# AIC.dglars: Akaike's An Information Criterion In dglars: Differential Geometric Least Angle Regression

## Description

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

## Usage

 ```1 2 3 4 5 6``` ```## 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 [email protected]

## 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.

`logLik.dglars`, `predict.dglars`, `dglars` and `summary.dglars`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36``` ```################################# # 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") ```