Soft Computing

, Volume 13, Issue 5, pp 451–465 | Cite as

Learning consistent, complete and compact sets of fuzzy rules in conjunctive normal form for regression problems

  • Jorge Casillas
  • Pedro Martínez
  • Alicia D. Benítez


When a flexible fuzzy rule structure such as those with antecedent in conjunctive normal form is used, the interpretability of the obtained fuzzy model is significantly improved. However, some important problems appear related to the interaction among this set of rules. Indeed, it is relatively easy to get inconsistencies, lack of completeness, redundancies, etc. Generally, these properties are ignored or mildly faced. This paper, however, focuses on the design of a multiobjective genetic algorithm that properly considers all these properties thus ensuring an effective search space exploration and generation of highly legible and accurate fuzzy models.


Genetic fuzzy systems Regression problems Multiobjective optimization Flexible fuzzy rules Interpretability constrains 



This work was supported in part by the Spanish Ministry of Education and Science under grant no. TIN2005-08386-C05-01 and the Andalusian Government under grants no. P05-TIC-00531 and P07-TIC-3185.


  1. Carmona P, Castro J (2005) Interpretability enhancement of fuzzy modeling using ant colony optimization. In: Proceedings of the 1st international workshop on genetic fuzzy systems (GFS 2005), Granada, Spain, pp 148–153Google Scholar
  2. Carmona P, Castro J, Zurita J (2004) Learning maximal structure fuzzy rules with exceptions. Fuzzy Sets Syst 146:63–77zbMATHCrossRefMathSciNetGoogle Scholar
  3. Casillas J, Martínez-López F (2008) Mining uncertain data with multiobjective genetic fuzzy systems to be applied in consumer behaviour modelling. Expert Syst Appl (in press). doi: 10.1016/j.eswa.2007.11.035
  4. Casillas J, Cordón O, Herrera F, Magdalena L (eds) (2003a) Accuracy improvements in linguistic fuzzy modeling. Springer, HeidelbergzbMATHGoogle Scholar
  5. Casillas J, Cordón O, Herrera F, Magdalena L (eds) (2003b) Interpretability issues in fuzzy modeling. Springer, HeidelbergzbMATHGoogle Scholar
  6. Casillas J, Cordón O, del Jesus M, Herrera F (2005a) Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction. IEEE Trans Fuzzy Syst 13(1):13–29CrossRefGoogle Scholar
  7. Casillas J, Cordón O, de Viana IF, Herrera F (2005b) Learning cooperative linguistic fuzzy rules using the best-worst ant system algorithm. Int J Intell Syst 20:433–452zbMATHCrossRefGoogle Scholar
  8. Castro J, Castro-Schez J, Zurita J (1999) Learning maximal structure rules in fuzzy logic for knowledge acquisition in expert systems. Fuzzy Sets Syst 101(3):331–342zbMATHCrossRefMathSciNetGoogle Scholar
  9. Cordón O, Herrera F, Sánchez L (1999) Solving electrical distribution problems using hybrid evolutionary data analysis techniques. Appl Intell 10(1):5–24CrossRefGoogle Scholar
  10. Cordón O, Herrera F, Hoffmann F, Magdalena L (2001) Genetic fuzzy systems: evolutionary tuning and learning of fuzzy knowledge bases. World Scientific, SingaporezbMATHGoogle Scholar
  11. Deb K, Pratap A, Agarwal S, Meyarevian T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evolut Comput 6(2):182–197CrossRefGoogle Scholar
  12. Fullér R (2000) Introduction to neuro-fuzzy systems. Physica-Verlag, HeidelbergzbMATHGoogle Scholar
  13. González A, Pérez R (1998) Completeness and consistency conditions for learning fuzzy rules. Fuzzy Sets Syst 96(1):37–51CrossRefGoogle Scholar
  14. González A, Pérez R (1999) SLAVE: a genetic learning system based on an iterative approach. IEEE Trans Fuzzy Syst 7(2):176–191CrossRefGoogle Scholar
  15. Hastie T, Tibshirani R (1990) Generalized additive models. Chapman & Hall, LondonzbMATHGoogle Scholar
