A0639
Title: Principal component analysis in Bayes spaces for sparsely sampled density functions
Authors: Lisa Steyer - Humboldt University of Berlin (Germany)
Sonja Greven - Humboldt University of Berlin (Germany) [presenting]
Abstract: A novel approach to functional principal component analysis in Bayes spaces is presented in the setting where densities are the object of analysis, but only a few individual samples from each density are observed. The observed data is used directly to account for all sources of uncertainty instead of relying on the prior estimation of underlying densities in a two-step approach, which can be inaccurate if small or heterogeneous numbers of samples per density are available. The approach is based on Bayes spaces, which extend the Aitchison geometry for compositional data to density functions to account for the constrained nature of densities. The isometric isomorphism is exploited between the Bayes space and the $L2$ subspace $L2_0$ with integration-to-zero constraint through the centered log-ratio transformation. As only discrete draws from each density are observed, the underlying functional densities are treated as latent variables within a maximum likelihood framework, and a Monte Carlo expectation maximization algorithm is employed for model estimation. Resulting estimates are useful for exploratory analyses of density data, for dimension reduction in subsequent analyses, and for improved preprocessing of sparsely sampled density data compared to existing methods. The proposed method is applied to analyze the distribution of maximum daily temperatures in Berlin during the summer months for the last 70 years and distributions of rental prices in Munich districts.
