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November 2014 The affinely invariant distance correlation
Johannes Dueck, Dominic Edelmann, Tilmann Gneiting, Donald Richards
Bernoulli 20(4): 2305-2330 (November 2014). DOI: 10.3150/13-BEJ558


Székely, Rizzo and Bakirov (Ann. Statist. 35 (2007) 2769–2794) and Székely 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.


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Johannes Dueck. Dominic Edelmann. Tilmann Gneiting. Donald Richards. "The affinely invariant distance correlation." Bernoulli 20 (4) 2305 - 2330, November 2014.


Published: November 2014
First available in Project Euclid: 19 September 2014

zbMATH: 1320.62133
MathSciNet: MR3263106
Digital Object Identifier: 10.3150/13-BEJ558

Keywords: affine invariance , Distance correlation , distance covariance , hypergeometric function of matrix argument , multivariate independence , multivariate normal distribution , vector time series , wind forecasting , zonal polynomial

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability


Vol.20 • No. 4 • November 2014
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