Fuzzy Partition Clustering Algorithms Based on Alternative Mahalanobis Distances

Abstract

The well-known fuzzy partition clustering algorithms are mainly based on Euclidean distance measure for partitioning, which can only be used for the clusters in the data set with the same super-spherical shape distribution. Instead of using Euclidean distance measure, Gustafson & Kessel (1979) proposed the G-K algorithm which employs the Mahalanobis distance. It is a fuzzy partition clustering algorithm which can be used for the clusters in the data set with different geometrical shapes. However, without the prior information of the shape volume for each class, the G-K algorithm can only be utilized for the clusters with the same volume in the data set. In other words, if any dimension of a class is greater than the number of samples in the class, the estimated covariance matrix of that class may not be fully ranked. Hence, the algorithm will induce the singular problem for the inverse covariance matrix. This is an important issue need be addressed when we use the G-K algorithm for clustering. To overcome the issues, a new solution is proposed. A regulating factor of the covariance matrix for each class and the alternative global scatter matrix are added in the objective function, besides, the constraint of the determinant of the covariance matrices used in the G-K algorithm is removed. This new proposed algorithm is called Liualgorithm. Based on the proposed Liu-algorithm, three well known fuzzy partition clustering algorithms using Euclidean distance measure; the Fuzzy C-Means (FCM), the Possibility C-Means (PCM), and the Fuzzy Possibility C-Means (FPCM), are extended by using the local and global Mahalanobis distance. They will be called the Fuzzy C-Means based on Alternative Mahalanobis distances (FCM-AM), the Possibility C-Means based on Alternative Mahalanobis distances (PCM-AM), the Fuzzy Possibility C-Means based on Alternative Mahalanobis distances (FPCM-AM), respectively.