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Approaches to select suitable subset of explanatory variables for establishing fuzzy regression models

Fuzzy regression models (FRMs) are used to describe the contribution of the corresponding fuzzy explanatory variables in explaining the fuzzy response variable. The selection of explanatory variables greatly affects the cost of establishing an FRM and its performance in applications. This paper investigates the quality of fit and suitable variable selection for building up FRMs. Based on the existing formulation of an FRM, a theorem and four related propositions are provided and proven. Then, two fitness measures, namely 2 and adjusted 2, are proposed to evaluate the fitting performance of potential FRMs for selecting a suitable model from all possible FRMs. In addition, based on the idea of the average marginal contribution, a stepwise selection procedure that includes forward and backward selections is developed to efficiently find a suitable subset of explanatory variables without requiring the fitting of all possible FRMs. Unlike the existing selection procedure that only includes the forward selection, the backward selection in the proposed stepwise procedure can avoid multicollinearity among explanatory variables. In addition, the proposed fitness measures and stepwise selection procedure are generalized to make them applicable to any data type of explanatory variables and response. The applicability and feasibility of the proposed measures and variable selection procedure are demonstrated using numerical examples and comparisons with existing approaches.