Abstract

Sparse functional data are common in longitudinal studies, where each subject is measured only a few times, at irregular time points and with measurement error. Functional principal component analysis (FPCA) is a key tool for summarising such data. However, likelihood based estimators that work directly under orthonormality constraints rely on optimisation over the Stiefel manifold. That optimisation is numerically demanding, and existing Newton Raphson and Riemannian conjugate gradient approaches can fail to converge.

We propose a new method for estimating functional principal components, which represents each eigenfunction with an unconstrained basis expansion and maps it to an orthonormal set using a modified Gram Schmidt process. The eigenfunctions, eigenvalues and error variance can then be estimated by minimising the negative log likelihood with standard quasi Newton methods, and an AIC criterion chooses the number of basis functions and components.

We compared our proposed approch with seven competing methods in simulation studies under Matérn and egg crate covariance structures. It converged in every replicate and recovered eigenfunctions and eigenvalues with accuracy as good as or better than the alternatives. We also illustrate the method on sparse CD4 cell count trajectories from HIV patients.