## Covariance Estimation via Adaptive Thresholding

### Description

Cai and Liu (2011) proposed an adaptive variant of Bickel and Levina (2008) - CovEst.hard. The idea of adaptive thresholding is to apply thresholding technique on correlation matrix in that it becomes adaptive in terms of each variable.

### Usage

CovEst.adaptive(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

Cai T, Liu W (2011). “Adaptive Thresholding for Sparse Covariance Matrix Estimation.” Journal of the American Statistical Association, 106(494), 672–684. ISSN 0162-1459, 1537-274X.

### 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 <- seq(from=0.01,to=0.99,length.out=10)

out1 <- CovEst.adaptive(data, thr=0.1)  # threshold value 0.1
out2 <- CovEst.adaptive(data, thr=0.5)  # threshold value 0.5
out3 <- CovEst.adaptive(data, thr=0.1)  # threshold value 0.9
out4 <- CovEst.adaptive(data, thr=mthr) # automatic threshold checking

## visualize 4 estimated matrices
image(out1$S[,pdim:1], col=gray((0:100)/100), main="thr=0.1") image(out2$S[,pdim:1], col=gray((0:100)/100), main="thr=0.5")
image(out3$S[,pdim:1], col=gray((0:100)/100), main="thr=0.9") image(out4$S[,pdim:1], col=gray((0:100)/100), main="automatic")