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Soft Computing

, Volume 15, Issue 10, pp 1981–1998 | Cite as

Learning concurrently data and rule bases of Mamdani fuzzy rule-based systems by exploiting a novel interpretability index

  • Michela Antonelli
  • Pietro Ducange
  • Beatrice Lazzerini
  • Francesco Marcelloni
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Abstract

Interpretability of Mamdani fuzzy rule-based systems (MFRBSs) has been widely discussed in the last years, especially in the framework of multi-objective evolutionary fuzzy systems (MOEFSs). Here, multi-objective evolutionary algorithms (MOEAs) are applied to generate a set of MFRBSs with different trade-offs between interpretability and accuracy. In MOEFSs interpretability has often been measured in terms of complexity of the rule base and only recently partition integrity has also been considered. In this paper, we introduce a novel index for evaluating the interpretability of MFRBSs, which takes both the rule base complexity and the data base integrity into account. We discuss the use of this index in MOEFSs, which generate MFRBSs by concurrently learning the rule base, the linguistic partition granularities and the membership function parameters during the evolutionary process. The proposed approach has been experimented on six real world regression problems and the results have been compared with those obtained by applying the same MOEA, with only accuracy and complexity of the rule base as objectives. We show that our approach achieves the best trade-offs between interpretability and accuracy.

Keywords

Accuracy-interpretability trade-off Granularity learning Interpretability index Multi-objective evolutionary fuzzy systems Piecewise linear transformation 

