Soft Computing

, Volume 13, Issue 5, pp 467–479 | Cite as

Obtaining linguistic fuzzy rule-based regression models from imprecise data with multiobjective genetic algorithms

  • Luciano Sánchez
  • José Otero
  • Inés Couso


Backfitting of fuzzy rules is an Iterative Rule Learning technique for obtaining the knowledge base of a fuzzy rule-based system in regression problems. It consists in fitting one fuzzy rule to the data, and replacing the whole training set by the residual of the approximation. The obtained rule is added to the knowledge base, and the process is repeated until the residual is zero, or near zero. Such a design has been extended to imprecise data for which the observation error is small. Nevertheless, when this error is moderate or high, the learning can stop early. In this kind of algorithms, the specificity of the residual might decrease when a new rule is added. There may happen that the residual grows so wide that it covers the value zero for all points (thus the algorithm stops), but we have not yet extracted all the information available in the dataset. Focusing on this problem, this paper is about datasets with medium to high discrepancies between the observed and the actual values of the variables, such as those containing missing values and coarsely discretized data. We will show that the quality of the iterative learning degrades in this kind of problems, because it does not make full use of all the available information. As an alternative to sequentially obtaining rules, we propose a new multiobjective Genetic Cooperative Competitive Learning (GCCL) algorithm. In our approach, each individual in the population codifies one rule, which competes in the population in terms of maximum coverage and fitting, while the individuals in the population cooperate to form the knowledge base.


Fuzzy Number Fuzzy Rule Rule Base Observation Error Fuzzy Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by the Spanish Ministry of Education and Science, under grants TIN2005-08036-C05-05 and TIN2005-08036-C05-01.


