Bayesian Estimation of Principal Components for Functional Data

Abstract

The area of principal components analysis (PCA) has seen relatively few contributions from the Bayesian school of inference. In this paper, we propose a Bayesian method for PCA in the case of functional data observed with error. We suggest modeling the covariance function by use of an approximate spectral decomposition, leading to easily interpretable parameters. We perform model selection, both over the number of principal components and the number of basis functions used in the approximation. We study in depth the choice of using the implied distributions arising from the inverse Wishart prior and prove a convergence theorem for the case of an exact finite dimensional representation. We also discuss computational issues as well as the care needed in choosing hyperparameters. A simulation study is used to demonstrate competitive performance against a recent frequentist procedure, particularly in terms of the principal component estimation. Finally, we apply the method to a real dataset, where we also incorporate model selection on the dimension of the finite basis used for modeling.