mdcov: Computation of the multidimensional distance covariance...
In IndependenceTests: Non-Parametric Tests of Independence Between Random Vectors

Description Usage Arguments Value Author(s) References Examples

View source: R/mdcov.R

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.

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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)

`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.

A list with the following components:

`mdcov`	the value of the statistic n Tn(w)* (this value has been normed if `norming = TRUE`)
`Hn`	the denominator of nTn(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)

Lafaye de Micheaux P.

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

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)