Support Vector Machines in Fuzzy Regression

Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 634)

Abstract

This paper presents methods of estimating fuzzy regression models based on support vector machines. Starting from the approaches known from the literature and dedicated to triangular fuzzy numbers and based on linear and quadratic loss, a new method applying loss function based on the Trutschnig distance is proposed. Furthermore, a generalization of those models for fuzzy numbers with trapezoidal membership function is given. Finally, the proposed models are illustrated and compared in the examples and some of their properties are discussed.

Keywords

Fuzzy numbers Fuzzy regression Loss function Support vector machines 

References

  1. 1.
    Arabpour, A.R., Tata, M.: Estimating the parameters of a fuzzy linear regression model. Iran. J. Fuzzy Syst. 5, 1–19 (2008)MathSciNetMATHGoogle Scholar
  2. 2.
    Bargiela, A., Pedrycz, W., Nakashima, T.: Multiple regression with fuzzy data. Fuzzy Sets Syst. 158, 2169–2188 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bertoluzza, C., Corral, N., Salas, A.: On a new class of distances between fuzzy numbers. Mathw. Soft Comput. 2, 71–84 (1995)MathSciNetMATHGoogle Scholar
  4. 4.
    Chang, Y.H.O., Ayyub, B.M.: Fuzzy regression methods-a comparative assessment. Fuzzy Sets Syst. 119, 187–203 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Diamond, P.: Fuzzy least squares. Inf. Sci. 46, 141–157 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Diamond, P.: Least squares and maximum likelihood regression for fuzzy linear models. In: Kacprzyk, J., Fedrizzi, M. (eds.) Fuzzy Regression Analysis, pp. 137–151. Wiley, New York (1992)Google Scholar
  7. 7.
    Diamond, P., Tanaka, H.: Fuzzy regression analysis. Fuzzy Sets Decis. Anal. Oper. Res. Stat. 1, 349–387 (1988)MathSciNetGoogle Scholar
  8. 8.
    Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9, 613–626 (1978)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fletcher T.: Support vector machines explained. www.cs.ucl.ac.uk/staff/T.Fletcher/
  10. 10.
    Grzegorzewski, P.: Metrics and orders in space of fuzzy numbers. Fuzzy Sets Syst. 97, 83–94 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grzegorzewski, P.: Trapezoidal approximations of fuzzy numbers preserving the expected interval-algorithms and properties. Fuzzy Sets Syst. 159, 1354–1364 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grzegorzewski, P.: Algorithms for trapezoidal approximations of fuzzy numbers preserving the expected interval. In: Bouchon-Meunier, B., Magdalena, L., Ojeda-Aciego, M., Verdegay, J.-L., Yager, R.R. (eds.) Foundations of Reasoning under Uncertainty, pp. 85–98. Springer, Berlin (2010)CrossRefGoogle Scholar
  13. 13.
    Grzegorzewski, P., Mrwka E.: Linear regression analysis for fuzzy data, In: Proceedings of the 10th IFSA World Congress–IFSA 2003, Istanbul, Turkey, pp. 228–231. June 29-July 2, 2003Google Scholar
  14. 14.
    Grzegorzewski, P., Mrwka, E.: Regression analysis with fuzzy data. In: Grzegorzewski, P., Krawczak, M., Zadrony, S. (eds.) Soft Computing-Tools, Techniques and Applications, pp. 65–76. Warszawa, Exit (2004)Google Scholar
  15. 15.
    Grzegorzewski P., Pasternak-Winiarska, K.: Weighted trapezoidal approximations of fuzzy numbers. In: Carvalho, J.P., Dubois, D., Kaymak, U., Sousa, J.M.C (eds.), Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, pp. 1531–1534, LisbonGoogle Scholar
  16. 16.
    Grzegorzewski, P., Pasternak-Winiarska, K.: Bi-symmetrically weighted trapezoidal approximations of fuzzy numbers. In: Abraham, A., Benitez Sanchez, J.M., Herrera, F., Loia, V., Marcelloni, F., Senatore, S. (eds.), Proceedings of Ninth International Conference on Intelligent Systems Design and Applications, pp. 318–323. Pisa (2009)Google Scholar
  17. 17.
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer, Berlin (2008)MATHGoogle Scholar
  18. 18.
    Hong, D.H., Hwang, C.: Support vector fuzzy regression machines. Fuzzy Sets Syst. 138, 271–281 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hong, D.H., Hwang C.: Fuzzy Nonlinear Regression Model Based on LS-SVM in Feature Space, pp. 208–216. Springer, Berlin (2006)Google Scholar
  20. 20.
    Kacprzyk, J., Fedrizzi, M.: Fuzzy Regression Analysis. Omnitech Press, Heidelberg (1992)MATHGoogle Scholar
  21. 21.
    Körner, R., Näther, W.: Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates. Inf. Sci. 109, 95–118 (1998)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kao, C., Chyyu, C.L.: A fuzzy linear regression model with better explanatory power. Fuzzy Sets Syst. 126, 401–409 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Näther, W.: Linear statistical inference for random fuzzy data. Statistics 29, 221–240 (1997)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Redden, D.T., Woodall, W.H.: Properties of certain fuzzy linear regression methods. Fuzzy Sets Syst. 64, 361–375 (1994)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Smola, A.J., Scholkopf, B.: A tutorial on support vector regression. Stat. Comput 14, 199–222 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tanaka, H.: Fuzzy data analysis by possibilistic linear models. Fuzzy Sets Syst. 24(1987), 363–375 (1987)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tanaka, H., Uegima, S., Asai, K.: Linear analysis with fuzzy model. IEEE Trans. Syst. Man Cybern. 12, 903–907 (1982)CrossRefMATHGoogle Scholar
  28. 28.
    Tanaka, H., Watada, J.: Possibilistic linear systems and their application to the linear regression model. Fuzzy Sets Syst. 27, 275–289 (1988)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Trutschnig, W., González-Rodríguez, G., Colubi, A., Gil, M.A.: A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread. Inf. Sci. 179, 3964–3972 (2009)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer, Berlin (1995)CrossRefMATHGoogle Scholar
  31. 31.
    Vapnik, V.N.: Statistical Learning Theory. Wiley, New York (1998)MATHGoogle Scholar
  32. 32.
    Zeng, W., Li, H.: Weighted triangular approximation of fuzzy numbers. Int. J. Approx. Reason. 46, 137–150 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland
  2. 2.Department of Gastroenterology, Hepatology and Clinical Oncology, Medical Center for Postgraduate EducationWarsawPoland
  3. 3.Systems Research Institute, Polish Academy of SciencesWarsawPoland

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