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Brownian distance covariance

Gábor J. Székely, Maria L. Rizzo

arXiv:1010.0297v2stat.AP

TL;DR

Classical covariance and correlation do not generally characterize dependence or independence for multivariate, nonlinear, or nonmonotone relationships, while some multivariate methods require restrictive dimensional or distributional conditions. The paper introduces distance and Brownian covariance, showing that distance correlation applies across arbitrary dimensions, characterizes independence, and coincides with covariance based on Brownian motion. Its associated independence test can be conservative for many distributions despite having a sharp significance bound.

  • Problem

    Pearson covariance and correlation do not characterize independence for general multivariate, nonlinear, or nonmonotone dependence, and some existing multivariate methods fail when dimensions exceed sample size or distributional assumptions do not hold.

  • Method

    The paper defines dependence coefficients for random vectors of arbitrary dimensions and introduces covariance with respect to stochastic processes, including Brownian motion.

  • Results

    Distance correlation is zero if and only if the random vectors are independent, and population distance covariance equals covariance with respect to Brownian motion.

  • Takeaways & Limitations

    Brownian covariance provides a natural multivariate counterpart to product-moment covariance, while distance correlation extends dependence assessment beyond linear or monotone relationships.

  • Takeaways & Limitations

    The proposed test criterion can be conservative for many distributions, although its upper significance bound is sharp.

Abstract

from arXiv · show

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 are analogous to product-moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. Distance correlation characterizes independence: it is zero if and only if the random vectors are independent. The notion of covariance with respect to a stochastic process is introduced, and it is shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product-moment covariance, as we obtain Pearson product-moment covariance by replacing the Brownian motion in the definition with identity. The corresponding statistic has an elegantly simple computing formula. Advantages of applying Brownian covariance and correlation vs the classical Pearson covariance and correlation are discussed and illustrated.

1. Introduction.

The paper introduces distance correlation and Brownian covariance as multivariate dependence measures for arbitrary-dimensional random vectors, addressing limitations of classical correlation and likelihood-based methods. These measures characterize independence and extend Pearson covariance and correlation through a stochastic-process framework.

  • Motivation: Classical correlation and covariance do not generally characterize independence, because nonlinear or nonmonotone dependence can remain undetected.This limitation persists beyond the bivariate normal setting, where zero correlation is equivalent to independence.
  • Motivation: Likelihood-ratio methods may fail when dimension exceeds sample size or distributional assumptions are violated, while many rank methods target only linear or monotone dependence.The paper therefore seeks a dependence measure applicable to broader multivariate settings.
  • Contributions: Distance correlation applies to random vectors of arbitrary dimension and measures dependence between vectors without requiring equal dimensions.The proposed statistics support sample sizes n ≥2, are not constrained by dimension, and avoid matrix inversion or parameter estimation.
  • Contributions: Covariance with respect to a stochastic process unifies distance covariance and Pearson covariance as distinct special cases.Brownian motion yields Cov_W(X,Y)=V²(X,Y), whereas the identity function yields Pearson covariance in the bivariate case.
  • Implications: Distance-correlation-based zero-dependence conditions support a central limit theorem for strongly stationary sequences, unlike uncorrelatedness alone.The paper also presents theoretical foundations through characteristic functions and Brownian-motion covariance.

2. Distance covariance and distance correlation.

Distance covariance measures dependence through characteristic-function distance, while distance correlation standardizes it and equals zero exactly under independence. The resulting statistics apply in arbitrary dimensions without distributional assumptions or matrix inversion.

  • Definitions and motivation: Distance covariance measures the distance between the joint characteristic function fX,Y and the product fXfY, yielding an empirical independence test.The test evaluates H0: fX,Y = fXfY versus H1: fX,Y ≠ fXfY.
  • Weight functions: A suitable nonintegrable weight function makes the dependence coefficient zero if and only if X and Y are independent.Integrable weights can produce values arbitrarily close to zero for dependent variables, whereas the selected weights retain the independence characterization.
  • Statistical construction: The coefficients are scale invariant and have simple product-average forms based on pairwise sample distances.The corresponding statistics work for n ≥ 2 without constraining sample size by dimension, matrix inversion, or parameter estimation.
  • Inference: The dCov independence test is statistically consistent against all dependent alternatives with finite first moments.Under independence the statistic has a proper limit distribution; under dependent alternatives it tends to infinity stochastically.
  • Inference: The asymptotic critical-value test can be conservative, although its upper significance-level bound is sharp.The bound is achieved for independent Bernoulli variables.

Results for the bivariate normal distribution.

