sparsePrec {sparseMatEst}

Sparse precision matrix estimator with error control

Description

Given a data matrix, sparsePrec estimates the precision matrix for the data under the assumption that the true precision matrix is sparse, i.e. that most of the off-diagonal entries are equal to zero.

Usage

sparsePrec(dat = 0, prec = 0, alf = 0.5, iter = 10, pnrm = Inf,
  THRSH = "hard", rho = 1, regMeth = "glasso")

Arguments

dat

nxk data matrix, n observations, k dimensions

prec

start with a precision estimator instead of dat

alf

false positive rate in [0,1], Default is 0.5

iter

number of iterates, Default is 10

pnrm

norm to use, Default = Inf

THRSH

Type of thresholding used; Takes values: hard, soft, adpt, scad.

rho

Penalization parameter for the initial estimator, Default is 1

regMeth

Type of initial estimator, Default is 'glasso'

Details

The algorithm begins with an initial precision matrix estimator being either regMeth = 'glasso' for debiased graphical lasso or regMeth = 'ridge' for debiased ridge estimator. It then iteratively computes covariance estimators with false positive rates of alf, alf^2, and so on until iter estimators have been constructed.

If dat is not included, but prec is, then the code runs as before but using the argument prec as the initial precision matrix estimator. In this case, the regMeth is not considered.

The norm chosen determines the topology on the space of matrices. pnorm defaults to Inf being the operator norm or maximal eigenvalue. This is theoretically justified to work in the references. Other norms could be considered, but their performance is not a strong.

Four thresholding methods are implemented. THRSH defaults to hard thresholding where small matrix entries are set to zero while large entries are not affected. The soft and adpt thresholds shrink all entries towards zero. The scad threshold interpolates between hard and soft thresholding. More details can be found in the references.

Value

a list of arrays containing iter+1 sparse precision matrices corresponding to false positive rates of 1, alf, alf^2,..., alf^iter. Each list corresponds to one type of thresholding chosen by THRSH.

Author(s)

Adam B Kashlak [e-mail]

References

Kashlak, Adam B. "Non-asymptotic error controlled sparse high dimensional precision matrix estimation." arXiv preprint arXiv:1903.10988 (2019).

Examples

# Generate four sparse covariance matrix estimators
# with false positive rates of 0.5, 0.25, 0.125,
# and 0.0625
n = 30
k = 50
dat = matrix(rnorm(n*k),n,k)
out = sparsePrec( dat, alf=0.5, iter=4, THRSH=c("hard","soft") )
  par(mfcol=c(2,2))
  lab = c(1,0.5,0.5^2,0.5^3,0.5^4);
  for( i in 2:5 )
    image( out$hard[,,i]!=0, main=lab[i] )
  for( i in 2:5 )
    image( log(abs(out$hard[,,i] - out$soft[,,i])), main=lab[i] )