  1. Alcala J et al (2008) KEEL: a software tool to assess evolutionary algorithms for data mining problems. Soft Comput (in press)Google Scholar
  2. Cordón O, Herrera F (2000) A proposal for improving the accuracy of linguistic modeling. IEEE Trans Fuzzy Syst 8(3):335–344CrossRefGoogle Scholar
  3. Cordón O, Herrera F, Sánchez L (1999) Solving electrical distribution problems using hybrid evolutionary data analysis techniques. Appl Intell 10(1):5–24CrossRefGoogle Scholar
  4. Cornelis C, Kerre E (2003) A fuzzy inference methodology based on the fuzzification of set inclusion. In: Recent advances in intelligent paradigms and applications, Physica-Verlag, pp 71–89 Google Scholar
  5. Couso I, Sánchez L (2008) Higher order models for fuzzy random variables. Fuzzy Sets Syst 159:237–258CrossRefGoogle Scholar
  6. 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
  7. del Jesus MJ, Hoffmann F, Junco L, Sánchez L (2004) Induction of fuzzy-rule-based classifiers with evolutionary boosting algorithms. IEEE Trans Fuzzy Syst 12(3):296–308CrossRefGoogle Scholar
  8. Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24(3):279–300zbMATHCrossRefGoogle Scholar
  9. Ein-Dor P, Feldmesser J (1987) Attributes of the performance of central processing units: a relative performance prediction model. Commun ACM 30(4):308–317CrossRefGoogle Scholar
  10. Ferson S, Kreinovich V, Hajagos J, Oberkampf W, Ginzburg L (2007) Experimental uncertainty estimation and statistics for data having interval uncertainty. Technical Report SAND2007-0939, Sandia National LaboratoriesGoogle Scholar
  11. Friedman J (1991) Multivariate adaptive regression splines. Ann Stat 19:1–141zbMATHCrossRefGoogle Scholar
  12. Friedman J, Hastie T, Tibshirani R (1998) Additive logistic regression: a statistical view of boosting. Mach LearnGoogle Scholar
  13. Greene DP, Smith SF (1993) Competition-based induction of decision models from examples. Mach Learn 3:229–257CrossRefGoogle Scholar
  14. Herrera F (2005) Genetic fuzzy systems: status, critical considerations and future directions. Int J Comput Intell Res 1(1):59–67Google Scholar
  15. Herrera F (2008) Genetic fuzzy systems: taxonomy, current research trends and prospects. Evol Intell 1:27–46CrossRefGoogle Scholar
  16. Ishibuchi H, Nakashima T, Murata T (1999) Performance evaluation of fuzzy classifier systems for multidimensional pattern classification problems. IEEE Trans Syst Man Cybern Cybern 29(5):601–618CrossRefGoogle Scholar
  17. Juang CF, Lin JY, Lin CT (2000) Genetic reinforcement learning through symbiotic evolution for fuzzy controller design. IEEE Trans Syst Man Cybern B Cybern 30(2):290–302CrossRefMathSciNetGoogle Scholar
  18. Koeppen M, Franke K, Nickolay B (2003) Fuzzy-Pareto-dominance driven multi-objective genetic algorithm. In: Proceedings of 10th international fuzzy systems assotiation world congress (IFSA), Istanbul, pp 450–453Google Scholar
  19. Limbourg P (2005) Multi-objective optimization of problems with epistemic uncertainty. EMO 2005:413–427Google Scholar
  20. Mallat S, Zhang Z (1993) Matching pursuits with time–frequency dictionaries. IEEE Trans Signal Process 41:3397–3415zbMATHCrossRefGoogle Scholar
  21. Marín E, Sánchez L (2004) Supply estimation using coevolutionary genetic algorithms in the Spanish electrical market. Appl Intell 21(1):7–24CrossRefGoogle Scholar
  22. Nozaki K, Ishibuchi H, Tanaka H (1997) A simple but powerful heuristic method for generating fuzzy rules from numerical data. Fuzzy Sets Syst 86:251–270CrossRefGoogle Scholar
  23. Otero J, Sanchez L (2006) Induction of descriptive fuzzy classifiers with the Logitboost algorithm. Soft Comput 10(9):825–835Google Scholar
  24. Prechelt L (1994) PROBEN1—a set of benchmarks and benchmarking rules for neural network training algorithms. Tech. Rep. 21/94, Fakultat fur Informatik, Universitat KarlsruheGoogle Scholar
  25. Press W et al (1992) Numerical recipes in C. The art of scientific computing. Cambridge University Press, New YorkGoogle Scholar
  26. Sánchez L, Couso I (2007) Advocating the use of imprecisely observed data in genetic fuzzy systems. IEEE Trans Fuzzy Syst 15(4):551–562CrossRefGoogle Scholar
  27. Sánchez L, Otero J (2004) A fast genetic method for inducting descriptive fuzzy models. Fuzzy Sets Syst 141(1):33–46zbMATHCrossRefGoogle Scholar
  28. Sánchez L, Otero J (2007) Boosting fuzzy rules in classification problems under single-winner inference. Int J Intell Syst 22(9):1021–1034zbMATHCrossRefGoogle Scholar
  29. Sánchez L, Villar JR (2008) Obtaining transparent models of chaotic systems with multiobjective simulated annealing algorithms. Inform Sci 178(4):952–970zbMATHCrossRefGoogle Scholar
  30. Sánchez L, Casillas J, Cordón O et al (2002) Some relationships between fuzzy and random set-based classifiers and models. Int J Approx Reason 29(2):175–213zbMATHCrossRefGoogle Scholar
  31. Sánchez L, Otero J, Villar JR (2006) Boosting of fuzzy models for high-dimensional imprecise datasets. In: Proceedings of IPMU 2006, Paris, pp 1965–1973Google Scholar
  32. Sánchez L, Couso I, Casillas J (2007) Modelling vague data with genetic fuzzy systems under a combination of crisp and imprecise criteria. In: Proceedings of 2007 IEEE symposium on Computational Intellignece in multicriteria decision making, Honolulu, pp 30–37Google Scholar
  33. Sánchez L, Couso I, Casillas J (2009) Genetic learning of fuzzy rules based on low quality data. Fuzzy Sets Syst (submitted)Google Scholar
  34. Takagi T, Sugeno M (1985) Fuzzy identification of systems and its application to modeling and control. IEEE Trans Syst Man Cybern 15(1):116–132zbMATHGoogle Scholar
  35. Teich J (2001) Pareto-front exploration with uncertain objectives. EMO 2001:314–328Google Scholar
  36. Wang LX, Mendel J (1992) Generating fuzzy rules by learning from examples. IEEE Trans Syst Man Cybern 25(2):353–361MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Computer Science DepartmentUniversity of OviedoGijónSpain
  2. 2.Statistics Department, Facultad de CienciasUniversity of OviedoOviedoSpain

Personalised recommendations