The affinely invariant distance correlation

By Johannes Dueck, Dominic Edelmann, Tilmann Gneiting, Donald St. P. Richards. 2012

43 citations recorded by the source corpus.

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

Subfields: Economics and Econometrics, Finance

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Citations in the corpus

NoPeek: Information leakage reduction to share activations in distributed deep learning (2020, methodology): “Our approach is based on minimizing a statistical dependency measure called distance correlation [10, 11, 12, 13, 14] between raw data and any intermediary communications across the clients or server participating in distributed deep learning.”
Applications of distance correlation to time series (2016, background): “In the 1970s and 1980s, Feuerverger and Mureika (1977), Csörgő (1981a,1981b,1981c) and many others proved fundamental asymptotic results for iid sequences, including Donsker-type theory for the empirical characteristic function.”
Partial Distance Correlation with Methods for Dissimilarities (2013, unknown): “Properties, extensions and applications of distance correlation have been discussed in the recent literature; see, for example, Dueck et al. [4], Lyons [16], Kong et al. [12] and Li, Zhong and Zhu [15].”
Statistical Dependence: Beyond Pearson’s ρ (2018, background): “It has been extended to time series and multiple dependencies by Davis et al. (2018), Fokianos and Pitsillou (2017), Zhou (2012), and Dueck et al. (2014), Dueck, Edelman, and Richards (2015) and Yao, Zhang, and Shao (2018).”
Partial martingale difference correlation (2015, background): “[2], Shao and Zhang [10], and Székely and Rizzo [14] among others for further extensions and applications of these concepts.”
R EDUCING LEAKAGE IN DISTRIBUTED DEEP LEARNING FOR SENSITIVE HEALTH DATA (2019, background): “and also alternate versions of distance correlation that achieve affine invariance exist Dueck et al. (2014). These are highly suitable properties for computer vision given that a leakage reduction measure should be able to find a representation beyond a simple orthogonal group transformation. (1)Se”