ridgePmultiT {porridge}

Ridge estimation of the inverse covariance matrix with multi-target shrinkage.

Description

Function that evaluates the ridge estimator of the inverse covariance matrix with multi-target shrinkage.

Usage

ridgePmultiT(S, lambda, targetList)

Arguments

S

Sample covariance matrix.

lambda

A numeric of positive penalty parameters. Values should be specified in the same order as the target's appearance in targetList.

targetList

A list of semi-positive definite target matrices towards which the precision matrix is potentially shrunken.

Details

This function generalizes the ridgeP-function in the sense that multiple shrinkage targets can be provided 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} ) - \sum\nolimits_{g=1}^G \lambda_g \| \boldsymbol{\Omega} - \mathbf{T}_g \|_F^2,

where \mathbf{S} the sample covariance matrix, \{ \lambda_g \}_{g=1}^G the penalty parameters of each target matrix, and the \{ \mathbf{T}_g \}_{g=1}^G the precision matrix' shrinkage targets. For more details see van Wieringen et al. (2020).

Value

The function returns a regularized inverse covariance matrix.

Author(s)

W.N. van Wieringen.

References

van Wieringen, W.N., Stam, K.A., Peeters, C.F.W., van de Wiel, M.A. (2020), "Updating of the Gaussian graphical model through targeted penalized estimation", Journal of Multivariate Analysis, 178, Article 104621.

See Also

ridgeP.

Examples

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

# specify vector of penalty parameters
lambda       <- c(2, 1)

# generate precision matrix
T1       <- matrix(0.7, p, p)
diag(T1) <- 1
T2       <- diag(rep(2, p))

# generate precision matrix
Omega       <- matrix(0.4, p, p)
diag(Omega) <- 2
Sigma       <- solve(Omega)

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

# unpenalized diagonal estimate
Phat <- ridgePmultiT(S, lambda, list(T1=T1, T2=T2))