Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance

TL;DR

This paper introduces distance multivariance as a unifying framework for dependence measures, extending existing methods, defining new dependence metrics, and providing visualization tools with practical R implementations.

Contribution

It unifies various dependence measures under a single concept, introduces new dependence metrics and tests, and offers visualization methods with an R package.

Findings

Distance multivariance unifies and extends dependence measures.

New dependence metrics and tests are computationally feasible and consistent.

A visualization scheme for higher order dependencies is proposed.

Abstract

Distance multivariance is a multivariate dependence measure, which can detect dependencies between an arbitrary number of random vectors each of which can have a distinct dimension. Here we discuss several new aspects, present a concise overview and use it as the basis for several new results and concepts: In particular, we show that distance multivariance unifies (and extends) distance covariance and the Hilbert-Schmidt independence criterion HSIC, moreover also the classical linear dependence measures: covariance, Pearson's correlation and the RV coefficient appear as limiting cases. Based on distance multivariance several new measures are defined: a multicorrelation which satisfies a natural set of multivariate dependence measure axioms and $m$-multivariance which is a dependence measure yielding tests for pairwise independence and independence of higher order. These tests are computationally feasible and under very mild moment conditions they are consistent against all alternatives. Moreover, a general visualization scheme for higher order dependencies is proposed, including consistent estimators (based on distance multivariance) for the dependence structure. Many illustrative examples are provided. All functions for the use of distance multivariance in applications are published in the R-package 'multivariance'.