AIC.dglars {dglars} | R Documentation

## Akaike's An Information Criterion

### Description

`AIC.dglars` is used to compute the Akaike's ‘An Information Criterion’ for the sequence of models estimated by “`dglars`”.

### Usage

    ## S3 method for class 'dglars'
    AIC(object, phi = c("pearson", "deviance", "mle", "grcv"),
    k = 2, complexity = c("df", "gdf"), g = NULL, ...)

    ## S3 method for class 'dglars'
    BIC(object, ...)

### Arguments

`object` | a fitted `dglars` object.
`phi` | a description of the estimator of the dispersion parameter (see below for more details).
`k` | non negative value used to weight the complexity of the fitted dglars model (see below for more details).
`complexity` | argument used to specify the method to measure the complexity of a fitted dglars model, i.e. the number of non-zero estimates (`complexity = "df"`) of the generalized degrees-of-freedom (`complexity = "gdf"`); see below for more details.
`g` | vector of values of the tuning parameter.
`...` | further arguments passed to the function `link{grcv}`.

### Details

The values returned by `AIC.dglars` are computed according to the following formula of a generic measure of Goodness-of-Fit (GoF):

-2 * log-likelihood + k * comp,

where “comp” represents the term used to measure the complexity of the fitted model, and k is the ‘weight’ of the complexity in the previous formula.

For binomial and Poisson family, the log-likelihood function is evaluated assuming that the dispersione parameter is known and equal to one while for the remaining families the dispersion parameter is estimated by the method specified by `phi` (see `phihat` for more details).

According to the results given in Augugliaro et. al. (2013), the complexity of a model fitted by dglars method can be measured by the classical notion of ‘Degrees-of-Freedom’ (`complexity = "df"`), i.e., the number of non-zero estimated, or by the notion of ‘Generalized Degrees-of-Freedom’ (`complexity = "gdf"`).

By the previous formula, it is easy to see that the standard AIC-values are obtained setting `k = 2` and `complexity = "df"` (default values for the function `AIC.dglars`) while the so-called BIC-values (Schwarz's Bayesian criterion) are obtained setting `k = log(n)`, where n denotes the sample size, and `complexity = "df"` (default values for the function `BIC.dglars`).

The optional argument `g` is used to specify the values of the tuning parameter; if not specified (default), the values of the measure of goodness-of-fit are computed for the sequence of models storage in `object` otherwise `predict.dglars` is used to compute the estimate of the parameters needed to evaluate the log-likelihood function (see the example below).

### Value

`AIC.dglars` and `BIC.dglars` return a named list with class “`gof_dglars`” and components:

`val` | the sequence of AIC/BIC-values;
`g` | the sequence of g-values;
`loglik` | the sequence of log-likelihood values used to compute the AIC or BIC;
`k` | the non negative value used to weight the complexity of the fitted dglars model;
`comp` | the measures of model complexity used to compute the measure of goodness-of-fit. It is equal to `npar` when codecomplexity = "df";
`npar` | the seqeunce of the number of non-zero estimates
`phi` | a description of the estimator used to estimate the dispersion pamater;
`phih` | the vector of penalized estimate of the dispersion parameter used to evaluate the log-likelihood function;
`complexity` | character specifying the method to measure the complexity of a fitted dglars model;
`object` | the fitted `dglars` object;
`type` | character specifying the type of used measure-of-goodness of fit, i.e., AIC, BIC or GoF.

In order to summarize the information about the AIC-valuse, a `print` method is available for an object with class “`gof_dglars`”.

### Author(s)

Luigi Augugliaro
Maintainer: Luigi Augugliaro luigi.augugliaro@unipa.it

### References

Augugliaro L., Mineo A.M. and Wit E.C. (2013) dgLARS: a differential geometric approach to sparse generalized linear models, Journal of the Royal Statistical Society. Series B., Vol 75(3), 471-498.

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

### See Also

`logLik.dglars`, `predict.dglars`, `dglars` and `summary.dglars`.

### Examples

    #################################
    # y ~ Pois

    library("dglars")
    set.seed(123)
    n <- 100
    p <- 5
    X <- matrix(abs(rnorm(n*p)),n,p)
    eta <- 1 + X[, 1]  + X[, 2]
    mu <- poisson()$linkinv(eta)
    y <- rpois(n, mu)
    out <- dglars(y ~ X, poisson)
    out
    AIC(out)
    AIC(out, g = seq(2, 1, by = -0.1))
    AIC(out, complexity = "gdf")
    AIC(out, k = log(n)) #BIC-values
    BIC(out)

    #################################
    # y ~ Gamma

    n <- 100
    p <- 50
    X <- matrix(abs(rnorm(n*p)),n,p)
    eta <- 1 + 2 * X[, 1L]
    mu <- drop(Gamma()$linkinv(eta))
    shape <- 0.5
    phi <- 1 / shape
    y <- rgamma(n, scale = mu / shape, shape = shape)
    out <- dglars(y ~ X, Gamma("log"))

    AIC(out, phi = "pearson")
    AIC(out, phi = "deviance")
    AIC(out, phi = "mle")
    AIC(out, phi = "grcv")

[Package dglars version 2.1.2 Index]