BIC.cglasso {cglasso}

Bayesian Information Criterion

Description

‘BIC’ computes the Bayesian Information Criterion.

Usage

## S3 method for class 'cglasso'
BIC(object, g = 0, type, mle, ...)

Arguments

object

an R object inheriting class ‘cglasso’, that is, the output of the model-fitting functions cglasso and cggm.

g

a value belonging to the interval [0, 1]. Classical BIC is returned by letting g = 0 (default value), whereas extended BIC corresponds to the case g = 0.5.

type

character; if g is not zero then the measure proposed in Foygel and other (2010) is returned by setting type = "FD", otherwise (type = "CC") returns the measure proposed in Chen and other (2008, 2012). See section ‘Details’ for more details.

mle

logical. TRUE if the measure of goodness-of-fit should be computed using the maximum likelihood estimates. Default depends on the class of the argument object: mle = FALSE for objects of class cglasso and mle = TRUE for objects of class cggm.

...

further arguments passed to the model-fitting function cggm.

Details

‘BIC’ computes the Bayesian Information Criterion (BIC) for models fitted by cglasso or cggm. As proposed in Ibrahim and other (2008), BIC computes the measure of goodness-of-fit by replacing the log-likelihood function with the Q-function, that is, the function maximized in the M-Step of the EM-algorithm. The values of the Q-function are computed using QFun. By default, for an object of class cglasso these values are computed using the penalized estimates whereas, if the object has class cggm, maximum likelihood estimates are used (see argument ‘mle’ in QFun).

By default, BIC computes the standard BIC measure (\gamma = 0):

-2\,\mbox{Q-function} + \log(n)\,\mbox{df},

where n is the sample size and \mbox{df} represents the number of unique non-zero parameters in the fitted model.

If \gamma \ne 0, the default depends on the number of predictors (q).

If q = 0, BIC computes the measure of goodness-of-fit proposed in Foygel and other (2010) (type = "FD"):

\mbox{eBIC} = -2\,\mbox{QFun} + (\log n + 4 \, \gamma \, log \, p)\,\mbox{df},

where \gamma is a value belonging to the interval [0, 1] and indexing the measure of goodness-of-fit.

If q \ne 0 , BIC computes the measure of goodness-of-fit proposed in Chen and other (2008, 2012) (type = "CC"):

\mbox{eBIC} = -2\,\mbox{QFun} + (\log n + 2 \, \gamma \, log \, q)\,\mbox{df},

BIC can be passed to the functions select_cglasso and summary.cglasso to select and print the optimal fitted model, respectively.

The function plot.GoF can be used to graphically evaluate the behaviour of the fitted models in terms of goodness-of-fit.

Value

‘BIC’ returns an R object of S3 class “GoF”, i.e. a named list containing the following components:

value_gof

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

df

a matrix storing the number of the estimated non-zero parameters.

dfB

a matrix storing the number of estimated non-zero regression coefficients.

dfTht

a matrix storing the number of estimated non-zero partial correlation coefficients.

value

a matrix storing the values of the Q-function.

n

the sample size.

p

the number of response variables.

q

the number of columns of the design matrix X used to fit the model.

lambda

the \lambda-values used to fit the model.

nlambda

the number of \lambda-values used.

rho

the \rho-values used to fit the model.

nrho

the number of \rho-values used.

type

a description of the computed measure of goodness-of-fit.

model

a description of the fitted model passed through the argument object.

Author(s)

Luigi Augugliaro ([e-mail])

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.

Chen, J. and Chen, Z. (2008) <doi:10.1093/biomet/asn034>. Extended Bayesian information criteria for model selection with large model spaces. Biometrika, Vol. 95(2), pp. 759–771.

Chen, J. and Chen, Z. (2012) <doi:10.5705/ss.2010.216>. Extended BIC for small-n-large-p sparse GLM. Statistica Sinica, Vol. 22, pp. 555–574.

Wit, E., Heuvel, E. V. D., & Romeijn, J. W. (2012) <doi:10.1111/j.1467-9574.2012.00530.x>. All models are wrong...?: an introduction to model uncertainty. Statistica Neerlandica, 66(3), 217-236.

See Also

cglasso, cggm, AIC.cglasso, QFun, plot.GoF and summary.cglasso

Examples

set.seed(123)

# Y ~ N(0, Sigma) and probability of left/right censored values equal to 0.05
n <- 100L
p <- 3L
rho <- 0.3
Sigma <- outer(1L:p, 1L:p, function(i, j) rho^abs(i - j))
Z <- rcggm(n = n, Sigma = Sigma, probl = 0.05, probr = 0.05)
out <- cglasso(. ~ ., data = Z)
BIC(out)                                    # standard BIC measure
BIC(out, mle = TRUE, g = 0.5, type = "FD")  # eBIC proposed in Foygel and other (2010)

# Y ~ N(b0 + XB, Sigma)  and probability of left/right censored values equal to 0.05
n <- 100L
p <- 3L
q <- 2
b0 <- runif(p)
B <- matrix(runif(q * p), nrow = q, ncol = p)
X <- matrix(rnorm(n * q), nrow = n, ncol = q)
rho <- 0.3
Sigma <- outer(1L:p, 1L:p, function(i, j) rho^abs(i - j))
Z <- rcggm(n = n, b0 = b0, X = X, B = B, Sigma = Sigma, probl = 0.05, probr = 0.05)
out <- cglasso(. ~ ., data = Z)
BIC(out)                                    # standard BIC measure
BIC(out, mle = TRUE, g = 0.5, type = "FD")  # eBIC proposed in Foygel and other (2010)
BIC(out, mle = TRUE, g = 0.5, type = "CC")  # eBIC proposed in Chen and other (2008, 2010)