Functional data analysis for multivariate distributions through Wasserstein slicing
Han Chen, Hans-Georg Müller
The functional modeling of samples of distributions is a major challenge since distributions do not form a vector space. While various approaches exist for univariate distributions, including transformations to a Hilbert space, there are far fewer tools available for multivariate distributions. We introduce a transformation approach to map multivariate distributions to a Hilbert space by a map that proceeds via a Wasserstein slicing approach and can be inverted. Then tools from functional data analysis,  including functional principal component analysis, functional modes of variation and functional regression can be adopted in the Hilbert space and the invertibility of the map makes it possible to obtain the distributions that correspond to the resulting functional objects in the Hilbert space. Convergence results for these Hilbert space representations can be obtained for a general class of such maps. The proposed method is illustrated by modeling joint distributions of systolic and diastolic blood pressure data.