CovEst.soft {CovTools} R Documentation

## Covariance Estimation via Soft Thresholding

### Description

Soft Thresholding method for covariance estimation takes off-diagonal elements z of sample covariance matrix and applies

h_{τ}(z) = \textrm{sgn}(z)(|z|-τ)_{+}

where \textrm{sgn}(z) is a sign of the value z, and (x)_+ = \textrm{max}(x,0). If thr is rather a vector of regularization parameters, it applies cross-validation scheme to select an optimal value.

### Usage

CovEst.soft(X, thr = 0.5, nCV = 10, parallel = FALSE)


### Arguments

 X an (n\times p) matrix where each row is an observation. thr user-defined threshold value. If it is a vector of regularization values, it automatically selects one that minimizes cross validation risk. nCV the number of repetitions for 2-fold random cross validations for each threshold value. parallel a logical; TRUE to use half of available cores, FALSE to do every computation sequentially.

### Value

a named list containing:

S

a (p\times p) covariance matrix estimate.

CV

a dataframe containing vector of tested threshold values(thr) and corresponding cross validation scores(CVscore).

### References

Antoniadis A, Fan J (2001). “Regularization of Wavelet Approximations.” Journal of the American Statistical Association, 96(455), 939–967. ISSN 0162-1459, 1537-274X.

Donoho DL, Johnstone IM, Kerkyacharian G, Picard D (1995). “Wavelet Shrinkage: Asymptopia?” Journal of the Royal Statistical Society. Series B (Methodological), 57(2), 301–369. ISSN 00359246.

### Examples

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

## apply 4 different schemes
#  mthr is a vector of regularization parameters to be tested
mthr <- exp(seq(from=log(0.1),to=log(10),length.out=10))

out1 <- CovEst.soft(data, thr=0.1)  # threshold value 0.1
out2 <- CovEst.soft(data, thr=1)    # threshold value 1
out3 <- CovEst.soft(data, thr=10)   # threshold value 10
out4 <- CovEst.soft(data, thr=mthr) # automatic threshold checking

## visualize 4 estimated matrices
gcol <- gray((0:100)/100)
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="automatic")