Fuzzy clustering and switching regression models using ambiguity and distance rejects

Abstract

This paper examines how reject options can be used in performing fuzzy clustering and switching regression models. We define 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 fit 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 suffer 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 define 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 defined in Demko et al. (Actes des sixièmes rencontres del la société francophone de classification, 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.