-dimensional systems: complexity reduction and interpretability improvement. IEEE Trans Fuzzy Systems 8(2):212–221
Johansen TA, Bab̌ska R (2003) Multiobjective identification of Takagi-Sugeno fuzzy models. IEEE Trans Fuzzy Systems 11(6):847–860
Knowles JD, Corne DW (2000) Approximating the non dominated front using the Pareto archived evolution strategy. Evolut Comput 8(2):149–172
Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141
Murata T, Ishibuchi H (1995) MOGA: multi-objective genetic algorithms. In: Proceedings of the second IEEE international conference on evolutionary computing, vol. 1, pp 289–294
Narukawa K, Nojima Y, Ishibuchi H (2005) Modification of multiobjective optimization algorithms for multiobjective design of fuzzy rule-based classification systems. In: Proceedings of 2005 IEEE international conference on fuzzy systems, pp 809–814
Palit AK, Popovic D (1999) Fuzzy logic based automatic rule generation and forecasting of time series. In: Proceedings of 1999 IEEE international conference on fuzzy sets, vol. 1, pp 360–365
Rojas L, Pomares H, Bernier JL et al. (2002) Time series analysis using normalized PG–RBF network with regression weights. Neurocomputing 42:267–285
Srinivas N, Deb K (1995) Multi-objective function optimization using non-dominated sorting genetic algorithms. Evolut Comput 2:221–248
Wang CH, Hong TP, Tseng SS (1996) Inductive learning from fuzzy examples. In: Proceedings of the Fifth IEEE international conference on fuzzy systems, vol. 1, pp 13–18
Wang LX, Mendel JM (1992) Generating fuzzy rules by learning from examples. IEEE Trans Systems Man Cybern 22(6):1414–1427
Watanabe S, Sakakibara K (2005) Multi-objective approaches in a single-objective optimization environment. In: Proceedings of 2005 IEEE congress on evolutionary computation (CEC’2005), vol. 2, pp 1714–1721
Yen J, Wang L (1999) Simplifying fuzzy rule-based models using orthogonal transformation methods. IEEE Trans Systems Man Cybern part B 29(1):13–24
Zhang BS, Edmunds JM (1991) On fuzzy logic controllers. In: Proceedings of the IEEE international conference on control, vol. 2, pp 961–965
Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evolut Comput 3(4):257–271
Zitzler E, Deb K, Thiele L (2000) Comparison of evolutionary algorithms: empirical results. Evolut Comput 8(2):173–195
Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Proceedings of EUROGEN2001 evolutionary methods for design, optimization and control with applications to industrial problems, pp 95–100
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Cococcioni, M., Ducange, P., Lazzerini, B. et al. A Pareto-based multi-objective evolutionary approach to the identification of Mamdani fuzzy systems. Soft Comput 11, 1013–1031 (2007). https://doi.org/10.1007/s00500-007-0150-6
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DOI: https://doi.org/10.1007/s00500-007-0150-6