Fast algorithm for smoothing parameter selection in multidimensional generalized P-splines

A new computational algorithm for estimating the smoothing parameters of a multidimensional penalized spline generalized model with anisotropic penalty is presented. This new proposal is based on the mixed model representation of a multidimensional P-spline, in which the smoothing parameter for each covariate is expressed in terms of variance components. On the basis of penalized quasi-likelihood methods (PQL), closed-form expressions for the estimates of the variance components are obtained. This formulation leads to an efficient implementation that can considerably reduce the computational load. The proposed algorithm can be seen as a generalization of the algorithm by Schall (1991) - for variance components estimation - to deal with non-standard structures of the covariance matrix of the random effects. The practical performance of the proposed computational algorithm is evaluated by means of simulations, and comparisons with alternative methods are made on the basis of the mean square error criterion and the computing time. Finally, we illustrate our proposal with the analysis of two real datasets: a two dimensional example of historical records of monthly precipitation data in USA and a three dimensional one of mortality data from respiratory disease according to the age at death, the year of death and the month of deathThe authors would like to express their gratitude for the support received in the form of the Spanish Ministry of Economy and Competitiveness grants MTM2011-28285-C02-01 and MTM2011-28285-C02-02. Work of Mar a Xose Rodríguez - Alvarez was supported by grant CA09/0053 from the Instituto de Salud Carlos III. The research of Dae-Jin Lee was funded by an NIH grant for the Superfund Metal Mixtures, Biomarkers and Neurodevelopment project 1PA2ES016454-01A