Journal Article

Properties of Principal Component Methods for Functional and Longitudinal Data Analysis

Wang, Jane-Ling, Hall, Peter, Müller, Hans-Georg

2006

Overview

The use of principal component methods to analyze 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 if each observation is encumbered by noise. However, estimation of eigenfunctions becomes a nonparametric problem when observations are sparse. The optimal convergence rates in this case are those which pertain to more familiar function-estimation settings. We also describe the effects of sampling at regularly spaced points, as opposed to random points. In particular, it is shown that there are often advantages in sampling randomly. However, even in the case of noisy data there is a threshold sampling rate (depending on the number of functions treated) above which the rate of sampling (either randomly or regularly) has negligible impact on estimator performance, no matter whether eigenfunctions or eigenvectors are being estimated.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 62G08 / 62H25 / 62M09 / Biomedical studies / Consistent estimators / Covariance / curse of dimensionality / Data analysis / Data smoothing / eigenfunction / Eigenfunctions / eigenvalue / Eigenvalues / eigenvector / Estimating techniques / Estimators / Exact sciences and technology / Inference from stochastic processes; time series analysis / Karhunen–Loève expansion / local polynomial methods / Longitudinal data / Mathematical functions / Mathematical models / Mathematics / Multivariate analysis / nonparametric / Nonparametric inference / operator theory / optimal convergence rate / principal component analysis / Principal Component Methodology / Principal components analysis / Probability and statistics / rate of convergence / Sampling / Sciences and techniques of general use / semiparametric / smoothing / sparse data / spectral decomposition / Statistical analysis / Statistics / Studies