Soft Computing

, Volume 17, Issue 1, pp 165–194 | Cite as

Multi-objective genetic learning of serial hierarchical fuzzy systems for large-scale problems

Original Paper

Abstract

When we face a problem with a high number of variables using a standard fuzzy system, the number of rules increases exponentially and the obtained fuzzy system is scarcely interpretable. This problem can be handled by arranging the inputs in hierarchical ways. This paper presents a multi-objective genetic algorithm that learns serial hierarchical fuzzy systems with the aim of coping with the curse of dimensionality. By means of an experimental study, we have observed that our algorithm obtains good results in interpretability and accuracy with problems in which the number of variables is relatively high.

Keywords

Curse of dimensionality Hierarchical fuzzy systems Multi-objective genetic algorithms Variable selection 

References

  1. Aja-Fernández S, Alberola-López C (2008) Matriz modeling of hierarchical fuzzy systems. IEEE Trans Fuzzy Syst 16(3):585–599CrossRefGoogle Scholar
  2. Alcalá R, Alcalá-Fdez J, Casillas J, Cordón O, Herrera F (2006) Hybrid learning models to get the interpretability-accuracy trade-off in fuzzy modeling. Soft Comput 10:717–734CrossRefGoogle Scholar
  3. Alcalá R, Alcalá-Fdez J, Gacto M, Herrera F (2007a) Rule base reduction and genetic tuning of fuzzy systems based on the linguistic 3-tuples representation. Soft Comput 11:401–419CrossRefGoogle Scholar
  4. Alcalá R, Alcalá-Fdez J, Herrera F, Otero J (2007b) Genetic learning of accurate and compact fuzzy rule based systems based on the 2-tuples linguistic representation. Int J Approx Reason 44(1):46–64CrossRefGoogle Scholar
  5. Alcalá R, Gacto M, Herrera F (2011a) A fast and scalable multiobjective genetic fuzzy system for linguistic fuzzy modeling in high-dimensional regression problems. IEEE Trans Fuzzy Syst 19:666–681CrossRefGoogle Scholar
  6. Alcalá R, Nojima Y, Herrera F, Ishibuchi H (2011b) Multiobjective genetic fuzzy rule selection of single granularity-based fuzzy classification rules and its interaction with the lateral tuning of membership functions. Soft Comput 15(12):2303–2318CrossRefGoogle Scholar
  7. Antonelli M, Ducange P, Lazzerini B, Marcelloni F (2011) Learning knowledge bases of multi-objective evolutionary fuzzy systems by simultaneously optimizing accuracy, complexity, and partition integrity. Soft Comput 15(12):2335–2354CrossRefGoogle Scholar
  8. Benftez A, Casillas J (2009) Genetic learning of serial hierarchical fuzzy systems for large-scale problems. In: Proceedings of Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference (IFSA-EUSFLAT 2009). Lisbon, pp 1751–1756Google Scholar
  9. Casillas J, Carse B (2009) Genetic fuzzy systems: recent developments and future directions. Soft Comput 13:417–418 (special issue)CrossRefGoogle Scholar
  10. Casillas J, Cordón O, del Jesús M, Herrera F (2005) 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
  11. Chen Y, Dong J, Yang B (2004), Automatic design of hierarchical ts-fs model using ant programming and pso algorithm. In: Bussler C, Fensel D (eds) Proceedings 12th international conference on artificial intelligence, methodology, systems and applications. Lecture notes on artificial inteligence, LNAI 3192, pp 285–294Google Scholar
  12. Chen Y, Yang B, Abraham A, Peng L (2007) Automatic design of hierarchical Takagi-Sugeno type fuzzy systems using evolutionary algorithms. IEEE Trans Fuzzy Syst 15(3):385–397MATHCrossRefGoogle Scholar
  13. Cheong F (2007) A hierarchical fuzzy system with high input dimensions for forecasting foreign exchange rates. In: IEEE Congress on Evolutionary Computation, CEC, pp 1642–1647Google Scholar
  14. Chiu S (1996) Selecting input variables for fuzzy models. J Intell Fuzzy Syst 4(4):243–256Google Scholar
  15. Cordón 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
  16. Cordón O, Herrera F, Magdalena 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 136:85–107MATHCrossRefGoogle Scholar
  17. Deb K, Pratap A, Agarwal S, Meyarevian T (2002) A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  18. Duan J, Chung F (2002) Multilevel fuzzy relational systems: structure and identification. Soft Comput 6(2):71–86MATHCrossRefGoogle Scholar
  19. Gacto M, 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 13:419–436CrossRefGoogle Scholar
  20. Gaweda A, Scherer R (2004) Fuzzy number-based hierarchical fuzzy system, vol 3070. Lecture notes in computer sciencess. Springer, Berlin, pp 302–307Google Scholar
  21. González A, Pérez R (2001) Selection of relevant features in a fuzzy genetic learning algorithm. IEEE Trans Syst Man Cybern Part B Cybern 31(3):417–425CrossRefGoogle Scholar
  22. Ho TK, Basu M (2002) Complexity measures of supervised classification problems. IEEE Trans Pattern Anal Mach Intell 24(3):289–300CrossRefGoogle Scholar
  23. Ho TK, Basu M, Law M (2006) Measures of geometrical complexity in classification problems. In: Data complexity in pattern recognition. Springer, Berlin, pp 1–23 Google Scholar
