Goodness of fit measures and model selection in a fuzzy least squares regression analysis

Abstract

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. © Springer-Verlag Berlin Heidelberg 2013.