Abstract
In this study, fuzzy regression (FR) models with fuzzy inputs and outputs are discussed. Some of the FR methods based on linear programming and fuzzy least squares in the literature are explained. Within this study, we propose a Fuzzy Radial Basis Function (FRBF) Network to obtain the estimations for FR model in the case that inputs and outputs are symmetric/nonsymmetric triangular fuzzy numbers. Proposed FRBF Network approach is a fuzzification of the inputs, outputs and weights of traditional RBF Network and it can be used as an alternative to FR methods. The FRBF Network approach is constructed on the basis of minimizing the square of the total difference between observed and estimated outputs. A simple training algorithm from the cost function of the FRBF Network through Backpropagation algorithm is developed in this study. The advantage of our proposed approach is its simplicity and easy computation as well as its performance. To compare the performance of the proposed method with those given in the literature, three numerical examples are presented.
Highlights
Regression analysis is one of the most widely used methods of estimation and it is applied to determine the functional relationship between independent and dependent variables
We have reviewed the relevant articles on Fuzzy Regression and provided an computation approach to estimate Fuzzy regression (FR) models with fuzzy input and fuzzy output
We presented a new estimation approach, Fuzzy Radial Basis Function Network, for Fuzzy Regression in the case that inputs and outputs are symmetric or nonsymmetric triangular fuzzy numbers
Summary
Regression analysis is one of the most widely used methods of estimation and it is applied to determine the functional relationship between independent and dependent variables. − [Wi j ]Uα v ζ U = −α [Ypk ]Uα − [Ypk ]Uα h pj (σpj )−3 × [X pi ]Uα − [Wi j ]Uα 2 vUj. From the above expressions, the training algorithm of the proposed FRBF Network can be summarized as follows: Step 1 Determine the fuzzy weights Wi j using modified FCM algorithm given in Eqs. We considered three numerical examples to demonstrate the proposed FRBF Network approach that performs well while handling with FR model when input and outputs are triangular fuzzy numbers Using these fuzzy data, we obtain an estimated fuzzy regression equation Y = A0 + A1 Xwith fuzzy parameters A0 = (a0, c0, c0) and A1 = (a1, c1, c1).
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