Fuzzy clustering and switching regression models using ambiguity and distance rejects
Marceau Michel · 2001
This paper examines how reject options can be used in performing fuzzy clustering and switching regression models. We de/ne an objective function in which reject options are introduced to optimization of certain clustering models. This approach can be directly applied to any clustering model which can be represented as a functional dependent upon a set of cluster centers. The approach can be further generalized for models that require parameters other than the cluster centers. Two types of reject have been included: (1) the ambiguity reject which concerns patterns lying near the cluster boundaries or in the case of switching regression problems, the data points which /t several models equally well; (2) the distance or error reject dealing with patterns that are far away from all the clusters. Clustering and fuzzy c-regression algorithms such as FcM (fuzzy c-means) and FcRM (fuzzy c-regression models) which use calculus-based optimization methods su5er from several drawbacks. They are very sensitive to the presence of noise. Moreover, the memberships are relative numbers. The membership of a point in a cluster depends on the membership of the point in all other clusters. So, the cluster centers or estimates for the parameters are poor. This can be a serious problem in situations where one wishes to generate membership functions from training data. This paper provides answers to these problems: to avoid the memberships to be spread across the clusters and to allow the distinction between “equally likely” and “unknown”, we de/ne partial ambiguity rejects which introduce a discounting process between the classical FcM or FcRM membership functions; to improve the performance of our algorithm in the presence of noise, we use an amorphous noise cluster de/ned in Demko et al. (Actes des sixi: emes rencontres del la soci% et% e francophone de classi/cation, Montpellier, France, September 1998). To compute these rejects, we propose an extension of FcRM algorithm (Hathaway and Bezdek, IEEE Trans. Fuzzy Systems 1 (3) (1993) 195–203). This algorithm is called the fuzzy (c+2)-regression model (Fc+2RM). Preliminary computational experiences on the developed algorithm are encouraging and compare favorably with results from other methods as FcRM and AFC algorithms on the same data sets. c © 2001 Elsevier Science B.V. All rights reserved.