Fuzzy Clustering and Robust Estimation

A modified version of the fuzzy k-means clustering is proposed for gelling the robustness (the resistance) against a few outliers. Some numerical examples are presented for illustrating the intuitively appropriate interpretation the modified method provides, and it is pointed out that the estimator of the typical value (the population mean) obtained in one-cluster case is equivalent to a kind of M -estimator. is added to (1)~(3), the resulting solution reduces to the usual partitioning of n observations to k clusters (hard clustering). The easiest-to-implement technique of the fuzzy clustering is the objective functional clustering which minimizes the appropriately chose functional J = J(U,X,some parameters), where V = ( v, v, "', v) is the set of the "typical val uc" vectors of ....... \ -.2 -.k reduces to that of the hard k-means clustering, and as m approaches to infinity, the solution approaches to the most vague status the clusters, d(x..', z:;) = 11x..' z:;11 is an arbitrary inner product norm (usually Enclidean distance) and m is the tuning constant which determines the vagueness (fuzziness) of the solution. When m = 1, the solution of and the technique is regarded as a generalization of its hard version such as k-means clustering (IS0DA T A) and Friedman and Rubin' s method (see, for example, Everitt (1974)). The fuzzy k-means clustering (Bezdek (1981), Chap. 3) is a direct generalization of the k-means clustering, and adopts as the functional the least squares error criterion