Fuzzy Sets and Systems 119 (2001) 291–301 www.elsevier.com/locate/fss Fuzzy regression with radial basis function network Chi-Bin Cheng a , E. Stanley Lee b ; ∗ a Department of Industrial Engineering and Management, Chao Yang University of Technology, Tai-Chung County, Taiwan b Department of Industrial and Manufacturing Systems Engineering, Kansas State University, Manhattan, KS 66506, USA Received June 1998; received in revised form December 1998 Abstract Radial basis function network is used in fuzzy regression analysis without predened functional relationship between the input and the output. The proposed approach is a fuzzication of the connection weights between the hidden and the output layers. This fuzzy network is trained by a hybrid learning algorithm, where self-organized learning is used for training the parameters of the hidden units and supervised learning is used for updating the weights between the hidden and the output layers. The c-mean clustering method and the k -nearest-neighbor heuristics are used for the self-organized learning. The supervised learning is carried out by solving a linear possibilistic programming problem. Techniques for the generalization of the network are also proposed. Numerical examples are used to illustrate and to test the performances of the approach. c  2001 Elsevier Science B.V. All rights reserved. Keywords: Regression analysis: Nonparametric fuzzy regression; Fuzzy radial basis network 1. Introduction Multilayered neural networks are very powerful function approximators. Many investigators have ap- plied these network, such as the back propagation network [19], radial basis function network [16], and the projection pursuit network [7], to the nonparamet- ric regression analysis. The traditional back propagation networks have also been directly used or modied for nonparametric fuzzy regression. In general, the node functions are dened by the use of the extension principle [27] and the learning algorithms are derived by using the gradi- ent descent approach based on error measurement be- tween the estimated and the target outputs. Based on * Corresponding author. the connection weights used, these approaches can be divided into the use of nonfuzzy connection weights and the use of fuzzy connection weights. The former approaches include the investigations by Ishibuchi and Tanaka [11], and Fedrizzi et al. [6]; and the latter approaches include the works of Ishibuchi and coworkers [12], Miyazaki et al. [15], Ishibuchi and Nii [10], and Ishibuchi et al. [9]. In addition, Pokorny [17] applied the Sugeno fuzzy model [22] to fuzzy nonlinear regression, where the crisp linear functions in the consequent section of the Sugeno model are replaced by the possibilistic linear functions. In this paper, the radial basis function (RBF) net- work is fuzzied and is applied to the fuzzy non- parametric regression analysis. This fuzzy network is trained by a hybrid learning algorithm, where the hid- den units are learned exclusively from the inputs as 0165-0114/01/$ - see front matter c  2001 Elsevier Science B.V. All rights reserved. PII:S0165-0114(99)00098-6