Unbiased dcov and bias-corrected dcor statistics

Description

These functions compute unbiased estimators of squared distance covariance, distance variance, and a bias-corrected estimator of (squared) distance correlation.

Usage

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bcdcor(x, y)
dcovU(x, y)
dcovU_stats(Dx, Dy)

Arguments

x

data or dist object of first sample

y

data or dist object of second sample

Dx

distance matrix of first sample

Dy

distance matrix of second sample

Details

The unbiased (squared) dcov is inner product definition of dCov, in the Hilbert space of U-centered distance matrices.

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.

Value

dcovU returns the unbiased estimator of squared dcov. bcdcor returns a bias-corrected estimator of squared dcor.

dcovU_stats returns a vector of the components of bias-corrected dcor: [dCovU, bcdcor, dVarXU, dVarYU].

Note

Unbiased distance covariance (SR2014) corresponds to the biased (original) dCov^2. Since dcovU is an unbiased statistic, it is signed and we do not take the square root. For the original distance covariance test of independence (SRB2007, SR2009), the distance covariance test statistic is the V-statistic n V_n^2 (not dCov). Similarly, bcdcor is bias-corrected, so we do not take the square root as with dCor.

Author(s)

Maria L. Rizzo mrizzo @ bgsu.edu and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities. Annals of Statistics, Vol. 42 No. 6, 2382-2412.

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.
http://dx.doi.org/10.1214/009053607000000505

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

Examples

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 x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 Dx <- as.matrix(dist(x))
 Dy <- as.matrix(dist(y))

 dcovU(x, y)
 bcdcor(x, y)
 dcovU_stats(Dx, Dy)