Monte Carlo methods for estimating Mallows's Cp and AIC criteria for PLSR models: Illustration on agronomic spectroscopic NIR data

Mallows's Cp and Akaike information criterion (AIC) are common criteria for selecting the dimensionality of regression models, as an alternative to cross-validation (CV) and nonparametric bootstrap. A key parameter in the calculation of Cp and AIC is the effective number of degrees of freedom of the model, or model complexity (d). Parameter d is generally easy to calculate for linear smoothers, that is, models for which the prediction of the training response y is given by  = S y where S is a projector matrix that does not involve y. Nevertheless, d is more difficult to estimate for nonlinear smoothers, such as partial least squares regression (PLSR). In this article, we present two algorithms for estimating d for PLSR based on Monte Carlo simulation methods (parametric bootstrap and perturbation analysis) and with the particular case of high dimensional data. We compare these Monte Carlo methods to three other algorithms already published. We used the d estimates to compute Cp and AIC and select PLSR model dimensionalities that we then compare to CV. Two real and heterogeneous agronomic near infrared (NIR) datasets were considered as examples.