Clustering with complex centers

Abstract

Clustering analysis is the process of separating data according to some similarity measure. A cluster consists of data which are more similar to each other than to other clusters. The similarity of a datum to a certain cluster can be defined as the distance of that datum to the prototype of that cluster. Typically, the prototype of a cluster is a real vector that is called the center of that cluster. In this paper, the prototype of a cluster is generalized to be a complex vector (complex center). A new distance measure is introduced. New formulas for the fuzzy membership and the fuzzy covariance matrix are introduced. Cluster validity measures are used to assess the goodness of the partitions obtained by the complex centers compared those obtained by the real centers. The validity measures used in this paper are the partition coefficient, classification entropy, partition index, separation index, Xie and Benis index, and Dunns index. It is shown in this paper that clustering with complex prototypes will give better partitions of the data than using real prototypes.