CovEst.hardPD: Covariance Estimation via Hard Thresholding under...

In CovTools: Statistical Tools for Covariance Analysis

Covariance Estimation via Hard Thresholding under Positive-Definiteness Constraint

Description

Sparse covariance estimation does not necessarily guarantee positive definiteness of an estimated covariance matrix. Fan et al. (2013) proposed to solve this issue by taking an iterative procedure to take an incremental decrease of threshold value until positive definiteness is preserved.

Usage

CovEst.hardPD(X)

Arguments

X	an (n\times p) matrix where each row is an observation.

Details

The selected estimate is verified to have minimum eigenvalue at least \sqrt{\epsilon}, where \epsilon is machine precision. Since hard thresholding preserves the diagonal, the function stops with an informative error when a column variance is below this tolerance and no qualifying estimate can be constructed.

Value

a named list containing:

S

a (p\times p) covariance matrix estimate.

optC

a nonnegative threshold value C_{min} that guarantees positive definiteness after thresholding. A value of zero means that the sample covariance already satisfies the tolerance.

References

\insertRef

fan_large_2013CovTools

Examples

## generate data from multivariate normal with Identity covariance.
pdim <- 5
data <- matrix(rnorm(10*pdim), ncol=pdim)

## apply 4 different schemes
out1 <- CovEst.hard(data, thr=0.1)  # threshold value 0.1
out2 <- CovEst.hard(data, thr=1)    # threshold value 1
out3 <- CovEst.hard(data, thr=10)   # threshold value 10
out4 <- CovEst.hardPD(data) # automatic threshold checking

## visualize 4 estimated matrices
mmessage <- paste("hardPD::optimal thr=",sprintf("%.2f",out4$optC),sep="")
gcol     <- gray((0:100)/100)
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(out1$S[,pdim:1], col=gcol, main="thr=0.1")
image(out2$S[,pdim:1], col=gcol, main="thr=1")
image(out3$S[,pdim:1], col=gcol, main="thr=10")
image(out4$S[,pdim:1], col=gcol, main=mmessage)
par(opar)

CovTools documentation built on July 29, 2026, 9:07 a.m.