Adaptive functional principal components analysis

TL;DR

This paper introduces an adaptive kernel smoothing method for functional principal components analysis, providing explicit risk bounds, refined bandwidth selection, and demonstrating effectiveness through simulations and real data application.

Contribution

It develops a new adaptive smoothing technique tailored for FPCA, with explicit risk bounds and efficient bandwidth rules, improving over existing methods.

Findings

Provides explicit risk bounds for eigen-elements

Develops refined, computationally efficient bandwidth rules

Demonstrates effectiveness through simulations and real data

Abstract

Functional data analysis almost always involves smoothing discrete observations into curves, because they are never observed in continuous time and rarely without error. Although smoothing parameters affect the subsequent inference, data-driven methods for selecting these parameters are not well-developed, frustrated by the difficulty of using all the information shared by curves while being computationally efficient. On the one hand, smoothing individual curves in an isolated, albeit sophisticated way, ignores useful signals present in other curves. On the other hand, bandwidth selection by automatic procedures such as cross-validation after pooling all the curves together quickly become computationally unfeasible due to the large number of data points. In this paper we propose a new data-driven, adaptive kernel smoothing, specifically tailored for functional principal components analysis through the derivation of sharp, explicit risk bounds for the eigen-elements. The minimization of these quadratic risk bounds provide refined, yet computationally efficient bandwidth rules for each eigen-element separately. Both common and independent design cases are allowed. Rates of convergence for the estimators are derived. An extensive simulation study, designed in a versatile manner to closely mimic the characteristics of real data sets supports our methodological contribution. An illustration on a real data application is provided.