Paper #154

Title:

Unfolding a symmetric matrix

Authors:

John C. Gower and Michael Greenacre

Date:

January 1996

Abstract:

Graphical displays which show inter--sample distances are important for the interpretation and presentation of multivariate data. Except when the displays are two--dimensional, however, they are often difficult to visualize as a whole. A device, based on multidimensional unfolding, is described for presenting some intrinsically high--dimensional displays in fewer, usually two, dimensions. This goal is achieved by representing each sample by a pair of points, say $R_i$ and $r_i$, so that a theoretical distance between the $i$-th and $j$-th samples is represented twice, once by the distance between $R_i$ and $r_j$ and once by the distance between $R_j$ and $r_i$. Self--distances between $R_i$ and $r_i$ need not be zero. The mathematical conditions for unfolding to exhibit symmetry are established. Algorithms for finding approximate fits, not constrained to be symmetric, are discussed and some examples are given.

Keywords:

Dimensionality reduction, distances, graphics, multidimensional scaling, symmetric matrices, unfolding

JEL codes:

C19, C88

Area of Research:

Statistics, Econometrics and Quantitative Methods

Published in:

Journal of Classification, 13, (1996), pp. 81-105

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