ridgePgen {porridge}

Ridge estimation of the inverse covariance matrix with element-wise penalization and shrinkage.

Description

Function that evaluates the generalized ridge estimator of the inverse covariance matrix with element-wise penalization and shrinkage.

Usage

ridgePgen(S, lambda, target, nInit=100, minSuccDiff=10^(-10))

Arguments

S

Sample covariance matrix.

lambda

A symmetric matrix with element-wise positive penalty parameters.

target

A semi-positive definite target matrix towards which the estimate is shrunken.

nInit

A numeric specifying the number of iteration.

minSuccDiff

A numeric: minimum distance between two succesive estimates to be achieved.

Details

This function generalizes the ridgeP-function in the sense that, besides element-wise shrinkage, it allows for element-wise penalization in the estimation of the precision matrix of a zero-mean multivariate normal distribution. Hence, it assumes that the data stem from \mathcal{N}(\mathbf{0}_p, \boldsymbol{\Omega}^{-1}). The estimator maximizes the following penalized loglikelihood:

\log( | \boldsymbol{\Omega} |) - \mbox{tr} ( \boldsymbol{\Omega} \mathbf{S} ) - \| \boldsymbol{\Lambda} \circ (\boldsymbol{\Omega} - \mathbf{T}) \|_F^2,

where \mathbf{S} the sample covariance matrix, \boldsymbol{\Lambda} a symmetric, positive matrix of penalty parameters, the \circ-operator represents the Hadamard or element-wise multipication, and \mathbf{T} the precision matrix' shrinkage target. For more details see van Wieringen (2019).

Value

The function returns a regularized inverse covariance matrix.

Author(s)

W.N. van Wieringen.

References

van Wieringen, W.N. (2019), "The generalized ridge estimator of the inverse covariance matrix", Journal of Computational and Graphical Statistics, 28(4), 932-942.

See Also

ridgeP.

Examples

# set dimension and sample size
p <- 10
n <- 10

# penalty parameter matrix
lambda       <- matrix(1, p, p)
diag(lambda) <- 0.1

# generate precision matrix
Omega       <- matrix(0.4, p, p)
diag(Omega) <- 1
Sigma       <- solve(Omega)

# data 
Y <- mvtnorm::rmvnorm(n, mean=rep(0,p), sigma=Sigma)
S <- cov(Y)

# unpenalized diagonal estimate
Phat <- ridgePgen(S, lambda, 0*S)