Estimation of DESP by penalized likelihood minimization

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Description

This function estimates the diagonal of the precision matrix by penalized likelihood minimization, when the true value of the coefficient matrix B is known or has already been estimated. The observations of the data matrix X are assumed to have zero mean.

Usage

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  DESP_PML(X, B, thresh, kappa, tol, Theta = NULL)

Arguments

X

The data matrix.

B

The coefficient matrix.

thresh

The threshold level.

kappa

The tunning paramater.

tol

The gradient magnitude tolerance.

Theta

The matrix orresponding to outliers.

Details

When Theta is not NULL, we consider an additive contamination model. We assume that X = Y + E is observed, denoting the outlier-free data by Y and the matrix of errors by E. In this case, the matrix Theta should be equal to E * B.

Value

This function returns the diagonal of the precision matrix associated with X as a vector.

Author(s)

Arnak Dalalyan and Samuel Balmand.

Examples

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## build the true precision matrix
p <- 12 # number of variables
Omega <- 2*diag(p)
Omega[1,1] <- p 
Omega[1,2:p] <- 2/sqrt(2)
Omega[2:p,1] <- 2/sqrt(2)
## compute the true diagonal of the precision matrix
Phi <- 1/diag(Omega)
## generate the design matrix from a zero-mean Gaussian distribution
require(MASS)
n <- 100 # sample size
X <- mvrnorm(n,rep.int(0,p),ginv(Omega))
## compute the sample mean
barX <-  apply(X,2,mean)
## subtract the mean from all the rows
X <- t(t(X)-barX)
## estimate the coefficient matrix 
B <- DESP_SRL_B(X,lambda=sqrt(2*log(p)))
## compute the squared partial correlations
SPC <- DESP_SqPartCorr(B,n)
## re-estimate the coefficient matrix by ordinary least squares
B_OLS <- DESP_OLS_B(X,SPC)
## estimate the diagonal of the precision matrix and get its inverse
hatPhiPML <- 1/DESP_PML(X,B_OLS,thresh=0.05,kappa=0.1, tol=1e-5)
## measure the performance of the estimation using l2 vector norm
sqrt(sum((Phi-hatPhiPML)^2))