Distance covariance in metric spaces

By Russell Lyons. 2011

102 citations recorded by the source corpus.

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Szekely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Minimax Estimation of Kernel Mean Embeddings (2016, background): “s (2π)−d/2e− √ −1h·,xi, eh·,xi, and (4π)−d/2e−k·−xk22,x∈ Rd respectively reduces µkto these notions. The mean embedding µkis closely related to the generalized energy distance (Sz´ekely et al., 2007; Lyons, 2013), since the latter is shown (Sejdinovic et al., 2013, Theorem 22) to be equivalent to th”
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 New Coefficient of Correlation (2019, background): “…1959; Schweizer and Wolff 1981; Dette, Siburg, and Stoimenov 2013; Lopez-Paz, Hennig, and Schölkopf 2013; Zhang 2019), and coefficients based on pairwise distances (Friedman and Rafsky 1983; Székely, Rizzo, and Bakirov 2007; Székely and Rizzo 2009; Heller, Heller, and Gorfine 2013; Lyons 2013).”
Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, background): “Moreover, the −dU(n) (u, u′) is symmetric, nonnegative definite (Lyons 2013, p. 3291), and equicontinuous since |−dU(n) (u, u′)− (−dU(n) (u′′′, u′′))| ≤ 2‖u − u′′′‖+ 2‖u′ − u′′‖.”
Fréchet analysis of variance for random objects (2017, background): “This framework was extended to the case of metric space valued random variables in Lyons (2013) for metric spaces of strong negative type, leading to the distinction of measures that have finite first moments with respect to the underlying metric.”
MicrobiomeAnalyst 2.0: comprehensive statistical, functional and integrative analysis of microbiome data (2023, unknown): “For statistical correlation analysis, the default option is the distance-based correlation method which can detect both linear and non-linear correlations ( 35 ).”
An Updated Literature Review of Distance Correlation and Its Applications to Time Series (2017, unknown): “Lyons (2013) showed that this measure characterizes independence if the underlying metric spaces are of strong negative type; in particular, this class of sets includes all separable Hilbert spaces.”
A Statistically and Numerically Efficient Independence Test Based on Random Projections and Distance Covariance (2017, background): “2 Fast Algorithm in the Univariate Cases The paper [10] gives an equivalent definition for the distance covariance between random variables X and Y:”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, background): “5) or results from degenerate V-statistics (as used in (Lyons 2013, theor.”