GENERALIZED ASSOCIATION PLOTS: INFORMATION VISUALIZATION VIA ITERATIVELY GENERATED CORRELATION MATRICES

Abstract

Given a p-dimensional proximity matrix Dp×p, a sequence of correlation matrices, R =( R (1) ,R (2) ,...), is iteratively formed from it. Here R (1) is the correlation matrix of the original proximity matrix D and R (n) is the correlation matrix of R (n−1) , n> 1. This sequence was first introduced by McQuitty (1968), Breiger, Boorman and Arabie (1975) developed an algorithm, CONCOR, based on their rediscovery of its convergence. The sequence R often converges to a matrix R (∞) whose elements are +1 or �1. This special pattern of R (∞) partitions the p objects into two disjoint groups and so can be recursively applied to generate a divisive hierarchical clustering tree. While convergence is itself useful, we are more concerned with what happens before convergence. Prior to convergence, we note a rank reduction property with elliptical structure: when the rank of R (n) reaches two, the column vectors of R (n) fall on an ellipse in a two-dimensional subspace. The 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 computer graphics to retrieve important information hidden in the data or proximity matrices.