A general framework for prediction in penalized regression

We present several methods for prediction of new observations in penalized regression using different methodologies, based on the methods proposed in: i) Currie et al. (2004), ii) Gilmour et al. (2004) and iii) Sacks et al. (1989). We extend the method introduced by Currie et al. (2004) to consider the prediction of new observations in the mixed model framework. In the context of penalties based on differences between adjacent coefficients (Eilers & Marx (1996)), the equivalence of the different methods is shown. We demonstrate several properties of the new coefficients in terms of the order of the penalty. We also introduce the concept memory of a P-spline, this new idea gives us information on how much past information we are using to predict. The methodology and the concept of memory of a P-spline are illustrated with three real data sets, two of them on the yearly mortality rates of Spanish men and other on rental prices.The first and the second authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness MTM2014-52184-P. The third author acknowledges financial support from the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323