Distance Correlation and Covariance Statistics
Computes distance covariance and distance correlation statistics, which are multivariate measures of dependence.
dcov(x, y, index = 1.0) dcor(x, y, index = 1.0)
x |
data or distances of first sample |
y |
data or distances of second sample |
index |
exponent on Euclidean distance, in (0,2] |
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 R generalizes the idea of correlation in two fundamental ways: (1) R(X,Y) is defined for X and Y in arbitrary dimension. (2) R(X,Y)=0 characterizes independence of X and Y.
Distance correlation satisfies 0 ≤ R ≤ 1, and R = 0 only if X and Y are independent. Distance covariance V provides a new approach to the problem of testing the joint independence of random vectors. The formal definitions of the population coefficients V and R are given in (SRB 2007). The definitions of the empirical coefficients are as follows.
The empirical distance covariance V_n(X,Y) with index 1 is the nonnegative number defined by
V^2_n (X,Y) = (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, b_{kl} = ||Y_k - Y_l||_q, k,l=1,…,n,
and the subscript . denotes that the mean is computed for the
index that it replaces. Similarly,
V_n(X) is the nonnegative number defined by
V^2_n (X) = V^2_n (X,X) = (1/n^2) sum_{k,l=1:n} A_{kl}^2.
The empirical distance correlation R(\mathbf{X,Y}) is the square root of
R^2_n(X,Y)= V^2_n(X,Y) / sqrt(V^2_n (X) V^2_n(Y)).
See dcov.test for a test of multivariate independence
based on the distance covariance statistic.
dcov returns the sample distance covariance and
dcor returns the sample distance correlation.
Two methods of computing the statistics are provided.
dcov and dcor provide R interfaces to the C
implementation, which is usually faster. dcov and dcor
call an internal function .dcov.
Note that it is inefficient to compute dCor by:
square root of
dcov(x,y)/sqrt(dcov(x,x)*dcov(y,y))
because the individual
calls to dcov involve unnecessary repetition of calculations.
Maria L. Rizzo mrizzo@bgsu.edu and Gabor J. Szekely
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.
doi: 10.1214/009053607000000505
Szekely, G.J. and Rizzo, M.L. (2009),
Brownian Distance Covariance,
Annals of Applied Statistics,
Vol. 3, No. 4, 1236-1265.
doi: 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.
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)
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