Fuzzy Clustering by Quadratic Regularization

Abstract

A new method of fuzzy c-means using a quadratic term of the membership metrix components as a regularizing function is proposed. This method is accordingly called a quadratic regularization. It should be noted that the standard fuzzy c-means can be ragarded as a regularization of the crisp k-means, and moreover another regularization method by using an entropy term has already been proposed. An algorithm for calculating membership values is derived, whereas the formula for cluster centers is similar to the standard method of fuzzy c-means. The present method of quadratic regularization yields a piecewise linear function of fuzzy classification. We have thus the third method of fuzzy c-means in addition to the standard method and the entropy method previously proposed.