pdsoft.cv {PDSCE}

Tuning parameter selection and computation for the positive definite and sparse covariance matrix estimator

Description

Computes and selects the tuning parameter for the sparse and positive definite covariance matrix estimator proposed by Rothman (2012).

Usage

pdsoft.cv(x, lam.vec = NULL, standard = TRUE, 
          init = c("diag", "soft", "dense"), tau = 1e-04, 
          nsplits = 10, n.tr = NULL, tolin = 1e-08, tolout = 1e-08, 
          maxitin = 10000, maxitout = 1000, quiet = TRUE)

Arguments

x

A data matrix with n rows and p columns. The rows are assumed to be a realization of n independent copies of a p-variate random vector.

lam.vec

An optional vector of candidate lasso-type penalty tuning parameter values. The default for standard=TRUE is seq(from=0, to=1, by=0.05) and the default for standard=FALSE is seq(from=0, to=m, length.out=20), where m is the maximum magnitude of the off-diagonal entries in s. Both of these default choices are far from excellent and are time consuming, particularly for values close to zero. The user should consider refining this set by increasing its resolution in a narrower range.

standard

Logical: standard=TRUE first computes the observed sample correlation matrix from s, then computes the sparse correlation matrix estimate, and finally rescales to return the sparse covariance matrix estimate. The strongly recommended default is standard=TRUE.

init

The type of initialization used for the estimate computed at the maximum element in lam.vec. Subsequent initializations use the final iterates for sigma and omega at the previous value in lam.vec. The default option init="diag" uses diagonal starting values. The second option init="soft" uses a positive definite version of the soft thresholded covariance or correlation estimate, depending on standard. The third option init="dense" uses the closed-form solution when lam=0.

tau

The logarithmic barrier parameter. The default is tau=1e-4, which works well when standard=TRUE with the default choices for the convergence tolerances.

nsplits

The number of random splits to use for the tuning parameter selection.

n.tr

Optional number of cases to use in the training set. The default is the nearest integer to n(1-1/\log(n)). The value must be in \{3, \ldots, n-2\}.

tolin

Convergence tolerance for the inner loop of the algorithm that solves the lasso regression.

tolout

Convergence tolerance for the outer loop of the algorithm.

maxitin

Maximum number of inner-loop iterations allowed

maxitout

Maximum number of outer-loop iterations allowed

quiet

Logical: quiet=TRUE suppresses the printing of progress updates.

Details

See Rothman (2012) for the objective function and more information.

Value

A list with

sigma

covariance estimate at the selected tuning parameter

omega

inverse covariance estimate at the selected tuning parameter

best.lam

the selected value of the tuning parameter

cv.err

a vector of the validation errors, one for each element in lam.vec

lam.vec

the vector of candidate tuning parameter values

n.tr

the number of cases used for the training set

Note

It is always the case that omega is positive definite. If tolin and tolout are too large, or maxitin and maxitout are too small, then sigma may be indefinite.

Author(s)

Adam J. Rothman

References

Rothman, A. J. (2012). Positive definite estimators of large covariance matrices. Biometrika 99(3): 733-740

See Also

pdsoft

Examples

set.seed(1)
n=50
p=20
true.cov=diag(p)
true.cov[cbind(1:(p-1), 2:p)]=0.4
true.cov[cbind(2:p, 1:(p-1))]=0.4
eo=eigen(true.cov, symmetric=TRUE)
z=matrix(rnorm(n*p), nrow=n, ncol=p)
x=z%*% tcrossprod(eo$vec*rep(eo$val^(0.5), each=p),eo$vec)
output=pdsoft.cv(x=x)
plot(output$lam.vec, output$cv.err)
output$best.lam
output$sigma