Abstract

Given a p-dimensional proximity matrix , a sequence^* of correlation matrices, =(,,…), is iteratively formed from it. Here is the correlation matrix of the original proximity matrix D and is the correlation matrix of , n > 1. The sequence R often converges to a matrix whose elements are +1 or -1. This special pattern of partitions the p objects into two disjoint groups and it can be recursively applied to generate a divisive hierarchical clustering tree. While convergence is itself useful, we are even more concerned with what happens before convergence. We discover that before convergence, there is a rank reduction property with elliptical structure. When rank of reaches two, the column vectors on fall on an ellipse on a two-dimensional subspace. This unique order of relative positions for the p points on the ellipse can be used to solve seriation problems such as the reordering of a Robinson matrix. A software package, Generalized Association Plots (GAP), is developed which utilizes modern computer's graphic ability to retrieve important information hidden in the data or proximity matrices.

KEY WORDS: Data visualization, Divisive clustering tree, Latent structure; Perfect symmetry; Proximity matrices, Seriation.

* This correlation sequence was first introduced by McQuitty (1968). Breiger, Boorman & Arabie (1975) also developed an algorithm, CONCOR, based on their rediscovery of the convergence of this sequence.

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