AIC.cglasso {cglasso}

Akaike Information Criterion

Description

‘AIC’ computes the ‘Akaike Information Criterion’.

Usage

## S3 method for class 'cglasso'
AIC(object, k = 2, mle, ...)

Arguments

object

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

k

the penalty parameter to be used; the default k = 2 is the classical AIC.

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 cggm.

Details

‘AIC’ computes the following measure of goodness-of-fit (Ibrahim and other, 2008):

-2\,\mbox{Q-function} + k\,\mbox{df},

where k is the penalty parameter and \mbox{df} represents the number of unique non-zero parameters in the fitted model.

The values of the Q-function function are computed using QFun. By default, for an object of class cglasso these values are computed using the penalized estimates whereas maximum likelihood estimates are used if the object is of class cggm (see argument ‘mle’ in QFun).

The Akaike Information Criterion (AIC) is returned by letting k = 2 (default value of the function AIC) whereas the ‘Bayesian Information Criterion’ (BIC) is returned by letting k = \log(n), where n is the sample size.

Function AIC 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

‘AIC’ return 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

Ibrahim, J.G., Zhu, H. and Tang, N. (2008) <doi:10.1198/016214508000001057>. Model selection criteria for missing-data problems using the EM algorithm. Journal of the American Statistical Association 103, 1648–1658.

Sakamoto, Y., Ishiguro, M., and Kitagawa, G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.

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

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

Examples

set.seed(123)

# Y ~ N(b0 + XB, Sigma) and probability of left/right censored values equal to 0.05
n <- 100L
p <- 3L
q <- 2L
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)
AIC(out)

out.mle <- cggm(out, lambda.id = 3L, rho.id = 3L)
AIC(out.mle)