CovEst.hardPD {CovTools}

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.

Value

a named list containing:

S

a (p\times p) covariance matrix estimate.

optC

an optimal threshold value C_{min} that guarantees positive definiteness after thresholding.

References

Fan J, Liao Y, Mincheva M (2013). “Large covariance estimation by thresholding principal orthogonal complements.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4), 603–680. ISSN 13697412.

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)