HDCovMatTest

Homogeneity tests of covariance matrices with high-dimensional longitudinal data

Overview

HDCovMatTest implements the methodology developed in "Homogeneity tests of covariance matrices with high-dimensional longitudinal data" by Ping-Shou Zhong, Runze Li, and Shawn Santo.

• test_covmat() performs the hypothesis testing procedure.
• cpi_covmat() identifies change points in the time series of covariance matrices.

Paper's abstract

This paper deals with the detection and identification of changepoints among covariances of high-dimensional longitudinal data, where the number of features is greater than both the sample size and the number of repeated measurements. The proposed methods are applicable under general temporal-spatial dependence. A new test statistic is introduced for changepoint detection, and its asymptotic distribution is established. If a changepoint is detected, an estimate of the location is provided. The rate of convergence of the estimator is shown to depend on the data dimension, sample size, and signal-to-noise ratio. Binary segmentation is used to estimate the locations of possibly multiple changepoints, and the corresponding estimator is shown to be consistent under mild conditions. Simulation studies provide the empirical size and power of the proposed test and the accuracy of the changepoint estimator. An application to a time-course microarray dataset identifies gene sets with significant gene interaction changes over time.

Install package HDCovMatTest

Package HDCovMatTest can be installed using the devtools package in R.

# install (if not available) and load the devtools package
if(!require(devtools)) {
    install.packages("devtools")
    library(devtools)
}

# install HDCovMatTest
install_github("shawnsanto/HDCovMatTest")

# load HDCovMatTest
library(HDCovMatTest)

Example

Consider a small simulated data example with a change point at time 2.

# set primary parameters
p <- 30 # data dimension
n <- 10 # number of individuals
TT <- 5 # total number of time points

# set secondary parameters for simulation and data generation
delta <- 0.35
k0 <- 2

m <- p + 20; L <- 3; w <- 0.2

Generate the data. Here delta is chosen to be large so we ensure a change point exists at time 2.

set.seed(928)

gamma_1 <- gamma_2 <- matrix(0, p, m * L)
y <- array(0, c(p, n, TT))

for (i in 1:p) {
  for (j in 1:p) {
    dij <- abs(i - j)

    if (dij < (p * w)) {
      gamma_1[i, j] <- (dij + 1) ^ (-2)
      gamma_2[i, j] <- (dij + 1 + delta) ^ (-2)
    }
  }
}

Z <- matrix(rnorm(m * (TT + L - 1) * n), m * (TT + L - 1), n)

for (t in 1:k0) {
  y[, , t] <- gamma_1 %*% Z[((t - 1) * m + 1):((t + L - 1) * m), ]
}

for (t in (k0 + 1):TT) {
  y[, , t] <- gamma_2 %*% Z[((t - 1) * m + 1):((t + L - 1) * m), ]
}

Perform the testing procedure.

test_covmat(y, n, p, TT, alpha = 0.01)

Results are provided in an R list.

$reject # rejection flag
[1] 1

$estcp # estimated change point
[1] 2

$tstat # test statistic
[1] 3.501253

$pvalue # p-vale
[1] 0.0005785431

$corrmat # corresponding correlation matrix
          [,1]      [,2]      [,3]      [,4]
[1,] 1.0000000 0.8423203 0.8191572 0.7686094
[2,] 0.8423203 1.0000000 0.9790518 0.9195994
[3,] 0.8191572 0.9790518 1.0000000 0.9471566
[4,] 0.7686094 0.9195994 0.9471566 1.0000000

Reference

Ping-Shou Zhong, Runze Li, Shawn Santo, Homogeneity tests of covariance matrices with high-dimensional longitudinal data, Biometrika, Volume 106, Issue 3, September 2019, Pages 619–634, https://doi.org/10.1093/biomet/asz011.