For bivariate normal variables, the distance correlation R is related deterministically to Pearson correlation ρ, and the relationship is illustrated in Figure 1.

  • Results for the bivariate normal distribution: For standard bivariate normal variables, the relation between distance correlation R and Pearson correlation ρ is deterministic.Figure 1 displays this relationship for the bivariate normal case.

3. Brownian covariance.

Brownian covariance generalizes squared product-moment covariance by centering variables with stochastic processes, especially independent Wiener processes. Its population form equals distance covariance and preserves the independence characterization.

  • Definition and construction: Brownian covariance extends squared classical covariance by using conditional centering with independent stochastic processes U and V.Replacing the process with the identity recovers centered variables and Pearson covariance.
  • Relation to Pearson covariance: Replacing Brownian motion by the identity yields the absolute value of Pearson covariance, while Brownian standardization measures general relationships.Using Brownian processes instead produces the square of distance covariance.
  • Definition and construction: For random vectors, U- and V-centered versions are defined using conditional expectations with respect to independent random processes.The general (U,V) covariance is defined as a nonnegative squared quantity when finite.
  • Brownian–distance covariance equivalence: With independent Brownian motions, Brownian covariance is defined for random vectors in arbitrary dimensions and coincides with distance covariance.The equality is established by showing that their squared expressions coincide.
  • Existence: Brownian covariance is finite for random vectors with finite second moments, with the relevant expectation nonnegative and finite.The theorem represents the quantity through independent identically distributed copies.

4. Extensions.

The paper extends distance dependence measures through exponent- and Hurst-indexed families, affine-invariant transformations, and rank-based tests, while preserving key theoretical properties under stated moment conditions.

  • α-distance dependence measures: For 0 < α < 2, distance dependence measures form a one-parameter family indexed equivalently by α or the Hurst parameter H.The relation is h = 2H, with 0 < H < 1.
  • α-distance dependence measures: Replacing Euclidean-distance exponent 1 with α defines corresponding distance statistics, with almost sure convergence, weak convergence, and consistency when α-moments are finite.The strict range 0 < α < 2 is required for independence characterization.
  • α-distance dependence measures: At α = 2, distance dependence reduces to classical product-moment dependence and does not characterize independence; for p = q = 1, R(2) = |ρ|.The associated covariance statistic satisfies V(2)_n = 2|σ̂_xy|.
  • Affine invariance: Affine-invariant distance correlation is constructed by scaling samples with their covariance matrices so that pairwise distances, rather than the scaled sample vectors, are affine invariant.Theoretical properties established for V_n and R_n also hold for the transformed statistics.
  • Rank test: Rank-based distance covariance testing is distribution free and invariant to monotone transformations, but usually has lower power than dCov(X,Y); discrete data require random tie-breaking.Critical values for nR^2_n are provided from Monte Carlo results.

5. Applications.

Applications show distance covariance detecting nonlinear and multivariate dependence that Pearson correlation can miss, while supporting independence testing, residual analysis, and influential-observation diagnostics.

  • Applications: Measurement errors and nonlinear transformations can make ordinary correlation nearly irrelevant while distance correlation remains relevant.The framework is presented for testing dependence when variables are observed with independent errors or through functions of latent variables.
  • Applications: In the Eckerle4 example, dCov detects dependence between wavelength and transmittance when Pearson and Spearman tests do not.The dCov test is significant at p-value = 0.021, whereas Pearson and Spearman p-values are 0.839 and 0.9718, respectively.
  • Applications: For deterministic but nonmonotone dependence, dCov test power increases to 1 with sample size while product-moment correlation tests remain approximately at level.Rank-based distance correlation is more powerful than either correlation test but less powerful than dCov on the original data.
  • Applications: For aircraft speed and wing span, dCov finds dependence despite a nonsignificant Pearson test.The dCov test has p-value = 0.001, compared with Pearson’s p-value = 0.8001; the sample estimates are Rn = 0.2805 and ρ̂ = 0.0168.
  • Applications: In the Freedman crime data, distance correlation identifies a significant association between crime and population density that Pearson correlation does not.The analysis uses 100 cities with complete data from 110 observations containing missing values.
  • Applications: Jackknife replicates use leave-one-out distance matrices to identify observations that disproportionately change dependence, even when they are not original-data outliers.In the crime example, Philadelphia appears unusual in the dCor jackknife plot but not in principal-component plots.
  • Applications: After extracting a linear component, dCov power increases to 1 with sample size against the nonlinear dependence remaining in regression residuals.The procedure extends to arbitrary dimension through linear multiple regression or models with multivariate response.

