Hybrid fuzzy least-squares regression analysis and its reliability measures

A method for hybrid fuzzy least-squares regression is developed in this paper. The method uses the new definition of weighted fuzzy-arithmetic and the well-accepted least-squares fitting criterion. First, a bivariate regression model using asymmetrical triangular fuzzy variables is derived. Three sets of normal equations are formulated to solve the three parts of hybrid regression coefficients: fuzzy center, left fuzzy width, and right fuzzy width. Then, the method is extended to multiple regression analysis. Three numerical examples are used to demonstrate the proposed method; two bivariate regression examples one for symmetrical triangular fuzzy numbers and the other for asymmetrical triangular fuzzy numbers, and one multiple regression example for symmetrical triangular fuzzy numbers. In each example, hybrid regression equation and their reliability measures are calculated. The results from hybrid regression are also compared with the corresponding results from ordinary regression and other fuzzy regression methods. Conclusions are drawn based on the reliability evaluations. Furthermore, hybrid fuzzy least-squares regression is extended to nonlinear models. Issues on the calibration of nonlinear equations and their reliability measures are also discussed. In conclusions, the merits and limitations of the proposed method are summarized. (C) 2001 Elsevier Science B.V. All rights reserved.