klcv {sglasso}

Cross-Validated Kullback-Leibler Divergence

Description

Model selection criterion based on the leave-one-out cross-validated Kullback-Leibler divergence.

Usage

klcv(object, X, scale = 1)

Arguments

object

fitted sglasso/fglasso object;

X

the matrix used to compute the empirical variance/covariance matrix. Its dimension is N \times p, where p is the number of random variables and N is the samlpe size;

scale

scalar value used to scale the estimated degrees-of-freedom. See below for more details.

Details

klcv function implements the leave-one-out cross-validate Kullback-Leibler divergence criterion proposed in Vujacic et al. (2015). For l_1-penalized Gaussian Graphical Models this measure of goodness-of-fit has the following form

klcv(\rho) = -\frac{\ell(\hat K(\rho))}{N} + \frac{\code{scale}}{2N} gdf(\hat K(\rho)),

where \hat K(\rho) is the glasso estimate of the concentration matrix, \ell(\hat K(\rho)) is the corresponding value of the log-likelihood function, scale is a scale factor for the complexity part, i.e. gdf(\hat K(\rho)), which is defined as

gdf(\hat K(\rho)) = \frac{1}{N-1}\sum_{k=1}^N vec\{(\hat K(\rho)^{-1} - S_k)\circ 1_\rho\}'vec[\hat K(\rho)\{(S-S_k)\circ 1_\rho\}\hat K(\rho)].

In the previous expression S is the empirical variance/covariance matrix, S_k = X_k X_k', 1_\rho is a matrix with entries I(\hat k_{ij}(\rho)\ne 0) and \circ is the Hadamard product operator.

Value

klcv returns an S3 object with calls klcv, i.e. a named list with the following components:

klcv

the vector with the leave-one-out cross-validated Kullback-Leibler divergence;

rho

the rho-values used to compute the leave-one-out cross-validated Kullback-Leibler divergence;

loglik

a vector with the log-likelihood computed for the sequence of weighted l1-penalized RCON(V, E);

gdf

a vector returning the generalized degrees-of-freedom;

scale

the scale value used to define the leave-one-out cross-validated Kullback-Leibler divergence;

min.klcv

minimum value of the leave-one-out cross-validated Kullback-Leibler divergence;

rho.opt

the rho-value corresponding to minimum leave-one-out cross-validated Kullback-Leibler divergence;

rhoid

the index of the rho-value identified by the leave-one-out cross-validated Kullback-Leibler divergence.

Author(s)

Luigi Augugliaro
Maintainer: Luigi Augugliaro [e-mail]

References

Vujacic, I., Abbruzzo, A. and Wit, E. C. (2015) A computationally fast alternative to cross-validation in penalized Gaussian graphical models. J. Stat. Comput. Simul.

See Also

sglasso, loglik functions and plot.klcv method.

Examples

N <- 100
p <- 5
X <- matrix(rnorm(N * p), N, p)
S <- crossprod(X) / N
mask <- outer(1:p, 1:p, function(i,j) 0.5^abs(i-j))
mask[1,5] <- mask[1,4] <- mask[2,5] <- NA
mask[5,1] <- mask[4,1] <- mask[5,2] <- NA
out.sglasso_path <- sglasso(S, mask, tol = 1.0e-13)
out.klcv <- klcv(out.sglasso_path, X)
out.klcv