Use of symmetry and stability for data clustering

Abstract

An important consideration in clustering is the determination of an algorithm appropriate for partitioning a given data set. Thereafter identification of the correct model order and determining the corresponding partitioning need to be performed. In this paper, at first the effectiveness of the recently developed symmetry based cluster validity index named Sym-index which provides a measure of “symmetricity” of the different partitionings of a data set is shown to address all the above mentioned issues, viz., identifying the appropriate clustering algorithm, determining the proper model order and evolving the proper partitioning as long as the clusters possess the property of symmetry. Results demonstrating the superiority of the proposed cluster validity measure in appropriately determining the proper clustering technique as well as appropriate model order as compared to five other recently proposed measures, namely PS-index, I-index, CS-index, well-known XB-index, and stability based index, are provided for several clustering methods viz., two recently developed genetic algorithm based clustering techniques, the average linkage clustering algorithm, self organizing map and the expectation maximization clustering algorithm. Five artificial data sets and three real life data sets, are considered for this purpose. In the second part of the paper, a new measure of stability of clustering solutions over different bootstrap samples of a data set is proposed. Thereafter a multiobjective optimization based clustering technique is developed which optimizes both Sym-index and the measure of stability simultaneously to automatically determine the appropriate number of clusters and the appropriate partitioning of the data sets having symmetrical shaped clusters. Results on five artificial and five real-life data sets show that the proposed technique is well-suited to detect the number of clusters from data sets having point symmetric clusters.