  24. Holve R (1998) Investigation of automatic rule generation for hierarchical fuzzy systems. In: Fuzzy systems proceedings, vol 2. IEEE World Congress on Computational Intelligence, pp 973–978Google Scholar
  25. Hong T, Chen J (1999) Finding relevant attributes and membership functions. Fuzzy Sets Syst 103(3):389–404CrossRefGoogle Scholar
  26. Hong X, Harris C (2001) Variable selection algorithm for the construction of mimo operating point dependent neurofuzzy networks. IEEE Trans Fuzzy Syst 9(1):88–101CrossRefGoogle Scholar
  27. Ishibuchi H, Nozaki K, Yamamoto N, Tanaka H (1995) Selecting fuzzy if-then rules for classification problems using genetic algorithms. IEEE Trans Fuzzy Syst 3(3):260–270CrossRefGoogle Scholar
  28. Jelleli T, Alimi A (2005) Improved hierarchical fuzzy control scheme. Adapt Natural Comput 1:128–131Google Scholar
  29. Jelleli T, Alimi A (2010) Automatic design of a least complicated hierarchical fuzzy system. In: 6th IEEE World Congress on computational intelligence, pp 1–7Google Scholar
  30. Jin Y (2000) Fuzzy modeling of high-dimensional systems: complexity reduction and interpretability improvement. IEEE Trans Fuzzy Syst 8(2):212–221CrossRefGoogle Scholar
  31. Joo M, Lee J (1999) Hierarchical fuzzy control scheme using structured Takagi-Sugeno type fuzzy inference. Proceedings of IEEE international fuzzy systems conference, Seoul, In, pp 78–83Google Scholar
  32. Joo M, Lee J (2002) Universal approximation by hierarchical fuzzy system with constrains on the fuzzy rule. Fuzzy Sets Syst 130(2):175–188MathSciNetMATHCrossRefGoogle Scholar
  33. Joo M, Sudkamp T (2009) A method of converting a fuzzy system to a two-layered hierarchical fuzzy system and its run-time efficiency. IEEE Trans Fuzzy Syst 17(1):93–103CrossRefGoogle Scholar
  34. Lee H, Chen C, Chen J, Jou Y (2001) An efficient fuzzy classifier with feature selection based on fuzzy entropy. IEEE Trans Syst Man Cybern Part B Cybern 31(3):426–432CrossRefGoogle Scholar
  35. Lee M, Chung H, Yu F (2003) Modeling of hierarchical fuzzy systems. Fuzzy Sets Syst 138(2):343–361MathSciNetCrossRefGoogle Scholar
  36. Maeda H (1996) An investigation on the spread of fuzziness in multi-fold multi-stage approximate reasoning by pictorial representation under sup-min composition and triangular type membership function. Fuzzy Sets Syst 80(2):133–148CrossRefGoogle Scholar
  37. Nojima Y, Alcalá R, Ishibuchi H, Herrera F (2011) Special issue on evolutionary fuzzy systems. Soft Comput 15(12):2299–2301CrossRefGoogle Scholar
  38. Raju G, Zhou J, Kisner R (1991) Hierarchical fuzzy control. Int J Control 54(5):1201–1216MathSciNetMATHCrossRefGoogle Scholar
  39. Salgado P (2008) Rule generation for hierarchical collaborative fuzzy system. Appl Math Modell Sci Direct 32(7):1159–1178MathSciNetMATHCrossRefGoogle Scholar
  40. Shimojima K, Fukuda T, Hasegawa Y (1995) Self-tuning fuzzy modeling with adaptive membership function, rules, and hierarchical structure based on genetic algorithm. Fuzzy Sets Syst 71(3):295–309CrossRefGoogle Scholar
  41. Tan F, Fu X, Zhang Y, Bourgeois A (2008) A genetic algorithm-based method for feature subset selection. Soft Comput 12:111–120CrossRefGoogle Scholar
  42. Taniguchi T, Tanaka K, Ohtake H, Wang H (2001) Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems. IEEE Trans Fuzzy Syst 9(4):525–538CrossRefGoogle Scholar
  43. Torra V (2002) A review of the construction of hierarchical fuzzy systems. Int J Intell Syst 17(5):531–543MathSciNetMATHCrossRefGoogle Scholar
  44. Wang L (1998) Universal approximation by hierarchical fuzzy systems. Fuzzy Sets Syst 93(2):223–230MATHCrossRefGoogle Scholar
  45. Wang L (1999) Analysis and design of hierarchical fuzzy systems. IEEE Trans Fuzzy Syst 7(5):617–624CrossRefGoogle Scholar
  46. Wang L, Mendel J (1992) Generating fuzzy rules by learning from examples. IEEE Trans Syst Man Cybern 22(6):1414–1427MathSciNetCrossRefGoogle Scholar
  47. Wang D, Zeng X, Keane J (2006) Learning for hierarchical fuzzy systems based on gradient-descent method. In: Proceedings of IEEE international conference on fuzzy systems, pp 92–99Google Scholar
  48. Xiong N, Funk P (2006) Construction of fuzzy knowledge bases incorporating feature selection. Soft Comput 10:796–804CrossRefGoogle Scholar
  49. Zajaczkowski J, Verma B (2012) Selection and impact of different topologies in multilayered hierarchical fuzzy systems. Appl Intell 36(3):564–584CrossRefGoogle Scholar
  50. Zeng X, Goulermas J, Liatsis P, Wang D, Keane J (2008) Hierarchical fuzzy systems for function approximation on discrete input spaces with application. IEEE Trans Fuzzy Syst 16(5):1197–1215CrossRefGoogle Scholar
  51. Zhang X, Zhang N (2006) Universal approximation of binary-tree hierarchical fuzzy system with typical FLUs. Lecture notes in computer science, vol 4114. Springer, Berlin, pp 177–182Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Calculation Centre, Astrophysics Institute of Andalusia (IAA)Spanish National Research Council (CSIC)GranadaSpain
  2. 2.Department of Computer Science and Artificial Intelligence, CITIC-UGR (Research Center on Information and Communication Technology)University of GranadaGranadaSpain

Personalised recommendations