Based on a simple nonparametric procedure for comparing two proximity matrices, a measure of concordance is introduced that is appropriate when K independent proximity matrices are available. In addition to the development of a general concept of concordance and specific techniques for its evaluation within and between the subsets of a partition of the K matrices, several methods are also suggested for comparing and/or for fitting a particular structure to the given data. Finally, brief indications are provided as to how the well-known notion of concordance for K rank orders can be included within the more general framework.

This paper discusses rowwise matrix correlation, based on the weighted sum of correlations between all pairs of corresponding rows of two proximity matrices, which may both be square (symmetric or asymmetric) or rectangular. Using the correlation coefficients usually associated with Pearson, Spearman, and Kendall, three different rowwise test statistics and their normalized coefficients are discussed, and subsequently compared with their nonrowwise alternatives like Mantel's Z. It is shown that the rowwise matrix correlation coefficient between two matrices X and Y is the partial correlation between the entries of X and Y controlled for the nominal variable that has the row objects as categories. Given this fact, partial rowwise correlations (as well as multiple regression extensions in the case of Pearson's approach) can be easily developed.

A combinatorial data analysis strategy is reviewed that is designed to compare two arbitrary measures of proximity defined between the objects from some set. Based on a particular cross-product definition of correspondence between these two numerically specified notions of proximity (typically represented in the form of matrices), extensions are then pursued to indices of partial association that relate the observed pattern of correspondence between the first two proximity measures to a third. The attendant issues of index normalization and significance testing are discussed; the latter is approached through a simple randomization model implemented either through a Monte Carlo procedure or distributional approximations based on the first three moments. Applications of the original comparison strategy and its extensions to partial association may be developed for a variety of methodological and substantive tasks. Besides rank correlation, we emphasize the topics of spatial autocorrelation for one variable and spatial association between two and mention the connection to the usual randomization approach for one-way analysis-of-variance.

A comprehensive statistical framework is presented which encompasses a wide range of existing nonparametric methods. The basic strategy, referred to as linear assignment (LA), depends on a simple index of correspondence defined between two object sets that have been matched in some a priori manner. In this broad sense, LA can be interpreted as a general correlational technique. A variety of extensions are discussed along with the attendant problems of significance testing and the construction of normalized descriptive indices.

Numerical estimators of differential entropy and mutual information can be slow to converge as sample size increases. The offset Kozachenko–Leonenko (KLo) method described here implements an offset version of the Kozachenko–Leonenko estimator that can markedly improve convergence. Its use is illustrated in applications to the comparison of trivariate data from successive scene color images and the comparison of univariate data from stereophonic music tracks. Publicly available code for KLo estimation of both differential entropy and mutual information is provided for R, Python, and MATLAB computing environments at https://github.com/imarinfr/klo.