Exact first moments of the RV coefficient by invariant orthogonal integration

TL;DR

This paper introduces an exact method to compute the first four moments of the RV coefficient under the null hypothesis using invariant orthogonal integration, applicable to various multivariate settings and weighted observations.

Contribution

It presents a novel approach, invariant orthogonal integration, to derive exact moments of the RV coefficient, including the third and fourth moments, for the first time.

Findings

Exact formulas for the first three moments of the RV coefficient.

Novel expressions for the third and fourth moments involving Weingarten calculus.

Applicable to weighted and unweighted multivariate configurations.

Abstract

The RV coefficient measures the similarity between two multivariate configurations, and its significance testing has attracted various proposals in the last decades. We present a new approach, the invariant orthogonal integration, permitting to obtain the exact first four moments of the RV coefficient under the null hypothesis. It consists in averaging along the Haar measure the respective orientations of the two configurations, and can be applied to any multivariate setting endowed with Euclidean distances between the observations. Our proposal also covers the weighted setting of observations of unequal importance, where the exchangeability assumption, justifying the usual permutation tests, breaks down. The proposed RV moments express as simple functions of the kernel eigenvalues occurring in the weighted multidimensional scaling of the two configurations. The expressions for the third and fourth moments seem original. The first three moments can be obtained by elementary means, but computing the fourth moment requires a more sophisticated apparatus, the Weingarten calculus for orthogonal groups. The central role of standard kernels and their spectral moments is emphasized.