Fuzzy C-mean algorithm based on “complete” Mahalanobis distances

Abstract

The well known fuzzy partition clustering algorithms are most based on Euclidean distance function, which can only be used to detect spherical structural clusters. Gustafson-Kessel (GK) clustering algorithm and Gath-Geva (GG) clustering algorithm, were developed to detect non-spherical structural clusters, but both of them based on semi-supervised Mahalanobis distance, these two algorithms fail to consider the relationships between cluster centers in the objective function, needing additional prior information. When some training cluster size is small than its dimensionality, it induces the singular problem of the inverse covariance matrix. It is an important issue. In our previous study, we developed a new unsupervised algorithm, FCM-M, to solve the singular problem of the inverse covariance matrix. But, the previous work only consider the local covariance matrix of each cluster. In this paper, an improved new unsupervised algorithm, ldquofuzzy c-mean based on complete Mahalanobis distance without any prior information (FCM-CM)rdquo, is proposed. The proposed new algorithm which is considered not only the local covariance matrix of each cluster but also the overall covariance matrix, which can get more information and higher accuracy by considering the overall covariance matrix. A real data set was applied to prove that the performance of the FCM-CM algorithm is better than those of the traditional FCM algorithm and our previous FCM-M. For choosing the better initial value of the same new algorithm, FCM-CM, the ratio method is still the best of all choosing methods by using in FCM-M and FCM algorithms in our previous works.