pdRankTests: Rank-based hypothesis tests for HPD matrices

In pdSpecEst: An Analysis Toolbox for Hermitian Positive Definite Matrices

Description

pdRankTests performs a number of generalized rank-based hypothesis tests in the metric space of HPD matrices equipped with the affine-invariant Riemannian metric or Log-Euclidean metric for samples of HPD matrices or samples of sequences (curves) of HPD matrices as described in Chapter 4 of \insertCiteC18pdSpecEst.

Usage

pdRankTests(data, sample_sizes, test = c("rank.sum", "krusk.wall",
  "signed.rank", "bartels"), depth = c("gdd", "zonoid", "spatial"),
  metric = c("Riemannian", "logEuclidean"))

Arguments

data	either a (d,d,S)-dimensional array corresponding to an array of pooled individual samples of (d,d)-dimensional HPD matrices, or a (d,d,n,S)-dimensional array corresponding to an array of pooled individual samples of length n sequences of (d,d)-dimensional HPD matrices.
sample_sizes	a numeric vector specifying the individual sample sizes in the pooled sample data, such that sum(sample_sizes) is equal to S. Not required for tests "signed-rank" and "bartels", as the sample sizes are automatically determined from the input array data.
test	rank-based hypothesis testing procedure, one of "rank.sum", "krusk.wall", "signed.rank", "bartels" explained in the Details section below.
depth	data depth measure used in the rank-based tests, one of "gdd", "zonoid", or "spatial" corresponding to the geodesic distance depth, intrinsic zonoid depth and intrinsic spatial depth respectively. Defaults to "gdd". Not required for test "signed.rank". See the documentation of the function pdDepth for additional details about the different depth measures.
metric	the metric that the space of HPD matrices is equipped with, either "Riemannian" or "logEuclidean". Defaults to "Riemannian".

Details

For samples of (d,d)-dimensional HPD matrices with pooled sample size S, the argument data is a (d,d,S)-dimensional array of (d,d)-dimensional HPD matrices, where the individual samples are combined along the third array dimension. For samples of sequences of (d,d)-dimensional HPD matrices with pooled sample size S, the argument data is a (d,d,n,S)-dimensional array of length n sequences of (d,d)-dimensional HPD matrices, where the individual samples are combined along the fourth array dimension. The argument sample_sizes specifies the sizes of the individual samples so that sum(sample_sizes) is equal to S.
The available generalized rank-based testing procedures (specified by the argument test) are:

"rank.sum"

Intrinsic Wilcoxon rank-sum test to test for homogeneity of distributions of two independent samples of HPD matrices or samples of sequences of HPD matrices. The usual univariate ranks are replaced by data depth induced ranks obtained with pdDepth.

"krusk.wall"

Intrinsic Kruskal-Wallis test to test for homogeneity of distributions of more than two independent samples of HPD matrices or samples of sequences of HPD matrices. The usual univariate ranks are replaced by data depth induced ranks obtained with pdDepth.

"signed.rank"

Intrinsic signed-rank test to test for homogeneity of distributions of independent paired or matched samples of HPD matrices. The intrinsic signed-rank test is not based on data depth induced ranks, but on a specific difference score in the Riemannian manifold of HPD matrices equipped with either the affine-invariant Riemannian or Log-Euclidean metric.

"bartels"

Intrinsic Bartels-von Neumann test to test for randomness (i.e., exchangeability) within a single independent sample of HPD matrices or a sample of sequences of HPD matrices. The usual univariate ranks are replaced by data depth induced ranks obtained with pdDepth.

The function computes the generalized rank-based test statistics in the complete metric space of HPD matrices equipped with one of the following metrics: (i) the Riemannian metric (default) as detailed in e.g., \insertCiteB09pdSpecEst[Chapter 6] or \insertCitePFA05pdSpecEst; or (ii) the Log-Euclidean metric, the Euclidean inner product between matrix logarithms. The default Riemannian metric is invariant under congruence transformation by any invertible matrix, whereas the Log-Euclidean metric is only invariant under congruence transformation by unitary matrices, see \insertCiteC18pdSpecEst[Chapter 4] for more details.

Value

The function returns a list with five components:

test	name of the rank-based test
p.value	p-value of the test
statistic	computed test statistic
null.distr	distribution of the test statistic under the null hypothesis
depth.values	computed data depth values (if available)

Note

The intrinsic signed-rank test also provides a valid test for equivalence of spectral matrices of two multivariate stationary time series based on the HPD periodogram matrices obtained via pdPgram, see \insertCiteC18pdSpecEst[Chapter 4] for the details.

The function does not check for positive definiteness of the input matrices, and may fail if matrices are close to being singular.

The data depth computations under the Riemannian metric are more involved than under the Log-Euclidean metric, and may therefore result in (significantly) higher computation times.

References

\insertAllCited

See Also

pdDepth, pdPgram

Examples

## null hypothesis is true
data <- replicate(100, Expm(diag(2), H.coeff(rnorm(4), inverse = TRUE)))
pdRankTests(data, sample_sizes = c(50, 50), test = "rank.sum") ## homogeneity 2 samples
pdRankTests(data, sample_sizes = rep(25, 4), test = "krusk.wall") ## homogeneity 4 samples
pdRankTests(data, test = "bartels") ## randomness

## null hypothesis is false
data1 <- array(c(data, replicate(50, Expm(diag(2), H.coeff(0.5 * rnorm(4), inverse = TRUE)))),
                 dim = c(2,2,150))
pdRankTests(data1, sample_sizes = c(100, 50), test = "rank.sum")
pdRankTests(data1, sample_sizes = rep(50, 3), test = "krusk.wall")
pdRankTests(data1, test = "bartels")

## Not run: 
## signed-rank test for equivalence of spectra of multivariate time series
## ARMA(1,1) process: Example 11.4.1 in (Brockwell and Davis, 1991)
Phi <- array(c(0.7, 0, 0, 0.6, rep(0, 4)), dim = c(2, 2, 2))
Theta <- array(c(0.5, -0.7, 0.6, 0.8, rep(0, 4)), dim = c(2, 2, 2))
Sigma <- matrix(c(1, 0.71, 0.71, 2), nrow = 2)
pgram <- function(Sigma) pdPgram(rARMA(2^8, 2, Phi, Theta, Sigma)$X)$P

## null is true
pdRankTests(array(c(pgram(Sigma), pgram(Sigma)), dim = c(2,2,2^8)), test = "signed.rank")
## null is false
pdRankTests(array(c(pgram(Sigma), pgram(0.5 * Sigma)), dim = c(2,2,2^8)), test = "signed.rank")

## End(Not run)

pdSpecEst documentation built on Jan. 8, 2020, 5:08 p.m.