solve for the inverse matrix

Description

Solve for a series of the inverse covariance matrix estimates at a grid of values for the constraint lambda.

Usage

clime(x, lambda=NULL, nlambda=ifelse(is.null(lambda),100,length(lambda)),
      lambda.max=0.8, lambda.min=ifelse(nrow(x)>ncol(x), 1e-4, 1e-2),
      sigma=FALSE, perturb=TRUE, standardize=TRUE, logspaced=TRUE,
      linsolver=c("primaldual", "simplex"), pdtol=1e-3, pdmaxiter=50)

Arguments

Details

A constrained \ell_1 minimization approach for sparse precision matrix estimation (details in references) is implemented here using linear programming (revised simplex or primal-dual interior point method). It solves a sequence of lambda values on the following objective function

\min | \Omega |_1 \quad \textrm{subject to: } || \Sigma_n \Omega - I ||_\infty \le \lambda

where \Sigma_n is the sample covariance matrix and \Omega is the inverse we want to estimate.

Value

An object with S3 class "clime". You can also use it as a regular R list with the following fields:

Author(s)

T. Tony Cai, Weidong Liu and Xi (Rossi) Luo
Maintainer: Xi (Rossi) Luo xi.rossi.luo@gmail.com

References

Cai, T.T., Liu, W., and Luo, X. (2011). A constrained \ell_1 minimization approach for sparse precision matrix estimation. Journal of the American Statistical Association 106(494): 594-607.

Examples

## trivial example
n <- 50
p <- 5
X <- matrix(rnorm(n*p), nrow=n)
re.clime <- clime(X)

## tridiagonal matrix example
bandMat <- function(p, k) {
  cM <- matrix(rep(1:p, each=p), nrow=p, ncol=p)
  return((abs(t(cM)-cM)<=k)*1)
}
## tridiagonal Omega with diagonal 1 and off-diagonal 0.5
Omega <- bandMat(p, 1)*0.5
diag(Omega) <- 1
Sigma <- solve(Omega)
X <- matrix(rnorm(n*p), nrow=n)%*%chol(Sigma)
re.clime <- clime(X, standardize=FALSE, linsolver="simplex")
re.cv <- cv.clime(re.clime)
re.clime.opt <- clime(X, standardize=FALSE, re.cv$lambdaopt)

## Compare Frobenius norm loss
## clime estimator
sqrt( sum( (Omega-re.clime.opt$Omegalist[[1]])^2 ) )
## Not run: 0.3438533
## Sample covariance matrix inversed
sqrt( sum( ( Omega-solve(cov(X)*(1-1/n)) )^2 ) )
## Not run: 0.874041
sqrt( sum( ( Omega-solve(cov(X)) )^2 ) )
## Not run: 0.8224296

internal clime functions

Description

Internal clime functions

Usage

likelihood(Sigma, Omega)
tracel2(Sigma, Omega)

Arguments

Details

There are not intended for use by users.

Author(s)

T. Tony Cai, Weidong Liu and Xi (Rossi) Luo
Maintainer: Xi (Rossi) Luo xi.rossi.luo@gmail.com

References

Cai, T.T., Liu, W., and Luo, X. (2011). A constrained \ell_1 minimization approach for sparse precision matrix estimation. Journal of the American Statistical Association 106(494): 594-607.

k-fold cross validation for clime object

Description

Perform a k-fold cross validation for selecting lambda

Usage

cv.clime(clime.obj, loss=c("likelihood", "tracel2"), fold=5) 

Arguments

Details

Perform a k-fold cross validation for selecting the tuning parameter lambda in clime. Two losses are implemented currently:

\textrm{likelihood: } Tr[\Sigma \Omega] - \log|\Omega| - p

\textrm{tracel2: } Tr[ diag(\Sigma \Omega - I)^2].

Value

An object with S3 class "cv.clime". You can use it as a regular R list with the following fields:

Author(s)

T. Tony Cai, Weidong Liu and Xi (Rossi) Luo
Maintainer: Xi (Rossi) Luo xi.rossi.luo@gmail.com

References

Cai, T.T., Liu, W., and Luo, X. (2011). A constrained \ell_1 minimization approach for sparse precision matrix estimation. Journal of the American Statistical Association 106(494): 594-607.

Examples

## trivial example
n <- 50
p <- 5
X <- matrix(rnorm(n*p), nrow=n)
re.clime <- clime(X)
re.cv <- cv.clime(re.clime)
re.clime.opt <- clime(X, re.cv$lambdaopt)

print a clime object

Description

Print a summary of the clime object.

Usage

## S3 method for class 'clime'
print(x,digits = max(3, getOption("digits") - 3), ... )

Arguments

Details

This call simply outlines the options used for computing a clime object.

Value

The output above is invisiblly returned.

Author(s)

T. Tony Cai, Weidong Liu and Xi (Rossi) Luo
Maintainer: Xi (Rossi) Luo xi.rossi.luo@gmail.com

References

Cai, T.T., Liu, W., and Luo, X. (2011). A constrained \ell_1 minimization approach for sparse precision matrix estimation. Journal of the American Statistical Association 106(494): 594-607.

Examples

## trivial example
n <- 50
p <- 5
X <- matrix(rnorm(n*p), nrow=n)
re.clime <- clime(X)
print(re.clime)

print a cross validated clime object

Description

Print a summary of the cv.clime object.

Usage

## S3 method for class 'cv.clime'
print(x,digits = max(3, getOption("digits") - 3), ... )

Arguments

Details

This call outputs first a three column matrix with lambda, mean and sd for the cross validation loss values. The actual loss used and the optimal lambda value picked by cv are printed.

Value

The output above is invisibly returned.

Author(s)

T. Tony Cai, Weidong Liu and Xi (Rossi) Luo
Maintainer: Xi (Rossi) Luo xi.rossi.luo@gmail.com

References

Cai, T.T., Liu, W., and Luo, X. (2011). A constrained \ell_1 minimization approach for sparse precision matrix estimation. Journal of the American Statistical Association 106(494): 594-607.

Examples

## trivial example
n <- 50
p <- 5
X <- matrix(rnorm(n*p), nrow=n)
re.clime <- clime(X)
re.cv <- cv.clime(re.clime)
print(re.cv)