Varying-smoother models for functional responses

Abstract

This paper studies estimation of a smooth function \(f(t,s)\) when we are given functional responses of the form \(f(t,\cdot)\) + error, but scientific interest centers on the collection of functions \(f(\cdot,s)\) for different \(s\). The motivation comes from studies of human brain development, in which \(t\) denotes age whereas \(s\) refers to brain locations. Analogously to varying-coefficient models, in which the mean response is linear in \(t\), the "varying-smoother" models that we consider exhibit nonlinear dependence on \(t\) that varies smoothly with \(s\). We discuss three approaches to estimating varying-smoother models: (a) methods that employ a tensor product penalty; (b) an approach based on smoothed functional principal component scores; and (c) two-step methods consisting of an initial smooth with respect to \(t\) at each \(s\), followed by a postprocessing step. For the first approach, we derive an exact expression for a penalty proposed by Wood, and an adaptive penalty that allows smoothness to vary more flexibly with \(s\). We also develop "pointwise degrees of freedom," a new tool for studying the complexity of estimates of \(f(\cdot,s)\) at each \(s\). The three approaches to varying-smoother models are compared in simulations and with a diffusion tensor imaging data set.