The affinely invariant distance correlation

TL;DR

This paper introduces an affinely invariant version of distance correlation, establishes its consistency, and provides exact expressions for normally distributed data, with applications to wind vector time series analysis.

Contribution

It develops an affinely invariant distance correlation measure, derives its exact formulas for Gaussian vectors, and demonstrates its application to meteorological time series data.

Findings

Exact expressions for affinely invariant distance correlation in Gaussian cases.

Consistency of the empirical affinely invariant distance correlation.

Application to wind vector time series analysis.

Abstract

Sz\'{e}kely, Rizzo and Bakirov (Ann. Statist. 35 (2007) 2769-2794) and Sz\'{e}kely and Rizzo (Ann. Appl. Statist. 3 (2009) 1236-1265), in two seminal papers, introduced the powerful concept of distance correlation as a measure of dependence between sets of random variables. We study in this paper an affinely invariant version of the distance correlation and an empirical version of that distance correlation, and we establish the consistency of the empirical quantity. In the case of subvectors of a multivariate normally distributed random vector, we provide exact expressions for the affinely invariant distance correlation in both finite-dimensional and asymptotic settings, and in the finite-dimensional case we find that the affinely invariant distance correlation is a function of the canonical correlation coefficients. To illustrate our results, we consider time series of wind vectors at the Stateline wind energy center in Oregon and Washington, and we derive the empirical auto and cross distance correlation functions between wind vectors at distinct meteorological stations.