Hierarchical Clustering via Joint Between-Within Distances: Extending Ward's Minimum Variance Method

Abstract

We propose a hierarchical clustering method that minimizes a joint between-within measure of distance between clusters. This method extends Ward's minimum variance method, by defining a cluster distance and objective function in terms of Euclidean distance, or any power of Euclidean distance in the interval (0,2]. Ward's method is obtained as the special case when the power is 2. The ability of the proposed extension to identify clusters with nearly equal centers is an important advantage over geometric or cluster center methods. The between-within distance statistic determines a clustering method that is ultrametric and space-dilating; and for powers strictly less than 2, determines a consistent test of homogeneity and a consistent clustering procedure. The clustering procedure is applied to three problems: classification of tumors by microarray gene expression data, classification of dermatology diseases by clinical and histopathological attributes, and classification of simulated multivariate normal data.