Properties of principal component methods for functional and longitudinal data analysis

Abstract

The use of principal components methods to analyse functional data is appropriate in a wide range of different settings. In studies of “functional data analysis”, it has often been assumed that a sample of random functions is observed precisely, in the continuum and without noise. While this has been the traditional setting for functional data analysis, in the context of longitudinal data analysis a random function typically represents a patient, or subject, who is observed at only a small number of randomly distributed points, with nonnegligible measurement error. Nevertheless, essentially the same methods can be used in both these cases, as well as in the vast number of settings that lie between them. How is performance affected by the sampling plan? In this paper we answer that question. We show that if there is a sample of n functions, or subjects, then estimation of eigenvalues is a semiparametric problem, with root-n consistent estimators, even if only a few observations are made of each function, and each observation is encumbered by noise. However, estimation of