A Clustering Validity Index With Multi-Granularity Fusion for Multiple Fuzzy Clustering Algorithms

Most clustering validity indexes (CVIs) for fuzzy clustering are based upon the fuzzy c-means (FCMs) algorithm, and the effect of these CVIs is limited due to the "uniform effect" of FCM. Besides, main existing CVIs have the problems of incompleteness characterization of separateness and weak performance for noisy datasets. To address these challenges, the multi-granularity fusion (MGF) index is proposed. First, MGF synthetically considers the FCM, possibilistic fuzzy c-means and kernel-based FCM algorithms, which is more comprehensive than just considering FCM. Second, we add a perturbation to the sum of the partition matrix as the fuzzy cardinality and combine it with the fuzzy weighted distance, which are helpful to grasp the compactness. Third, four elements are considered together to characterize the separateness, incorporating the minimum distance, the maximum distance, the mean distance, and the sample variance of cluster center, where the last one can make the separateness unbiased from the macroscopic perspective. Besides, the convergence of MGF is proved. Finally, we test MGF for five algorithms on 36 datasets comparing with 14 CVIs, validating the accuracy and stability of MGF. It is observed that MGF can get superior results than other CVIs, especially for high-dimensional datasets and noisy datasets.