Fuzzy and Crisp Mahalanobis Fixed Point Clusters

Abstract

Fixed point clusters (FPCs) are based on the idea that local optima of redescending M-estimators can be used to locate clusters. FPCs satisfy a fixed point condition which means that they are data subsets that do not contain any outlier and with respect to that all other points in the data set are outliers. In this paper, outliers are defined in terms of the Mahalanobis distance. Crisp FPCs (where outlyingness is defined with a 0-1 weight function) are compared to fuzzy FPCs where outliers are smoothly downweighted. An algorithm to find substantial crisp and fuzzy FPCs is proposed, the results of a simulation study and a data example are discussed.