T. Kowalczyk, M. Niewiadomska-Bugaj

Decomposition of Kendall's tau: implications for clustering

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Abstract

A decomposition of a generalized Kendall's tau into three components ("within", "between" and "remainder" terms) is presented. We show the maximization of the "between" term can be used in clustering and that the optimal decomposition in the case of a regular dependence of variables is non-overlapping (remainder term is equal to zero). Characterization of admissible solutions to maximization problem is proved. Finally an efficient computer-intensive procedure of optimal clustering is suggested. In the Appendix the necessary conditions for maximizing tau are formulated. Moreover, the description and justification of the proposed procedure for maximizing tau is given.

Key words: concentration index, clustering, decomposition, Kendall's tau.