RIDGEsigma {ADMMsigma}

Ridge penalized precision matrix estimation

Description

Ridge penalized matrix estimation via closed-form solution. If one is only interested in the ridge penalty, this function will be faster and provide a more precise estimate than using ADMMsigma.
Consider the case where X_{1}, ..., X_{n} are iid N_{p}(\mu, \Sigma) and we are tasked with estimating the precision matrix, denoted \Omega \equiv \Sigma^{-1}. This function solves the following optimization problem:

Objective:

\hat{\Omega}_{\lambda} = \arg\min_{\Omega \in S_{+}^{p}} \left\{ Tr\left(S\Omega\right) - \log \det\left(\Omega \right) + \frac{\lambda}{2}\left\| \Omega \right\|_{F}^{2} \right\}

where \lambda > 0 and \left\|\cdot \right\|_{F}^{2} is the Frobenius norm.

Usage

RIDGEsigma(X = NULL, S = NULL, lam = 10^seq(-2, 2, 0.1), path = FALSE,
  K = 5, cores = 1, trace = c("none", "progress", "print"))

Arguments

X

option to provide a nxp data matrix. Each row corresponds to a single observation and each column contains n observations of a single feature/variable.

S

option to provide a pxp sample covariance matrix (denominator n). If argument is NULL and X is provided instead then S will be computed automatically.

lam

positive tuning parameters for ridge penalty. If a vector of parameters is provided, they should be in increasing order. Defaults to grid of values 10^seq(-2, 2, 0.1).

path

option to return the regularization path. This option should be used with extreme care if the dimension is large. If set to TRUE, cores will be set to 1 and errors and optimal tuning parameters will based on the full sample. Defaults to FALSE.

K

specify the number of folds for cross validation.

cores

option to run CV in parallel. Defaults to cores = 1.

trace

option to display progress of CV. Choose one of progress to print a progress bar, print to print completed tuning parameters, or none.

Value

returns class object RIDGEsigma which includes:

Lambda

optimal tuning parameter.

Lambdas

grid of lambda values for CV.

Omega

estimated penalized precision matrix.

Sigma

estimated covariance matrix from the penalized precision matrix (inverse of Omega).

Path

array containing the solution path. Solutions are ordered dense to sparse.

Gradient

gradient of optimization function (penalized gaussian likelihood).

MIN.error

minimum average cross validation error (cv.crit) for optimal parameters.

AVG.error

average cross validation error (cv.crit) across all folds.

CV.error

cross validation errors (cv.crit).

Author(s)

Matt Galloway [e-mail]

References

• Rothman, Adam. 2017. 'STAT 8931 notes on an algorithm to compute the Lasso-penalized Gaussian likelihood precision matrix estimator.'

See Also

plot.RIDGE, ADMMsigma

Examples

# generate data from a sparse matrix
# first compute covariance matrix
S = matrix(0.7, nrow = 5, ncol = 5)
for (i in 1:5){
 for (j in 1:5){
   S[i, j] = S[i, j]^abs(i - j)
 }
 }

# generate 100 x 5 matrix with rows drawn from iid N_p(0, S)
set.seed(123)
Z = matrix(rnorm(100*5), nrow = 100, ncol = 5)
out = eigen(S, symmetric = TRUE)
S.sqrt = out$vectors %*% diag(out$values^0.5)
S.sqrt = S.sqrt %*% t(out$vectors)
X = Z %*% S.sqrt

# ridge penalty no ADMM
RIDGEsigma(X, lam = 10^seq(-5, 5, 0.5))