Measuring and testing dependence by correlation of distances

By G'abor J. Sz'ekely, Maria L. Rizzo, N. K. Bakirov. 2007

2,979 citations recorded by the source corpus.

Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented.

Subfields: Finance, Food Science, Statistics and Probability

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “The distance covariance was later introduced in Székely et al. (2007), Székely and Rizzo (2009) for an independence test as a weighted L2-distance between characteristic functions of the joint and product distributions.”
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.”
Supervised Dimensionality Reduction via Distance Correlation Maximization (2016, methodology): “Our contribution in this paper is as follows: We propose a new formulation for supervised dimensionality reduction that is based on a dependency criterion called Distance Correlation, [Szekely et al., 2007].”
Invariant Causal Prediction for Nonlinear Models (2017, background): “There has been some progress towards nonparametric independence tests (Bergsma and Dassios, 2014; Hoeﬀding, 1948; Blum et al., 1961; R´enyi, 1959; Sz´ekely et al., 2007; Zhang et al., 2011).”
A low variance consistent test of relative dependency (2014, background): “, 2008), distance covariances (which are instances of kernel tests) (Székely et al., 2007; Sejdinovic et al., 2013b), kernel regression tests (Cortes et al.”
On the Decreasing Power of Kernel and Distance Based Nonparametric Hypothesis Tests in High Dimensions (2014, methodology): “…developed in parallel based on pairwise distances between points, as exempliﬁed for independence testing by distance correlation, introduced in (Sz´ekely, Rizzo, and Bakirov 2007) and further discussed or extended in (Sz´ekely and Rizzo 2009; Lyons 2013; Sz´ekely and Rizzo 2013; Sejdinovic et…”
On the optimal estimation of probability measures in weak and strong topologies (2013, background): “ Pand Qfor the choices of k(·,x),x∈ Rd in (12). Recently, it has been shown [30, Theorem 22] that choosing K = ˆ k(x,y) = 1 2 (ρ(x,z) +ρ(y,z)−ρ(x,y)),x,y,z∈ X ˙ yields the generalized energy distance [38, 24] where ρis a semimetric of negative type on X (see [24] and [30] for details) and zis somear”
A Kernel Test for Three-Variable Interactions (2013, methodology): “Much of the work in this area focuses on measuring or testing pairwise interaction: for instance, the Hilbert-Schmidt Independence Criterion (HSIC) or Distance Covariance [1, 2, 3], kernel canonical correlation [4, 5, 6], and mutual information [7].”
Hypothesis testing using pairwise distances and associated kernels (2012, methodology): “Second, results from Gretton et al. (2009a); Zhang et al. (2011) may be applied to get consistent two-sample and independence tests for the energy distance, without using bootstrap, which perform much better than the upper bound proposed by Székely et al. (2007) as an alternative to the bootstrap.”
A simpler condition for consistency of a kernel independence test (2015, background): “The distance covariance of [12] is a special case, for a particular choice of kernel [8].”
B-tests: Low Variance Kernel Two-Sample Tests (2013, methodology): “While we confine ourselves to the MMD in this paper, we emphasize that the block approach applies to a much broader variety of test situations where the null distribution cannot easily be computed, including the energy distance and distance covariance [18, 2, 22] and Fisher statistic [12] in the cas”
Large-scale kernel methods for independence testing (2016, methodology): “A closely related dependence coefficient that measures all types of dependence between two random vectors of arbitrary dimensions is the distance covariance (dCov) of [44, 43], which measures distances between empirical characteristic functions or equivalently measures covariances with respect to a ”
Multivariate nonparametric test of independence (2017, methodology): “Here Vn is the empirical distance covariance and is discussed in [31].”