Abstract
In this paper, we propose a multi-objective evolutionary algorithm (MOEA) to generate Mamdani fuzzy rule-based systems with different trade-offs between accuracy and complexity by learning concurrently granularities of the input and output partitions, membership function (MF) parameters and rules. To this aim, we introduce the concept of virtual and concrete partitions: the former is defined by uniformly partitioning each linguistic variable with a fixed maximum number of fuzzy sets; the latter takes into account, for each variable, the number of fuzzy sets determined by the evolutionary process. Rule bases and MF parameters are defined on the virtual partitions and, whenever a fitness evaluation is required, mapped to the concrete partitions by employing appropriate mapping strategies. The implementation of the MOEA relies on a chromosome composed of three parts, which codify the partition granularities, the virtual rule base and the membership function parameters, respectively, and on purposely-defined genetic operators. The MOEA has been tested on three real-world regression problems achieving very promising results. In particular, we highlight how starting from randomly generated solutions, the MOEA is able to determine different granularities for different variables achieving good trade-offs between complexity and accuracy.
Similar content being viewed by others
References
Casillas J, Cordon O, Herrera F, Magdalena L (eds) (2003) Interpretability issues in fuzzy modeling. Springer, Berlin
Casillas J, Herrera F, Pérez R, Del Jesus MJ, Villar P (2007) Special issue on genetic fuzzy systems and the interpretability-accuracy trade-off. Int J Approx Reason 44(1):1–3
Herrera F (2008) Genetic fuzzy systems: taxonomy, current research trends and prospects. Evol Intell 1:27–46
Ishibuchi H (2007) Multiobjective genetic fuzzy systems: review and future research directions. In: Proceedings of the 2007 international conference on fuzzy systems, London, 23–26 July, pp 1–6
Cordón O, Herrera F, Villar P (2000) Analysis and guidelines to obtain a good uniform fuzzy partition granularity for fuzzy rule-based systems using simulated annealing. Int J Approx Reason 25(3):187–215
Botta A, Lazzerini B, Marcelloni F, Stefanescu D (2009) Context adaptation of fuzzy systems through a multi-objective evolutionary approach based on a novel interpretability index. Soft Comput 13(5):437–449
Alcalá R, Gacto MJ, Herrera F, Alcalá-Fdez J (2007) A multi-objective genetic algorithm for tuning and rule selection to obtain accurate and compact linguistic fuzzy rule-based systems. Int J Uncertain Fuzziness Knowl Based Syst 15(5):521–537
Gacto MJ, Alcalá R, Herrera F (2009) Adaptation and application of multi-objective evolutionary algorithms for rule reduction and parameter tuning of fuzzy rule-based systems. Soft Comput 13(5):419–436
Alcalá R, Ducange P, Herrera F, Lazzerini B, Marcelloni F (2009) A multi-objective evolutionary approach to concurrently learn rule and data bases of linguistic fuzzy rule-based systems. IEEE Trans Fuzzy Syst 17(5):1106–1122
Ishibuchi H, Nojima Y (2007) Analysis of interpretability-accuracy tradeoff of fuzzy systems by multiobjective fuzzy genetics-based machine learning. Int J Approx Reason 44(1):4–31
Antonelli M, Ducange P, Lazzerini B, Marcelloni F (2009) Learning concurrently partition granularities and rule bases of Mamdani fuzzy systems in a multi-objective evolutionary framework. Int J Approx Reason 50(7):1066–1080
Mamdani EH, Assilian S (1975) An experiment in linguistic synthesis with a fuzzy logic controller. Int J Man Mach Stud 7(1):1–13
Klawonn F (2006) Reducing the number of parameters of a fuzzy system using scaling functions. Soft Comput 10(9):749–756
Pedrycz W, Gomide F (2007) Fuzzy systems engineering: toward human-centric computing. Wiley-IEEE Press, New Jersey
Cococcioni M, Ducange P, Lazzerini B, Marcelloni F (2007) A Pareto-based multi-objective evolutionary approach to the identification of Mamdani fuzzy systems. Soft Comput 11(11):1013–1031
Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, Chichester
Coello Coello CA, Lamont GB (2004) Applications of multi-objective evolutionary algorithms. World Scientific, Singapore
Ishibuchi H, Murata T, Turksen IB (1997) Single-objective and two-objective genetic algorithms for selecting linguistic rules for pattern classification problems. Fuzzy Sets Syst 89(2):135–150
Ishibuchi H, Yamamoto T (2004) Fuzzy rule selection by multi-objective genetic local search algorithms and rule evaluation measures in data mining. Fuzzy Sets Syst 141(1):59–88
Pulkkinen P, Koivisto H (2008) Fuzzy classifier identification using decision tree and multiobjective evolutionary algorithms. Int J Approx Reason 48:526–543
Pulkkinen P, Hytönen J, Koivisto H (2008) Developing a bioaerosol detector using hybrid genetic fuzzy systems. Eng Appl Artif Intell 21(8):1330–1346
Ducange P, Lazzerini B, Marcelloni F (2009) Multi-objective genetic fuzzy classifiers for imbalanced and cost-sensitive datasets. Soft Comput. doi: 10.1007/s00500-009-0460-y
Knowles JD, Corne DW (2002) Approximating the non dominated front using the Pareto archived evolution strategy. Evol Comput 8(2):149–172
Alcalá R, Alcalá-Fdez J, Herrera F (2007) A proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection. IEEE Trans Fuzzy Syst 15(4):616–635
Herrera F, Martinez L (2000) A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752
Deb K, Pratab A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197
Alcalá R, Alcalá-Fdez J, Herrera F, Otero J (2007) Genetic learning of accurate and compact fuzzy rule based systems based on the 2-tuples linguistic representation. Intern J Approx Reason 44:45–64
Ishibuchi H et al (1995) Selecting fuzzy if–then rules for classification problems using genetic algorithms. IEEE Trans Fuzzy Syst 3(3):260–270
Miller GA (1956) The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychol Rev 63(2):81–97
Cordón O, Herrera F, Sanchez L (1999) Solving electrical distribution problems using hybrid evolutionary data analysis techniques. Appl Intell 10:5–24
Cordón O, Herrera F, Villar P (2001) Generating the knowledge base of a fuzzy rule-based system by genetic learning of the data base. IEEE Trans Fuzzy Syst 9(4):667–674
Cordón O, Herrera F, Magdalena L, Villar P (2001) A genetic learning process for the scaling factors, granularity and contexts of the fuzzy rule-based system data base. Inf Sci 136:85–107
Altay Guvenir H, Uysal I (2000) Bilkent University function approximation repository. http://funapp.cs.bilkent.edu.tr
Acknowledgments
We would like to thank Dr. Rafael Alcalà and Dr. Jesus Alcalà-Fdez who very kindly have performed the experiments with the algorithms shown in Table 3, thus allowing us to reliably compare our approach with these algorithms.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Antonelli, M., Ducange, P., Lazzerini, B. et al. Multi-objective evolutionary learning of granularity, membership function parameters and rules of Mamdani fuzzy systems. Evol. Intel. 2, 21 (2009). https://doi.org/10.1007/s12065-009-0022-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12065-009-0022-3