Determining the Number of Clusters Using Multivariate Ranks

Abstract

Determining number of clusters in a multivariate data has become one of the most important issues in very diversified areas of scientific disciplines. The forward search algorithm is a graphical approach that helps us in this task. The traditional forward search approach based on Mahalanobis distances has been introduced by Hadi (1992), Atkinson (1994), while Atkinson et al. (2004) used it as a clustering method. But like many other Mahalanobis distance-based methods, it cannot be correctly applied to asymmetric distributions and more generally, to distributions which depart from the elliptical symmetry assumption. We propose a new forward search methodology based on spatial ranks, where clusters are grown with one data point at a time sequentially, using spatial ranks with respect to the points already in the subsample. The algorithm starts from a randomly chosen initial subsample. We illustrate with simulated data that the proposed algorithm is robust to the choice of initial subsample and it performs well in different mixture multivariate distributions. We also propose a modified algorithm based on the volume of central rank regions. Our numerical examples show that it produces the best results under elliptic symmetry.