  We introduce the method of Geodesic Principal Component Analysis (GPCA) on
the space of probability measures on the line, with finite second moment,
endowed with the Wasserstein metric. We discuss the advantages of this
approach, over a standard functional PCA of probability densities in the
Hilbert space of square-integrable functions. We establish the consistency of
the method by showing that the empirical GPCA converges to its population
counterpart, as the sample size tends to infinity. A key property in the study
of GPCA is the isometry between the Wasserstein space and a closed convex
subset of the space of square-integrable functions, with respect to an
appropriate measure. Therefore, we consider the general problem of PCA in a
closed convex subset of a separable Hilbert space, which serves as basis for
the analysis of GPCA and also has interest in its own right. We provide
illustrative examples on simple statistical models, to show the benefits of
this approach for data analysis. The method is also applied to a real dataset
of population pyramids.
