  This paper presents a novel approach to functional principal component
analysis (FPCA) in Bayes spaces in the setting 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 $\mathbb{L}^2$
subspace $\mathbb{L}_0^2$ with integration-to-zero constraint through the
centered log-ratio transformation. As only discrete draws from each density are
observed, we treat the underlying functional densities as latent variables
within a maximum likelihood framework and employ a Monte Carlo Expectation
Maximization (MCEM) algorithm for model estimation. Resulting estimates are
useful for exploratory analyses of density data, for dimension reduction in
subsequent analyses, as well as 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, as well as the distribution of rental
prices in the districts of Munich.
