Singular additive models for function to function regression

Abstract

In various functional regression settings one observes i.i.d. samples of paired stochastic processes (X,Y) and aims at predicting the trajectory of Y, given the trajectory X. For example, one may wish to predict the future segment of a process from observing an initial segment of its trajectory. Commonly used functional regression models are based on representations that are obtained separately for X and Y. In contrast to these established methods, often implemented with functional principal components, we base our approach on a singular expansion of the paired processes X,Y with singular functions that are derived from the cross-covariance surface between X and Y. The motivation for this approach is that the resulting singular components may better reflect the association between X and Y. The regression relationship is then based on the assumption that each singular component of Y follows an additive regression model with the singular components of X as predictors. To handle the inherent dependency of these predictors, we develop singular additive models with smooth backfitting. We discuss asymptotic properties of the estimates as well as their practical behavior in simulations and data analysis.