Weighted Least Squares and Least Median Squares estimation for the fuzzy linear regression analysis
In this paper, we discuss the problem of regression analysis in a fuzzy domain. By considering an iterative Weighted Least Squares estimation approach, we propose a general linear regression model for studying the dependence of a general class of fuzzy response variable, i.e., <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$LR_2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>L</mi> <msub> <mi>R</mi> <mn>2</mn> </msub> </mrow> </math> </EquationSource> </InlineEquation> fuzzy variable or trapezoidal fuzzy variable,on a set of crisp or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$LR_2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>L</mi> <msub> <mi>R</mi> <mn>2</mn> </msub> </mrow> </math> </EquationSource> </InlineEquation> fuzzy explanatory variables. We also show some theoretical properties and a suitable generalization of the determination coefficient in order to investigate the goodness of fit of the regression model. Furthermore, we discuss some theoretical issues and an assessment of imprecision of the regression function. Finally, we suggest a robust version of the fuzzy regression model based on the Least Median Squares estimation approach which is able to neutralize and/or smooth the disruptive effects of possible crisp or fuzzy outliers in the estimation process. A simulation study and two empirical applications are presented. Copyright Sapienza Università di Roma 2013
