IndependenceTests (version 0.5)

dependogram: Nonparametric tests of independence between random vectors

Description

This function can be used for the following two problems: 1) testing mutual independence between some numerical random vectors, and 2) testing for serial independence of a multivariate stationary quantitative time series. The proposed test does not assume continuous marginals. It is valid for any probability distribution. It is also invariant with respect to the affine general linear group of transformations on the vectors. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of M<U+00F6>bius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence can not be distribution free. The critical values of the proposed test are thus computed with the bootstrap which was shown to be consistent in this context.

Usage

dependogram(X, vecd.or.p, N = 10, B = 2000, alpha = 0.05, display =
            TRUE, graphics = TRUE, nbclus = 1)

Arguments

X

Data.frame or matrix with observations corresponding to rows and variables to columns.

vecd.or.p

For the mutual independence problem 1), a vector giving the sizes of each subvector. For the serial independence problem 2), an integer indicating the number of consecutive observations.

N

Integer. Number of points of the discretization to obtain directions on the sphere in order to evaluate the value of the test statistic.

B

Integer. Number of bootstrap samples. Note that B can be slightly modified if nbclus > 1

alpha

Double. Global significance level of the test.

display

Logical. TRUE to display values of the \(A\)-dependence statistics.

graphics

Logical. TRUE to plot the dependogram.

nbclus

Integer. Number of nodes in the cluster. Used only for parallel computations.

Value

A list with the following components:

In the mutual independence case:

norm.RnA

Supremum norm (Kolmogorov). Test statistic is \(\|R_{n,A}\|\) and is computed from the M<U+00F6>bius independence half space processes \(R_{n,A}\).

Rn

Maximum value of norm.RnA over all subsets \(A\) of variables.

rA

Critical value of the bootstrap distribution of the test statistic \(\|R_{n,A}\|\).

r

Critical value of the bootstrap distribution of the test statistic \(R_n\).

RnAsstar

Matrix of size \((2 ^ p - p - 1)\times B\) which contains, for each of the \(B\) bootstrap samples, the statistics norm.RnA for all subsets \(A\) of variables.

In the serial case:

norm.SnA

Supremum norm (Kolmogorov). Test statistic is \(\|S_{n,A}\|\) and is computed from the M<U+00F6>bius independence half space processes \(S_{n,A}\).

Sn

Maximum value of norm.SnA over all subsets \(A\) of variables.

sA

Critical value of the bootstrap distribution of the test statistic \(\|S_{n,A}\|\).

s

Critical value of the bootstrap distribution of the test statistic \(S_n\).

SnAsstar

Matrix of size \((2 ^ {p - 1} - 1)\times B\) which contains, for each of the \(B\) bootstrap samples, the statistics norm.SnA for all subsets \(A\) of variables.

References

Beran R., Bilodeau M., Lafaye de Micheaux P. (2007). Nonparametric tests of independence between random vectors, Journal of Multivariate Analysis, 98, 1805-1824.

Examples

# NOT RUN {
<!-- %% All these examples (but the last one) are taken from the paper -->
# }
# NOT RUN {
<!-- %% "Nonparametric tests of independence between random vectors" -->
# }
# NOT RUN {
<!-- %%  R. Beran, M. Bilodeau, P. Lafaye de Micheaux -->
# }
# NOT RUN {
<!-- %%  Journal of Multivariate Analysis 98 (2007) 1805-1824 -->
# }
# NOT RUN {
# NOTE: In real applications, B should be set to at least 1000.

# Example 4.1: Test of mutual independence between four discrete Poisson
# variables. The pair (X1,X2) is independent of the pair (X3,X4), with
# each pair having a correlation of 3/4.
# NOTE: with B=1000, this one took 65s with nbclus=1 and 15s with nbclus=7 on my computer.
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)
dependogram(X, vecd.or.p = c(1, 1, 1, 1), N = 10, B = 20, alpha = 0.05,
            display = TRUE, graphics = TRUE)

# Example 4.2: Test of mutual independence between three bivariate
#  vectors. The block-diagonal structure of the covariance matrix is
#  such that only the second and third subvectors are dependent.
# NOTE: with B=2000, this one took 3.8h with nbclus=1 on my computer.
n <- 50
mu <- rep(0,6)
Psi <- 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, byrow = TRUE)
X <- mvrnorm(n, mu, Psi)
# }
# NOT RUN {
dependogram(X, vecd.or.p = c(2, 2, 2), N = 10, B = 20, alpha = 0.05,
            display = TRUE, graphics = TRUE)
# }
# NOT RUN {
# Example 4.3: Test of mutual independence between 4 dependent binary
# variables which are 2-independent (pairwise) and also 3-independent
# (any 3 of the 4 variables are mutually independent).
n <- 100
W <- sample(x = 1:8, size = n, 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)
dependogram(X, vecd.or.p = c(1, 1, 1, 1), N = 10, B = 20, alpha = 0.05,
            display = TRUE, graphics = TRUE)

# Example 4.4: Test of serial independence of binary sequences of zeros
# and ones. The sequence W is an i.i.d. sequence. The sequence Y is
# dependent at lag 3.
n <- 100 ; lag <- 3
W <- rbinom(n, 1, 0.8)
Y <- W[1:(n - lag)] * W[(1 + lag):n]
dependogram(W, vecd.or.p = 4, N = 10, B = 20, alpha = 0.05, display =
            TRUE, graphics = TRUE)
dependogram(Y, vecd.or.p = 4, N = 10, B = 20, alpha = 0.05, display =
            TRUE, graphics = TRUE)

# Example 4.5: Test of serial independence of sequences of directional
# data on the 2-dimensional sphere. The sequence W is an
# i.i.d. sequence. The sequence Y is dependent at lag 1.
# NOTE: with B=2000, this one took 7.9h with nbclus=1 on my computer.
n <- 75 ; lag <- 1
U <- matrix(rnorm(2 * n), nrow = n, ncol = 2)
W <- U[1:(n - lag),] + sqrt(2) * U[(1 + lag):n,]
Y <- W / apply(W, MARGIN = 1, FUN = function(x) {sqrt(x[1] ^ 2 + x[2] ^ 2)})
# }
# NOT RUN {
dependogram(Y, vecd.or.p = 3, N = 10, B = 20, alpha = 0.05, display =
            TRUE, graphics = TRUE)
# }
# NOT RUN {
# This one always gives the same value of the test statistic:
x <- rnorm(100)
dependogram(X = cbind(x, x), vecd.or.p = c(1, 1), N = 2, B = 2, alpha =
0.05, display = FALSE, graphics = FALSE, nbclus = 1)$Rn
# This is correct because this is equivalent to computing:
I <- 1:100
n <- 100
sqrt(n) * max(I/n - I ^ 2 / n ^ 2)

# }