mdcov: Computation of the multidimensional distance covariance...

In IndependenceTests: Non-Parametric Tests of Independence Between Random Vectors

Computation of the multidimensional distance covariance statistic for mutual independence using characteristic functions.

Description

Computation of the multidimensional distance covariance statistic for mutual independence using characteristic functions. Compute the eigenvalues associated with the empirical covariance of the limiting Gaussian procces. Compute the p-value associated with the test statistic, using the Imhof procedure.

Usage

mdcov(X, vecd, a = 1, weight.choice = 1, N = 200, cubature = FALSE, K =
100, epsrel = 10 ^ -6, norming = TRUE, thresh.eigen = 10 ^ -8, estim.a =
FALSE, Cpp = TRUE, pval.comp = TRUE)

Arguments

X	Data.frame or matrix with observations corresponding to rows and variables to columns.
vecd	a vector giving the sizes of each subvector.
a	parameter for the weight function.
weight.choice	Integer value in 1, 2, 3, 4, 5 corresponding to the choice in our paper.
N	Number of Monte-Carlo samples.
cubature	Logical. If FALSE, a Monte-Carlo approach is used. If TRUE, a cubature approach is used.
K	Number of eigenvalues to compute.
epsrel	relative accuracy requested for the Imhof procedure.
norming	Logical. Should we normalize the test statistic with H_n.
thresh.eigen	We will not compute eigenvalues (involved in the limiting distribution) below that threshold.
estim.a	Logical. Should we automatically estimate the value of a.
Cpp	Logical. If TRUE computations will be done using a fast C code. The use of FALSE is only useful to compare the results with the one given by the C code.
pval.comp	Logical. If FALSE do not compute the p-values and lambdas.

Value

A list with the following components:

mdcov	the value of the statistic nT_n(w) (this value has been normed if norming = TRUE)
Hn	the denominator of nT_n(w), namely H_n
pvalue	the p-value of the test
lambdas	the vector of eigenvalues computed (they have not been divided by their sum)

Author(s)

Lafaye de Micheaux P.

References

Fan Y., Lafaye de Micheaux P., Penev S. and Salopek D. (2017). Multivariate nonparametric test of independence, Journal of Multivariate Analysis, 153, 189-210.

Examples

a <- 1

# 4.1 Dependence among four discrete variables
set.seed(1)
n <- 100
w1 <- rpois(n, 1)
w3 <- rpois(n, 1)
w4 <- rpois(n, 1)
w6 <- rpois(n, 1)
w2 <- rpois(n, 3)
w5 <- rpois(n, 3)
x1 <- w1 + w2
x2 <- w2 + w3
x3 <- w4 + w5
x4 <- w5 + w6

X <- cbind(x1, x2, x3, x4)
mdcov(X, vecd = c(2, 2), a, weight.choice = 1, N = 100, cubature = TRUE)
mdcov(X, vecd = c(1, 1, 1, 1), a, weight.choice = 1, N = 100, cubature = TRUE)

X <- cbind(x1, x2)
mdcov(X, vecd = c(1, 1), a, weight.choice = 1, N = 100, cubature = TRUE)

X <- cbind(x3, x4)
mdcov(X, vecd = c(1, 1), a, weight.choice = 1, N = 100, cubature = TRUE)

# 4.2 Dependence between three bivariate vectors
set.seed(2)
n <- 200
Sigma <- matrix(c(
1 , 0 ,  0 ,  0 ,  0 ,  0 ,
0 , 1 ,  0 ,  0 ,  0 ,  0 ,
0 , 0 ,  1 ,  0 , .4 , .5 ,
0 , 0 ,  0 ,  1 , .1 , .2 ,
0 , 0 , .4 , .1 ,  1 ,  0 ,
0 , 0 , .5 , .2 ,  0 ,  1 ) ,
nrow = 6 , ncol = 6)   
W <- mvrnorm(n = n, mu = rep(0,6), Sigma = Sigma, tol = 1e-6, empirical = FALSE, EISPACK = FALSE)
mdcov(W, vecd = c(2, 2, 2), a, weight.choice = 1, N = 100, cubature = TRUE, epsrel = 10 ^ -7)

 # X^{(1)} with X^{(2)}^2
mdcov(W[,1:4], vecd = c(2, 2), a, weight.choice = 1, N = 100, cubature = TRUE)

# X^{(2)} with X^{(3)}^2
mdcov(W[,2:6], vecd = c(2, 2), a, weight.choice = 1, N = 100, cubature = TRUE)

# X^{(1)} with X^{(3)}^2
mdcov(W[,c(1:2, 4:6)], vecd = c(2, 2), a, weight.choice = 1, N = 100, cubature = TRUE)

# 4.3 Four-dependent variables which are 2-independent and 3-independent
set.seed(3)
n <- 300
W <- sample(1:8, n, replace = TRUE)
X1 <- W %in% c(1, 2, 3, 5)
X2 <- W %in% c(1, 2, 4, 6)
X3 <- W %in% c(1, 3, 4, 7)
X4 <- W %in% c(2, 3, 4, 8)
X <- cbind(X1, X2, X3, X4)
   # pairwise independence
mdcov(X[,c(1, 2)], vecd = c(1, 1), a, weight.choice = 1, cubature = TRUE)
mdcov(X[,c(1, 3)], vecd = c(1, 1), a, weight.choice = 1, N = 100, cubature = TRUE)
mdcov(X[,c(1, 4)], vecd = c(1, 1), a, weight.choice = 1, cubature = TRUE)
mdcov(X[,c(2, 3)], vecd = c(1, 1), a, weight.choice = 1, cubature = TRUE)
mdcov(X[,c(2, 4)], vecd = c(1, 1), a, weight.choice = 1, cubature = TRUE)
mdcov(X[,c(3, 4)], vecd = c(1, 1), a, weight.choice = 1, cubature = TRUE)
   # 3-independence
mdcov(X[,c(1, 2, 3)], vecd = c(1, 1, 1), a, weight.choice =
                  1, cubature = TRUE)
mdcov(X[,c(1, 2, 4)], vecd = c(1, 1, 1), a, weight.choice =
                  1, cubature = TRUE)
mdcov(X[,c(1, 3, 4)], vecd = c(1, 1, 1), a, weight.choice =
                  1, cubature = TRUE)
mdcov(X[,c(2, 3, 4)], vecd = c(1, 1, 1), a, weight.choice =
                  1, cubature = TRUE)
   # 4-dependence
mdcov(X, vecd = c(1, 1, 1, 1), a, weight.choice = 1, cubature = TRUE)

IndependenceTests documentation built on July 17, 2026, 9:07 a.m.