6. Summary.

Distance correlation extends Pearson-style dependence measurement to arbitrary-dimensional random vectors and all dependence types, while retaining a scalar, computationally simple representation. Its Brownian-motion foundation links it naturally to Pearson covariance, and applications include multivariate exploratory analysis and testing linearity.

  • 6. Summary.: Distance covariance and correlation extend bivariate linear-association measures to all dependence relations and arbitrary-dimensional random vectors.They provide a single scalar measure rather than an array of bivariate statistics.
  • 6. Summary.: dCov and dCor statistics are computationally simple and applicable in arbitrary dimension without sample-size constraints.This practical advantage is stated alongside their theoretical generalization of classical correlation.
  • 6. Summary.: The extension is natural because Brownian-motion covariance yields distance covariance, whereas identity functions yield Pearson covariance.The paper presents both as covariance with respect to a pair of random processes.
  • 6. Summary.: Distance correlation can provide multivariate dependence information in exploratory analysis, including dependence among lower-dimensional marginal distributions.The paper contrasts this flexibility with classical correlation or arrays of bivariate statistics.
  • 6. Summary.: dCov can test linearity beyond simple linear regression for i.i.d. observations with longitudinal or multivariate predictors and responses.The basic method is described as applicable across these observation settings.
  • 6. Summary.: Distance correlation is presented as a practical tool extending classical correlation to multivariate analysis and general independence hypotheses.This is the paper’s concluding summary of its supported scope.

APPENDIX A: PROOFS OF STATEMENTS

The appendix uses Euclidean norms for random variables valued in R^d and suppresses the dimension index when it is clear from context.

  • APPENDIX A: PROOFS OF STATEMENTS: For R^d-valued random variables, |·|_d denotes the Euclidean norm.
  • APPENDIX A: PROOFS OF STATEMENTS: The dimension subscript on the norm is omitted whenever the dimension is self-evident.
  • APPENDIX A: PROOFS OF STATEMENTS: The notation applies specifically to random variables whose values lie in R^d.

A.1. Proof of Theorem 3(iii) and (vi).

The proof establishes equality conditions for an additivity inequality involving distance covariance by analyzing constant and independence cases.

  • A.1. Proof of Theorem 3(iii) and (vi).: The proof begins from the additivity expression V(X1+X2,Y1+Y2)=V(X1,Y1)+V(X2,Y2).
  • A.1. Proof of Theorem 3(iii) and (vi).: Equality holds when either paired variables are both constant or all four variables are mutually independent.
  • A.1. Proof of Theorem 3(iii) and (vi).: Under the stated independence of the pairs, equality without constant pairs requires X1,Y1 and X2,Y2 to be independently paired.
  • A.1. Proof of Theorem 3(iii) and (vi).: For the special case X1=Y1=X and X2=Y2=Y, the equality conditions reduce to both variables being constant.

A.2. Existence of W(X, Y ).

The appendix proves existence of the Brownian covariance construction by establishing finite fourth moments and almost-sure integrability of the relevant Brownian terms.

  • A.2. Existence of W(X, Y ).: The proof requires all factors in CovW(X,Y) to have finite fourth moments.
  • A.2. Existence of W(X, Y ).: Its fourth moment is E[W^4(t)]=12|t|^2, which supports the required moment bounds.
  • A.2. Existence of W(X, Y ).: The proof applies a fourth-power inequality and Jensen’s inequality to show finite fourth moments for the Brownian-weighted variables.
  • A.2. Existence of W(X, Y ).: The integral ∫W(t)dF_X(t) exists almost surely using Borel–Cantelli and small tails of suprema of centered Gaussian processes.
  • A.2. Existence of W(X, Y ).: Brownian motion satisfies E[W(t)W(s)]=|t|+|s|−|t−s| and E[W^2(t)]=2|t|.
  • A.2. Existence of W(X, Y ).: A similar argument for Y completes the existence proof.

APPENDIX B: CRITICAL VALUES

Appendix B provides critical values for the rank-dCov statistic nR2_n(rank(X), rank(Y)), using exact significance levels for small samples and Monte Carlo estimates for larger samples.

  • APPENDIX B: CRITICAL VALUES: Table 2 reports critical values for nR2_n(rank(X), rank(Y)) at 5% and 10% significance levels.These values estimate the 95th and 90th quantiles of the sampling distribution.
  • APPENDIX B: CRITICAL VALUES: For sample sizes n ≤10, achieved significance levels are exact because all possible rank permutations are generated.
  • APPENDIX B: CRITICAL VALUES: For n ≥11, critical values are estimated using 100,000 Monte Carlo replicates for each sample size.
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