energy (version 1.7-9)

distance correlation: Distance Correlation and Covariance Statistics

Description

Computes distance covariance and distance correlation statistics, which are multivariate measures of dependence.

Usage

dcov(x, y, index = 1.0)
dcor(x, y, index = 1.0)

Arguments

x

data or distances of first sample

y

data or distances of second sample

index

exponent on Euclidean distance, in (0,2]

Value

dcov returns the sample distance covariance and dcor returns the sample distance correlation.

Details

dcov and dcor compute distance covariance and distance correlation statistics.

The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values. Arguments x, y can optionally be dist objects; otherwise these arguments are treated as data.

Distance correlation is a new measure of dependence between random vectors introduced by Szekely, Rizzo, and Bakirov (2007). For all distributions with finite first moments, distance correlation \(\mathcal R\) generalizes the idea of correlation in two fundamental ways: (1) \(\mathcal R(X,Y)\) is defined for \(X\) and \(Y\) in arbitrary dimension. (2) \(\mathcal R(X,Y)=0\) characterizes independence of \(X\) and \(Y\).

Distance correlation satisfies \(0 \le \mathcal R \le 1\), and \(\mathcal R = 0\) only if \(X\) and \(Y\) are independent. Distance covariance \(\mathcal V\) provides a new approach to the problem of testing the joint independence of random vectors. The formal definitions of the population coefficients \(\mathcal V\) and \(\mathcal R\) are given in (SRB 2007). The definitions of the empirical coefficients are as follows.

The empirical distance covariance \(\mathcal{V}_n(\mathbf{X,Y})\) with index 1 is the nonnegative number defined by $$ \mathcal{V}^2_n (\mathbf{X,Y}) = \frac{1}{n^2} \sum_{k,\,l=1}^n A_{kl}B_{kl} $$ where \(A_{kl}\) and \(B_{kl}\) are $$ A_{kl} = a_{kl}-\bar a_{k.}- \bar a_{.l} + \bar a_{..} $$ $$ B_{kl} = b_{kl}-\bar b_{k.}- \bar b_{.l} + \bar b_{..}. $$ Here $$ a_{kl} = \|X_k - X_l\|_p, \quad b_{kl} = \|Y_k - Y_l\|_q, \quad k,l=1,\dots,n, $$ and the subscript . denotes that the mean is computed for the index that it replaces. Similarly, \(\mathcal{V}_n(\mathbf{X})\) is the nonnegative number defined by $$ \mathcal{V}^2_n (\mathbf{X}) = \mathcal{V}^2_n (\mathbf{X,X}) = \frac{1}{n^2} \sum_{k,\,l=1}^n A_{kl}^2. $$

The empirical distance correlation \(\mathcal{R}_n(\mathbf{X,Y})\) is the square root of $$ \mathcal{R}^2_n(\mathbf{X,Y})= \frac {\mathcal{V}^2_n(\mathbf{X,Y})} {\sqrt{ \mathcal{V}^2_n (\mathbf{X}) \mathcal{V}^2_n(\mathbf{Y})}}. $$ See dcov.test for a test of multivariate independence based on the distance covariance statistic.

References

Szekely, G.J., Rizzo, M.L., and Bakirov, N.K. (2007), Measuring and Testing Dependence by Correlation of Distances, Annals of Statistics, Vol. 35 No. 6, pp. 2769-2794. 10.1214/009053607000000505

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265. 10.1214/09-AOAS312

Szekely, G.J. and Rizzo, M.L. (2009), Rejoinder: Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1303-1308.

See Also

bcdcor dcovU pdcor dcov.test dcor.ttest pdcor.test

Examples

# NOT RUN {
 x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 dcov(x, y)
 dcov(dist(x), dist(y))  #same thing

 ## C implementation
 dcov(x, y, 1.5)
 dcor(x, y, 1.5)
# }