cglasso (version 1.1.2)

ebic: Extended Bayesian Information Criterion

Description

‘ebic’ function computes the extended Bayesian Information Criterion.

Usage

ebic(object, g)
# S3 method for glasso
ebic(object, g = 0.5)
# S3 method for mglasso
ebic(object, g = 0.5)
# S3 method for cglasso
ebic(object, g = 0.5)

Arguments

object

a fitted model object.

g

the parameter indexing the extended BIC: a value belonging to the interval \([0, 1]\). Default is 0.5

Value

‘ebic’ returns an object with S3 class “gof”, i.e. a list containing the following components:

value_gof

the values of the measure of goodness-of-fit used to evaluate the fitted models.

rho

the values of the tuning parameter used to fit the models.

value

the values of the log-likelihood or Q-function.

df

the number of the estimated non-zero parameters, i.e. the number of non-zero partial correlations plus \(2p\).

n

the sample size.

p

the number of variables.

model

the name of the fitted model.

type

the measure of goodness-of-fit used to evaluate the fitted models.

Details

The measure of goodness-of-fit (gof) returned by the function ‘ebic’ depends on the class of the fitted model.

If ‘object’ has class ‘glasso’ or ‘ggm’, then ‘ebic’ computes the extended Bayesian Information Criterion (eBIC) proposed in Foygel and others (2010): $$\mbox{eBIC} = -2\,\mbox{log-likelihood} + a(\rho)(\log n + 4 \gamma log p),$$ where \(a(\rho)\) denotes the number of non-zero off-diagonal elements in \(\hat{\Theta}^\rho\) and \(\gamma\) is a value belonging to the interval \([0, 1]\) indexing the measure of goodness-of-fit. As explained in Foygel and others (2010), the log-likelihood function is evaluated using the maximum likelihood estimates of the model select by glasso. For this reason, ‘ebic’ calls the generic function mle to fit the Gaussian graphical model (GGM) selected by glasso.

For the remaining models, eBIC is defined as: $$\mbox{eBIC} = -2\,Q\mbox{-function} + a(\rho)(\log n + 4 \gamma log p),$$ where the \(Q\)-function is evaluated at the M-step of the EM-like algorithm describted in mle.

‘ebic’ returns an object with S3 class ‘gof’ for which are available the method functions ‘print.gof’ and ‘plot.gof’. These method functions are developed with the aim of helping the user in finding the optimal value of the tuning parameter, defined as the \(\rho\)-value minimizing the eBIC meaure. For this reason, ‘print.gof’ shows also the ranking of the fitted models (the best model is pointed out with an arrow) whereas ‘plot.gof’ points out the optimal \(\rho\)-value by a vertical dashed line (see below for some examples).

References

Foygel, R. and Drton, M. (2010). Extended Bayesian Information Criteria for Gaussian Graphical Models. In: Lafferty, J., Williams, C., Shawe-taylor, J., Zemel, R.s. and Culott, A. (editors), Advances in Neural Information Processing Systems 23. pp. 604--612.

See Also

loglik, cglasso, mglasso, glasso, mle, aic, bic and the method funtions plot and summary.

Examples

# NOT RUN {
library("cglasso")
set.seed(123)

#################
# cglasso model #
#################
n <- 100L
p <- 5L
mu <- rep.int(0L, times = p)
X <- rdatacggm(n = n, mu = mu, probr = 0.05)
out <- cglasso(X = X)
out_ebic <- ebic(out)
out_ebic
plot(out_ebic)

##############
# cggm model #
##############
out_mle <- mle(out)
out_ebic <- ebic(out_mle)
out_ebic
plot(out_ebic)

#################
# mglasso model #
#################
X <- rnorm(n * p)
id.na <- sample.int(n = n * p, size = n * p * 0.05)
X[id.na] <- NA
dim(X) <- c(n, p)
out <- mglasso(X = X)
out_ebic <- ebic(out)
out_ebic
plot(out_ebic)

##############
# mggm model #
##############
out_mle <- mle(out)
out_ebic <- ebic(out_mle)
out_ebic
plot(out_ebic)

#################
# glasso model #
#################
X <- rnorm(n * p)
dim(X) <- c(n, p)
out <- glasso(X)
out_ebic <- ebic(out)
out_ebic
plot(out_ebic)

#############
# ggm model #
#############
out_mle <- mle(out)
out_ebic <- ebic(out_mle)
out_ebic
plot(out_ebic)
# }