Deterministic annealing Gustafson-Kessel fuzzy clustering algorithm

The Gustafson and Kessel (GK) fuzzy clustering algorithm, proposed by Gustafson and Kessel in 1979, was the first important extension to the fuzzy c-means (FCM) algorithm. Up to now, the GK algorithm had become one of the most commonly used fuzzy clustering algorithms, where the Mahalanobis distance is used as a dissimilarity measure to provide more effectiveness and robustness than the FCM algorithm. Recently, Chaomurilige et al. (2015) proposed a theoretical analysis on the parameter selection for the GK algorithm in which they indicated that the parameter of the fuzziness index heavily influences the performance of the GK algorithm. In this paper we propose a novel GK fuzzy clustering algorithm based on the deterministic annealing approach for decreasing the effect of parameters. We first consider maximizing the Shannon's entropy of membership functions to the GK objective function, and then use deterministic annealing to adjust the annealing parameter. We also mathematically provide a theoretical initialization lower bound for the annealing parameter of the proposed deterministic annealing GK (DA-GK) algorithm. Comparisons between the DA-GK algorithm and other methods are made. The computational complexity of the proposed method is also provided. Experimental results and comparisons actually verify theoretical results and also indicate the superiority and effectiveness of the proposed DA-GK algorithm.