Least-squares estimates in fuzzy regression analysis

Regression is a very powerful methodology for forecasting, which is considered as an essential component of successful OR applications. In this paper an idea stemmed from the classical least squares is proposed to handle fuzzy observations in regression analysis. Based on the extension principle, the membership function of the sum of squared errors is constructed. The fuzzy sum of squared errors is a function of the regression coefficients to be determined, which can be minimized via a nonlinear program formulated under the structure of the Chen–Klein method for ranking fuzzy numbers. To illustrate how the proposed method is applied, three cases, one crisp input–fuzzy output, one fuzzy input–fuzzy output, and one non-triangular fuzzy observations, are exemplified. The results show that the least-squares method of this paper is able to determine the regression coefficients with better explanatory power. Most important, it works for all types of fuzzy observations, not restricted to the triangular one.