In statistics and in probability theory, distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal, dimension. The population distance correlation coefficient is zero if and only if the random vectors are independent. Thus, distance correlation measures both linear and nonlinear association between two random variables or random vectors. This is in contrast to Pearson's correlation, which can only detect linear association between two random variables.
Distance correlation can be used to perform a statistical test of dependence with a permutation test. One first computes the distance correlation (involving the re-centering of Euclidean distance matrices) between two random vectors, and then compares this value to the distance correlations of many shuffles of the data.
Background
The classical measure of dependence, the Pearson correlation coefficient, is mainly sensitive to a linear relationship between two variables. Distance correlation was introduced in 2005 by Gábor J. Székely in several lectures to address this deficiency of Pearson's correlation, namely that it can easily be zero for dependent variables. Correlation = 0 (uncorrelatedness) does not imply independence while distance correlation = 0 does imply independence. The first results on distance correlation were published in 2007 and 2009. It was proved that distance covariance is the same as the Brownian covariance. These measures are examples of energy distances.
The distance correlation is derived from a number of other quantities that are used in its specification, specifically: distance variance, distance standard deviation, and distance covariance. These quantities take the same roles as the ordinary moments with corresponding names in the specification of the Pearson product-moment correlation coefficient.
Definitions
Distance covariance
Let us start with the definition of the sample distance covariance. Let (Xk, Yk), k = 1, 2, ..., n be a statistical sample from a pair of real valued or vector valued random variables (X, Y). First, compute the n by n distance matrices (aj, k) and (bj, k) containing all pairwise distances
where ||⋅ ||denotes Euclidean norm. Then take all doubly centered distances
where is the -th row mean, is the -th column mean, and is the grand mean of the distance matrix of the sample. The notation is similar for the values. (In the matrices of centered distances (Aj, k) and (Bj,k) all rows and all columns sum to zero.) The squared sample distance covariance (a scalar) is simply the arithmetic average of the products Aj, k Bj, k:
The statistic Tn = n dCov2n(X, Y) determines a consistent multivariate test of independence of random vectors in arbitrary dimensions. For an implementation see dcov.test function in the energy package for R.
The population value of distance covariance can be defined along the same lines. Let X be a random variable that takes values in a p-dimensional Euclidean space with probability distribution and let Y be a random variable that takes values in a q-dimensional Euclidean space with probability distribution , and suppose that X and Y have finite expectations. Write
Finally, define the population value of squared distance covariance of X and Y as
One can show that this is equivalent to the following definition:
where E denotes expected value, and and are independent and identically distributed. The primed random variables and denote independent and identically distributed (iid) copies of the variables and and are similarly iid. Distance covariance can be expressed in terms of the classical Pearson's covariance, cov, as follows:
This identity shows that the distance covariance is not the same as the covariance of distances, ). This can be zero even if X and Y are not independent.
Alternatively, the distance covariance can be defined as the weighted L2 norm of the distance between the joint characteristic function of the random variables and the product of their marginal characteristic functions: