# Distance covariance in metric spaces

@article{Lyons2013DistanceCI, title={Distance covariance in metric spaces}, author={Russell Lyons}, journal={Annals of Probability}, year={2013}, volume={41}, pages={3284-3305} }

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… Expand

#### 112 Citations

On distance covariance in metric and Hilbert spaces

- Mathematics
- 2019

Distance covariance is a measure of dependence between two random variables that take values in two, in general different, metric spaces, see Szekely, Rizzo and Bakirov (2007) and Lyons (2013). It is… Expand

On the uniqueness of distance covariance

- Mathematics
- 2012

Distance covariance and distance correlation are non-negative real numbers that characterize the independence of random vectors in arbitrary dimensions. In this work we prove that distance covariance… Expand

Distance-based and RKHS-based dependence metrics in high dimension

- Mathematics
- 2019

In this paper, we study distance covariance, Hilbert-Schmidt covariance (aka Hilbert-Schmidt independence criterion [Gretton et al. (2008)]) and related independence tests under the high dimensional… Expand

A New Framework for Distance and Kernel-based Metrics in High Dimensions

- Mathematics
- 2019

The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based… Expand

Partial Distance Correlation with Methods for Dissimilarities

- Mathematics
- 2013

Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions, and applications of distance… Expand

Fréchet analysis of variance for random objects

- Mathematics
- Biometrika
- 2019

Fréchet mean and variance provide a way of obtaining a mean and variance for metric space-valued random variables, and can be used for statistical analysis of data objects that lie in abstract… Expand

Distance correlation coefficients for Lancaster distributions

- Computer Science, Mathematics
- J. Multivar. Anal.
- 2017

This work derives under mild convergence conditions a general series representation for the distance covariance and distance correlation coefficients for the bivariate normal distribution and its generalizations of Lancaster type, the multivariate normal distributions, and the b correlations between random vectors whose joint distributions belong to the class of Lancaster distributions. Expand

Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation

- Mathematics
- Journal of the American Statistical Association
- 2021

In this paper, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure… Expand

Expectations of Random Sets

- Mathematics
- 2017

The space \(\mathbb{F}\) of closed sets (and also the space \(\mathbb{K}\) of compact sets) is non-linear, so that conventional concepts of expectations in linear spaces are not directly applicable… Expand

Hotelling's T2 in separable Hilbert spaces

- Computer Science, Mathematics
- J. Multivar. Anal.
- 2018

We address the problem of finite-sample null hypothesis significance testing on the mean element of a random variable that takes value in a generic separable Hilbert space. For this purpose, we pro… Expand

#### References

SHOWING 1-10 OF 48 REFERENCES

On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space

- Mathematics
- 1937

1. W. A. Wilson ([9])2 has recently investigated those metric spaces which arise from a metric space by taking as its new metric a suitable (one variable) function of the old one. He considered in… Expand

On potentials of measures in Banach spaces

- Mathematics
- 1987

in M with respect to the Kobayashi metric, such that the closure of V in M is compact (the existence of such a ball follows from the condition that M is hyperbolic), U be a ball of the same radius in… Expand

Metric spaces and positive definite functions

- Mathematics
- 1938

As poo we get the space Em with the distance function maxi-, ... I xi X. Let, furthermore, lP stand for the space of real sequences with the series of pth powers of the absolute values convergent.… Expand

Positive definite metric spaces

- Mathematics
- 2013

Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of… Expand

Isometric operators in vector-valued LP-spaces

- Mathematics
- 1987

The note is devoted to the generalization of A. I. Plotkin's theorem on the equimeasurability of a function and of its image under a LP-isometry to the case of isometric operators, acting in certain… Expand

L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry

- Mathematics, Computer Science
- ArXiv
- 2010

This work surveys connections between the theory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization and explains how the key ideas evolved over the past 20 years, emphasizing the interactions with Banach space theory, geometric measure theory, and geometric group theory. Expand

DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I

- Mathematics
- 2009

Let ( X , d ) be a compact metric space and let ℳ( X ) denote the space of all finite signed Borel measures on X . Define I :ℳ( X )→ℝ by and set M ( X )=sup I ( μ ), where μ ranges over the… Expand

Brownian distance covariance

- Mathematics
- 2009

Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation… Expand

Convergence of probability measures

- Mathematics
- 2011

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the… Expand

Measuring and testing dependence by correlation of distances

- Mathematics
- 2007

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… Expand