Goodness of Fit Measures and Model Selection in a Fuzzy Least Squares Regression Analysis

  • Francesco Campobasso
  • Annarita Fanizzi
Part of the Studies in Computational Intelligence book series (SCI, volume 465)


Market researches and opinion polls usually include customers’ responses as verbal labels of sets with vague and uncertain borders. Recently we generalized the estimation procedure of a simple regression model with triangular fuzzy numbers, into the space of which Diamond introduced a metrics, to the case of a multivariate model with an asymmetric intercept also fuzzy.

In this paper we show under what conditions the sum of squares of the dependent variable can be decomposed in exactly the same way as the classical OLS estimation and we propose a fuzzy version of the coefficient of determination, which takes into account the corresponding freedom degrees. Furthermore we introduce a stepwise procedure designed not only to include only one independent variable at a time, but also to eliminate in each iteration that variable whose explanatory contribution is subrogated by the combination of the other ones included after it was.


Fuzzy least square regression multivariate generalization asymmetric fuzzy intercept total sum of squares goodness of fit stepwise selection of independent variables 


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  1. 1.
    Kao, C., Chyu, C.L.: Least-squares estimates in fuzzy regression analysis. European Journal of Operational Research (2003)Google Scholar
  2. 2.
    Takemura, K.: Fuzzy least squares regression analysis for social judgment study. Journal of Advanced Intelligent Computing and Intelligent Informatics (2005)Google Scholar
  3. 3.
    Diamond, P.M.: Fuzzy Least Square. Information Sciences (1988)Google Scholar
  4. 4.
    Bilancia, M., Campobasso, F., Fanizzi, A.: The pricing of risky securities in a Fuzzy Least Square Regression model. In: Locarek-Junge, H., Weihs, C. (eds.) Studies in Classification, Data Analysis and Knowledge Organization - Classification as a Tool for Research, pp. 639–646. Springer, Heidelberg (2010)Google Scholar
  5. 5.
    Zadeh, L.A.: Fuzzy sets. Information and Control 8(3), 338–353 (1965)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Campobasso, F., Fanizzi, A., Tarantini, M.: Some results on a multivariate generalization of the Fuzzy Least Square Regression. In: Proceedings of the International Conference on Fuzzy Computation, Madeira - Portugal (2009)Google Scholar
  7. 7.
    Montrone, S., Campobasso, F., Perchinunno, P., Fanizzi, A.: An Analysis of Poverty in Italy through a Fuzzy Regression Model. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2011, Part I. LNCS, vol. 6782, pp. 342–355. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Campobasso, F., Fanizzi, A.: A Fuzzy Approach to the Least Squares Regression Model With a Symmetric Fuzzy Intercept. In: Proceedings of the 14th Applied Stochastic Model and Data Analysis Conference, Roma (2011)Google Scholar
  9. 9.
    Campobasso, F., Fanizzi, A.: A stepwise procedure to select variables in a fuzzy least square regression model. In: Proceedings of the International Conference on International Conference on Fuzzy Computation Theory and Applications, pp. 417–426 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Economics and MathematicsUniversity of BariBariItaly

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