Combinatorial data analysis: Confirmatory comparisons between sets of matrices

Abstract

AWhat's promised, you'll enjoy with naught subtracted, With naught unduly snipped off or exacted.-Mephistopheles (Translation by G. M. Priest, 1941) FUUS~ 1, 1416-1417 SUMMARYThe task of assessing the similarity of pattern between the entries of two square matrices has been discussed extensively over the last decade, as a unifying strategy for approaching a variety of seemingly disparate statistical problems. As typically defined, the comparison depends on a measure of matrix correspondence, usually a normalized cross-product measure of some form, that is evaluated for relative size by the use of a reference distribution constructed through an equally likely permutation hypothesis defined at the level of the objects corresponding to the rows and columns of the two matrices. The extreme generality provided by this very simple framework subsumes a variety of different statistical problems, ranging from the study of spatial autocorrelation for variables observed over a set of geographic locations,to the topics of analysis of variance, the measurement of rank correlation, and confirmation techniques concerned with various conjectures of combinatorial structure that might be posited for an empirically determined measure of relationship between pairs of a given set of objects. The comparison strategies extant always assume that both matrices are fixed, and in those cases where one of the matrices codifies a given theoretical structure to be evaluated according to a second, this assumption can lead to substantial arbitrariness in how matrix similarity might be indexed, and thus, in how the comparison is implemented.As developed in this paper, exactly the same principles appropriate for use in the fixed comparison context can be extended to include matrices constructed through optimally weighted linear combinations of other sets of matrices. This generalization provides one mechanism for developing comparison strategies that allow assessment against very broad classes of matrices, which in turn serve to represent very general conjectures of possible combinatorial structure. This paper reviews some of these extensions in detail, with a particular emphasis on categorical and ordered categorical variables and whether they may reflect an empirically generated measure of object relationship.

Structure and meaning in the verb lexicon: Input for a syntax-aided verb learning procedure

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Sensor-based proximity metrics for team research. A validation study across three organizational contexts

“…In order to assess the significance of the correlation coefficients between observed BT detections and selfreported measures, several reference distributions have been generated using the QAP (Brusco & Steinley, 2015;Hubert & Arabie, 1989). These simulated reference distributions provide a benchmark for the likelihood to obtain the observed correlation coefficients between BT detections and self-reported measures by chance only.…”

Global networks of trade and bits

“…Figure 6 shows these findings by displaying a comparison of the distribution of distances in the two networks. Lastly, another way of testing for the correlation between network structure and distance across countries is by a quadratic assignment procedure (QAP, see for instance Krackhardt 1987, Hubert & Arabie 1989. This entails computing the correlation between the matrix of hypergeometric cumulative probabilities P and the matrix of physical distances across countries, and comparing it with the values obtained once the rows and columns of P are randomly shuffled.…”