On some symmetry based validity indices

Identification of the correct number of clusters and the corresponding partitioning are two important considerations in clustering. In this paper, a newly developed point symmetry based distance is used to propose symmetry based versions of six cluster validity indices namely, DB-index, Dunn-index, Generalized Dunn-index, PS-index, I-index and XB-index. These indices provide measures of "symmetricity" of the different partitionings of a data set. A Kd-tree-based data structure is used to reduce the complexity of computing the symmetry distance. A newly developed genetic point symmetry based clustering technique, GAPS-clustering is used as the underlying partitioning algorithm. The number of clusters are varied from 2 to root n where n is the total number of data points present in the data set and the values of all the validity indices are noted down. The optimum value of a validity index over these root n - 1 partitions corresponds to the appropriate partitioning and the number of partitions as indicated by the validity index. Results on five artificially generated and four real-life data sets show that symmetry distance based I-index performs the best compared to all the other five indices.