Out-of-sample prediction in multidimensional P-spline models

Prediction of out-of-sample values is a problem of interest in any regression model. In the context of penalized smooth mixed model regression Carballo et al. (2017) have proposed a general framework for prediction in additive models without interaction terms. The aim of this paper is to extend this work, based on the methodology proposed in Currie et al. (2004), to models that include interaction terms, i.e. prediction is needed in multidimensional setting. Our approach fits the data and predicts the new observations simultaneously and uses constraints to ensure a coherent fit or to impose further restrictions on the predictions. We also develop this methodology for the so called smooth-ANOVA models which allow us to include interaction terms that can be decomposed as a sum of several smooth functions. To illustrate the methodology two real data sets are used, one to predict log mortality rates in the Spanish population and another to predict aboveground biomass in Populus trees as a smooth function of height and diameter. We examine the performance of the interaction models in comparison to the Smooth-ANOVA models (both models with and without the restriction the fit has to be maintained) through a simulation study.