Partial Distance Correlation with Methods for Dissimilarities

By G'abor J. Sz'ekely, Maria L. Rizzo. 2013

349 citations recorded by the source corpus.

Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions, and applications of distance correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation we introduce a new Hilbert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hilbert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.

Subfields: Statistics and Probability

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Citations in the corpus

Kernel‐based tests for joint independence (2016, background): “…et al., 2005, 2007), and other related approaches for estimating or testing pairwise (in)dependence including distance correlations (Székely and Rizzo, 2009, 2014), rank-based correlations (Bergsma and Dassios, 2014; Leung and Drton, 2016; Nandy et al., 2016) or also non-parametric and…”
The conditional permutation test (2018, background): “There is also a large literature on extending measures of marginal independence to the conditional setting, including partial distance covariance [24]; conditional mutual information [15]; characteristic functions [21]; Hellinger distances [22]; and smoothed empirical likelihoods [23].”
Energy distance (2016, methodology): “The inner product notation is due to the fact that this statistic is an inner product in the Hilbert space of U-centered distance matrices.(31) A bias corrected R(2)n is defined by normalizing the inner product statistic with the bias corrected dVar statistics.”
Applications of distance correlation to time series (2016, methodology): “Székely et al.1 (Székely et al. (2007), Székely and Rizzo (2014), Székely and Rizzo (2009), see also the references therein) developed asymptotic techniques for the empirical distance covariance and correlation of iid sequences for the infinite measure µ given by µ(ds, dt) =…”
Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, background): “Definition 1 in Székely and Rizzo (2013), Equation (3.2) in Székely and Rizzo (2014), Definition 5.3 (Ustatistic) in Jakobsen (2017), and Definition 3.2 are equivalent.”
The conditional permutation test for independence while controlling for confounders (2018, background): “…a large literature on extending measures of marginal independence to the conditional setting, including partial distance covariance (Székely and Rizzo, 2014), conditional mutual information (Runge, 2018), characteristic functions (Su and White, 2007), Hellinger distances (Su and White, 2008)…”
Testing mutual independence in high dimension via distance covariance (2016, methodology): “X, Y/ for i = 1, : : : , n, we adopt the idea of U-centring in Székely and Rizzo (2014) and Park et al. (2015) to construct an unbiased distance covariance estimator.”
On Azadkia–Chatterjee’s conditional dependence coefficient (2021, background): “20) in Tsybakov (2009) states that total variation and Hellinger distances satisfy TV(H1,n(∆0),H0) ≤ HL(H1,n(∆0),H0).”