Principal component analysis in Bayes spaces for sparsely sampled density functions

Abstract

This paper presents a novel approach to functional principal component analysis (FPCA) in Bayes spaces, where densities are the object of analysis, but only few individual samples from each density are observed. We use the observed data directly to account for all sources of uncertainty, instead of relying on prior estimation of the underlying densities in a two-step approach, which can be inaccurate if small or heterogeneous numbers of samples per density are available. To account for the constrained nature of densities, we base our approach on Bayes spaces, which extend the Aitchison geometry for compositional data to density functions. For modeling, we exploit the isometric isomorphism between the Bayes space and the L2 subspace L02 with integration-to-zero constraint through the centered log-ratio (clr) transformation. As only discrete draws from each density are observed, we treat the underlying densities as latent variables within a maximum likelihood framework, based on a Gaussian process assumption with finite basis expansion for their clr-transformations, and employ a Monte Carlo Expectation Maximization (MCEM) algorithm for model estimation. We apply the method to rental price distributions across Munich districts and maximum daily summer temperatures in Berlin over the past 70 years.