Akaike and Bayesian Information Criteria and Generalized minimum description length

This function computes the Akaike and Bayesian Information Criteria and the Generalized minimum description length.

aic.dof(RSS, n, DoF, sigmahat)

bic.dof(RSS, n, DoF, sigmahat)

gmdl.dof(sigmahat, n, DoF, yhat)

Arguments

RSS	vector of residual sum of squares.
n	number of observations.
DoF	vector of Degrees of Freedom. The length of DoF is the same as the length of RSS.
sigmahat	Estimated model error. The length of sigmahat is the same as the length of RSS.
yhat	vector of squared norm of Yhat. The length of yhat is the same as the length of sigmahat.

Value

vector

numerical values of the requested AIC, BIC or GMDL.

Details

The gmdl criterion is defined as $$gmdl=\frac{n}{2}log(S)+\frac{DoF}{2}log(F)+\frac{1}{2}log(n)$$ with $$S=\hat\sigma^2$$

References

M. Hansen, B. Yu. (2001). Model Selection and Minimum Descripion Length Principle, Journal of the American Statistical Association, 96, 746-774.
N. Kraemer, M. Sugiyama. (2011). The Degrees of Freedom of Partial Least Squares Regression. Journal of the American Statistical Association, 106(494), 697-705.
N. Kraemer, M.L. Braun, Kernelizing PLS, Degrees of Freedom, and Efficient Model Selection, Proceedings of the 24th International Conference on Machine Learning, Omni Press, (2007) 441-448.

See also

plsR.dof for degrees of freedom computation and infcrit.dof for computing information criteria directly from a previously fitted plsR model.

Author

Frédéric Bertrand
frederic.bertrand@utt.fr
https://fbertran.github.io/homepage/

Examples

data(Cornell)
XCornell<-Cornell[,1:7]
yCornell<-Cornell[,8]
modpls <- plsR(yCornell,XCornell,4)
#> ____************************************************____
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Component____ 4 ____
#> ____Predicting X without NA neither in X nor in Y____
#> ****________________________________________________****
#> 
dof.object <- plsR.dof(modpls)
aic.dof(modpls$RSS,modpls$nr,dof.object$DoF,dof.object$sigmahat)
#>          [,1]     [,2]     [,3]      [,4]      [,5]
#> [1,] 46.07088 4.569969 2.107546 0.8467795 0.8232505
bic.dof(modpls$RSS,modpls$nr,dof.object$DoF,dof.object$sigmahat)
#>          [,1]     [,2]     [,3]      [,4]      [,5]
#> [1,] 47.78935 4.955816 2.394933 0.9628191 0.9357846
gmdl.dof(dof.object$sigmahat,modpls$nr,dof.object$DoF,dof.object$yhat)
#> [1] 27.59461 21.34020 27.40202 24.40842 24.23105
naive.object <- plsR.dof(modpls,naive=TRUE)
aic.dof(modpls$RSS,modpls$nr,naive.object$DoF,naive.object$sigmahat)
#>          [,1]     [,2]     [,3]      [,4]      [,5]
#> [1,] 46.07088 4.169957 1.537029 0.7363469 0.8721072
bic.dof(modpls$RSS,modpls$nr,naive.object$DoF,naive.object$sigmahat)
#>          [,1]    [,2]     [,3]      [,4]      [,5]
#> [1,] 47.78935 4.45882 1.686092 0.8256118 0.9964867
gmdl.dof(naive.object$sigmahat,modpls$nr,naive.object$DoF,naive.object$yhat)
#> [1] 27.59461 18.37545 17.71117 19.01033 24.16510