Complex fuzzy c-means algorithm

Abstract

In this paper a new clustering algorithm is presented: A complex-based Fuzzy c-means (CFCM) algorithm. While the Fuzzy c-means uses a real vector as a prototype characterizing a cluster, the CFCM's prototype is generalized to be a complex vector (complex center). CFCM uses a new real distance measure which is derived from a complex one. CFCM's formulas for the fuzzy membership are derived. These formulas are extended to derive the complex Gustafson---Kessel algorithm (CGK). 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 Beni's Index, Dunn's Index. It is shown in this paper that the CFCM give better partitions of the data than the FCM and the GK algorithms. It is also shown that the CGK algorithm outperforms the CFCM but at the expense of much higher computational complexity.