Abstract
Multiple discrete (non-spatial) and continuous (spatial) structures can be fitted to a proximity matrix to increase the information extracted about the relations among the row and column objects vis-a-vis a representation featuring only a single structure. However, using multiple discrete and continuous structures often leads to ambiguous results that make it difficult to determine the most faithful representation of the proximity matrix in question. We propose to resolve this dilemma by using a nonmetric analogue of spectral matrix decomposition, namely, the decomposition of the proximity matrix into a sum of equally-sized matrices, restricted only to display an order-constrained patterning, the anti-Robinson (AR) form. Each AR matrix captures a unique amount of the total variability of the original data. As our ultimate goal, we seek to extract a small number of matrices in AR form such that their sum allows for a parsimonious, but faithful reconstruction of the total variability among the original proximity entries. Subsequently, the AR matrices are treated as separate proximity matrices. Their specific patterning lends them immediately to the representation by a single (discrete non-spatial) ultrametric cluster dendrogram and a single (continuous spatial) unidimensional scale. Because both models refer to the same data base and involve the same number of parameters, estimated through least-squares, a direct comparison of their differential fit is legitimate. Thus, one can readily determine whether the amount of variability associated which each AR matrix is most faithfully represented by a discrete or a continuous structure, and which model provides in sum the most appropriate representation of the original proximity matrix. We propose an extension of the order-constrained anti-Robinson decomposition of square-symmetric proximity matrices to the analysis of individual differences of three-way data, with the third way representing individual data sources. An application to judgments of schematic face stimuli illustrates the method.
Highlights
The structural representation of proximity data has always been an important topic in multivariate data analysis
We propose to resolve this ambiguity by using a nonmetric analogue of spectral matrix decomposition, namely, the additive decomposition of a proximity matrix into -sized matrices, constrained only to display a specific patterning among the cell entries called the anti-Robinson (AR) form
Why use the AR decomposition of the proximity matrix instead of the well-known spectral decomposition? A remarkable result in combinatorial data analysis states that fitting a unidimensional scale or an ultrametric tree structure to a square-symmetric proximity matrix generally produces matrices of distance and path length estimates that can be permuted into a perfect AR form [1]
Summary
The structural representation of proximity data has always been an important topic in multivariate data analysis (see, for example, the monograph by Hubert, Arabie, and Meulman [1]). A remarkable result in combinatorial data analysis states that fitting a unidimensional scale or an ultrametric tree structure to a square-symmetric proximity matrix generally produces matrices of distance and path length estimates that can be permuted into a perfect AR form [1] Within this context, order-constrained AR matrix decomposition attains the status of a combinatorial data analytic meta-technique. Each extracted AR component relates immediately to the representation by a single unidimensional scale and a single ultrametric tree As both models are directly comparable for each AR matrix, we can determine which structure provides a superior representation of the associated amount of variability—a feature not available when fitting multiple structures to a data matrix without order-constrained AR decomposition. We conclude with an illustrative application of three-way order-constrained AR matrix decomposition for analyzing the structure of individual differences in judgments of schematic face stimuli
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