  16. 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):4–31zbMATHCrossRefMathSciNetGoogle Scholar
  17. Ishibuchi H, Yamamoto T, Nakashima T (2006) An approach to fuzzy default reasoning for function approximation. Soft Comput 10:850–864CrossRefGoogle Scholar
  18. Jin Y, von Seelen W, Sendhoff B (1999) On generating FC3 fuzzy rule systems from data using evolution strategies. IEEE Trans Syst Man Cybern B Cybern 29(4):829–845Google Scholar
  19. Karr C (1991) Genetic algorithms for fuzzy controllers. AI Expert 6(2):26–33Google Scholar
  20. Lavrač N, Cestnik B, Gamberger D, Flach P (2004) Decision support through subgroup discovery: three case studies and the lessons learned. Mach Learn 57(1–2):115–143zbMATHCrossRefGoogle Scholar
  21. Magdalena L (1997) Adapting the gain of an FLC with genetic algorithms. Int J Approx Reason 17(4):327–349zbMATHCrossRefGoogle Scholar
  22. Nauck D, Klawonn F, Kruse R (1997) Foundations of neuro-fuzzy systems. Wiley, New YorkGoogle Scholar
  23. Nozaki K, Ishibuchi H, Tanaka H (1997) A simple but powerful heuristic method for generating fuzzy rules from numerical data. Fuzzy Sets Syst 86(3):251–270CrossRefGoogle Scholar
  24. Otero J, Sánchez L (2006) Induction of descriptive fuzzy classifiers with the logitboost algorithm. Soft Comput 10(9):825–835CrossRefGoogle Scholar
  25. Sánchez L, Couso I (2000) Fuzzy random variables-based modeling with GA-P algorithms. In: In Information, uncertainty and fusion. Kluwer, Dordrecht, pp 245–256Google Scholar
  26. Sánchez L, Couso I, Corrales JA (2001) Combining GP operators with SA search to evolve fuzzy rule based classifiers. Inf Sci 136(1–4):175–191zbMATHCrossRefGoogle Scholar
  27. Thrift P (1991) Fuzzy logic synthesis with genetic algorithms. In: Belew R, Booker L (eds) Proceedings of the 4th international conference on genetic algorithms. Morgan Kaufmann Publishers, San Mateo, pp 509–513Google Scholar
  28. Valenzuela-Rendón M (1991) The fuzzy classifier system: a classifier system for continuously varying variables. In: Proceedings of the 4th international conference on genetic algorithms. Morgan Kaufmann Publishers, San Mateo, pp 346–353Google Scholar
  29. Wang C, Hong T, Tseng S, Liao C (1998) Automatically integrating multiple rules sets in a distributed-knowledge environment. IEEE Trans Syst Man Cybern C Appl Rev 28(3):471–476zbMATHCrossRefGoogle Scholar
  30. Wang H, Kwong S, Jin Y, Wei W, Man KF (2005) Agent-based evolutionary approach for interpretable rule-based knowledge extraction. IEEE Trans Syst Man Cybern 35(2):143–155CrossRefGoogle Scholar
  31. Wang LX (2003) The WM method completed: a flexible fuzzy system approach to data mining. IEEE Trans Fuzzy Syst 11(6):768–782CrossRefGoogle Scholar
  32. Wang LX, Mendel J (1992) Generating fuzzy rules by learning from examples. IEEE Trans Syst Man Cybern 22(6):1414–1427CrossRefMathSciNetGoogle Scholar
  33. Weigend A, Gershenfeld N (eds) (1993) Time series prediction: forecasting the future and understanding the past. In: 1992 NATO Advanced Research Workshop on Comparative Time Series Analysis. Addison-Wesley, Santa FeGoogle Scholar
  34. Xiong N, Litz L (2000) Fuzzy modeling based on premise optimization. In: Proceedings of the 9th IEEE international conference on fuzzy systems, San Antonio, TX, USA, pp 859–864Google Scholar
  35. Yager R (1993) On a hierarchical structure for fuzzy modeling and control. IEEE Trans Syst Man Cybern 23(4):1189–1197CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Jorge Casillas
    • 1
  • Pedro Martínez
    • 1
  • Alicia D. Benítez
    • 1
  1. 1.Department of Computer Science and Artificial IntelligenceUniversity of GranadaGranadaSpain

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