Analysis of Parameter Selection for Gustafson–Kessel Fuzzy Clustering Using Jacobian Matrix

In fuzzy clustering, the fuzzy c-means (FCM) is the most known algorithm. Several extensions and variations of FCM had been proposed in the literature. The first important extension to FCM was proposed by Gustafson and Kessel (GK). In the GK fuzzy clustering, they considered the effect of different cluster shapes except for spherical shapes by replacing the Euclidean distance of the FCM objective function with the Mahalanobis distance. The GK algorithm has become one of the most frequently used clustering algorithms. Just like FCM, the fuzziness index m is a parameter in which the value will greatly influence the performance of the GK algorithm. However, there is no theoretical work on the parameter selection for the fuzziness index m of GK. In this paper, we reveal the relation between the stable fixed points of the GK algorithm and the datasets using Jacobian matrix analysis, and then provide a theoretical base for selecting the fuzziness index m in the GK algorithm. Some experimental results verify the effectiveness of our theoretical results.