References

  1. Alcalá R, Alcalá-Fdez J, Herrera F, Otero J (2007a) Genetic learning of accurate and compact fuzzy rule based systems based on the 2-Tuples linguistic representation. Int J Approx Reason 44:45–64zbMATHCrossRefGoogle Scholar
  2. Alcalá R, Gacto MJ, Herrera F, Alcalá-Fdez J (2007b) A multi-objective genetic algorithm for tuning and rule selection to obtain accurate and compact linguistic fuzzy rule-based systems. Int J Uncertain Fuzz Knowl Based Syst 15(5):521–537CrossRefGoogle Scholar
  3. 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):1106–1122CrossRefGoogle Scholar
  4. Alonso JM, Magdalena L, Guillaume S (2008) HILK: a new methodology for designing highly interpretable linguistic knowledge bases using the fuzzy logic formalism. Int J Intell Syst 23:761–794zbMATHCrossRefGoogle Scholar
  5. Alonso JM, Magdalena L, González-Rodríguez G (2009) Looking for a good fuzzy system interpretability index: an experimental approach. Int J Approx Reason 51(1):115–134CrossRefGoogle Scholar
  6. Antonelli M, Ducange P, Lazzerini B, Marcelloni F (2009a) Learning concurrently partition granularities and rule bases of Mamdani fuzzy systems in a multi-objective evolutionary framework. Int J Approx Reason 50(7):1066–1080CrossRefGoogle Scholar
  7. Antonelli M, Ducange P, Lazzerini B, Marcelloni F (2009b) Multi-objective evolutionary learning of granularity, membership function parameters and rules of Mamdani fuzzy systems. Evol Intel 2(1–2):21–37CrossRefGoogle Scholar
  8. 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):437–449CrossRefGoogle Scholar
  9. Casillas J, Cordón O, Herrera F (2002) COR: a methodology to improve ad hoc data-driven linguistic rule learning methods by inducing cooperation among rules. IEEE Trans Syst Man Cybern 32(4):526–537Google Scholar
  10. Casillas J, Cordon O, Herrera F, Magdalena L (eds) (2003) Interpretability issues in fuzzy modeling. Springer, HeidelbergzbMATHGoogle Scholar
  11. 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):1013–1031CrossRefGoogle Scholar
  12. Coello Coello CA, Lamont GB (2004) Applications of multi-objective evolutionary algorithms. World Scientific, SingaporezbMATHGoogle Scholar
  13. Cordon O, Herrera F, Villar P (2001a) Generating the knowledge base of a fuzzy rule-based system by the genetic learning of the data base. IEEE Trans Fuzzy Syst 9(4):667–674CrossRefGoogle Scholar
  14. Cordon O, Herrera F, Magadalena L, Villar P (2001b) A genetic learning process for the scaling factors, granularity and contexts of the fuzzy rule-based system data base. Inf Sci 136:85–107zbMATHCrossRefGoogle Scholar
  15. de Oliveira JV (1999) Semantic constraints for membership function optimization. IEEE Trans Syst Man Cybern Part A 29(1):128–138CrossRefGoogle Scholar
  16. Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, ChichesterzbMATHGoogle Scholar
  17. Ducange P, Lazzerini B, Marcelloni F (2009) Multi-objective genetic fuzzy classifiers for imbalanced and cost-sensitive datasets. Soft Comput 14(7):713–728CrossRefGoogle Scholar
  18. 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):419–436CrossRefGoogle Scholar
  19. Gacto MJ, Alcalá R, Herrera F (2010) Integration of an index to preserve the semantic interpretability in the multi-objective evolutionary rule selection and tuning of linguistic fuzzy systems. IEEE Trans Fuzzy Syst. doi: 10.1109/TFUZZ.2010.2041008
  20. González A, Pérez R (1999) SLAVE: a genetic learning system based on the iterative approach. IEEE Trans Fuzzy Syst 7:176–191CrossRefGoogle Scholar
  21. Guillaume S (2001) Designing fuzzy inference systems from data: An interpretability-oriented review. IEEE Trans Fuzzy Syst 9(3):426–443MathSciNetCrossRefGoogle Scholar
  22. Herrera F (2008) Genetic fuzzy systems: taxonomy, current research trends and prospects. Evol Intel 1:27–46CrossRefGoogle Scholar
  23. Ishibuchi H (2007) Multiobjective genetic fuzzy systems: review and future research direction. In: Proceedings of FUZZ-IEEE 2007 international conference on fuzzy systems, London, 23–26 JulyGoogle Scholar
  24. 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):4–31MathSciNetzbMATHCrossRefGoogle Scholar
  25. 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):59–88MathSciNetzbMATHCrossRefGoogle Scholar
  26. 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):135–150CrossRefGoogle Scholar
  27. Klawonn F (2006) Reducing the number of parameters of a fuzzy system using scaling functions. Soft Comput 10(9):749–756CrossRefGoogle Scholar
  28. Knowles JD, Corne DW (2002) Approximating the non dominated front using the Pareto archived evolution strategy. Evol Comput 8(2):149–172CrossRefGoogle Scholar
  29. Mamdani EH, Assilian S (1975) An experiment in linguistic synthesis with a fuzzy logic controller. Int J Man Mach Stud 7(1):1–13zbMATHCrossRefGoogle Scholar
  30. Massey FJ (1951) The Kolmogorov-Smirnov test for goodness of fit. J Am Stat Assoc 46(253):68–78zbMATHCrossRefGoogle Scholar
  31. Mencar C, Fanelli AM (2008) Interpretability constraints for fuzzy information granulation. Inf Sci 178:4585–4618MathSciNetCrossRefGoogle Scholar
  32. Mencar C, Castellano G, Fanelli AM (2007) Distinguishability quantification of fuzzy sets. Inf Sci 177:130–149MathSciNetzbMATHCrossRefGoogle Scholar
  33. Pedrycz W, Gomide F (2007) Fuzzy systems engineering: toward human-centric computing. Wiley-IEEE Press, NJGoogle Scholar
  34. Pulkkinen P, Koivisto H (2010) A dynamically constrained multiobjective genetic fuzzy system for regression problems. IEEE Trans Fuzzy Syst 18(1):161–177CrossRefGoogle Scholar
  35. Ruspini EH (1969) A new approach to clustering. Inform Control 15(1):22–32zbMATHCrossRefGoogle Scholar
  36. Teng Y, Wang W (2004) Constructing a user-friendly ga-based fuzzy system directly from numerical data. IEEE Trans Syst Man Cybern B 34(5):2060–2070CrossRefGoogle Scholar
  37. Wang LX, Mendel JM (1992) Generating fuzzy rules by learning from examples. IEEE Trans Syst Man Cybern 22(6):1414–1427MathSciNetCrossRefGoogle Scholar
  38. Zhou SM, Gan JQ (2008) Low-level interpretability and high-level interpretability: a unified view of data-driven interpretable fuzzy system modelling. Fuzzy Sets Syst 159:3091–3131MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Michela Antonelli
    • 1
  • Pietro Ducange
    • 1
  • Beatrice Lazzerini
    • 1
  • Francesco Marcelloni
    • 1
  1. 1.Dipartimento di Ingegneria dell’Informazione: Elettronica, Informatica, TelecomunicazioniUniversity of PisaPisaItaly

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