Fuzzy Regression Analysis

Fuzzy regresion analysis gives a fuzzy functional relationship between dependent and independent variables where vagueness is present in some form. The input data may be crisp or fuzzy. This chapter considers two types of fuzzy regression. The first is based on possibilistic concepts and the second upon a least squares approach. However, in both the notion of “best fit ” incorporates the optimization of a functional associated with he problem. In possibilistic regression, this functional takes the form of a measure of the spreads of the estimated output, either as a weighted linear sum involving the estimated coefficients in linear regression, or as quadratic form in the case of exponential possibilistic regression. These optimization problems reduce to linear programming. For the least squares approach, the functional to be minimized is an L₂ distance between the observed and estimated outputs. This reduces to a class of quadratic optimization problems and constrained quadratic optimization. The method can incorporate stochastic fuzzy input and fuzzy kriging uses covariances to obtain BLUE estimators.