A geometric approach to maximum likelihood estimation of

the functional principal components from sparse longitudinal

data

Jie Peng and Debashis Paul

Department of Statistics, University of California, Davis, CA 95616

Correspondence author: jie@wald.ucdavis.edu

Abstract:

In this paper, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. We then approach this problem through a restricted maximum likelihood method. The estimation scheme is based on a Newton-Raphson procedure on the Stiefel manifold using the fact that the basis coefficient matrix for representing the eigenfunctions has orthonormal columns. We also address the selection of the number of basis functions, as well as that of the dimension of the covariance kernel by a second order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4+ counts data set. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing, and another using an EM algorithm.

Keywords : longitudinal data, covariance kernel, functional principal components, Stiefel manifold,Newton-Raphson algorithm, cross-validation.