Fréchet Regression and Wasserstein Covariance for Random Density Data

Samples of density functions appear in a variety of disciplines, including connectivity distributions of voxel-to-voxel correlations of fMRI signals or distributions of voxel-specific attenuation coefficients from CT scans across subjects. The nonlinear nature of the space of densities necessitates adaptations and new methodologies for the analysis of random densities. We define our geometry using the Wasserstein metric, an increasingly popular choice in theory and application, and investigate two modeling problems. First, when densities appear as responses in a regression relationship with vector covariates, we consider the Fréchet regression model, which provides a general purpose methodology for response objects in a generic metric space. Importantly, we enlarge the scope of this regression framework for density data by placing distributional assumptions on the error processes (in this case, random optimal transport maps) that allow for further inference beyond estimation, specifically submodel testing. Second, when multiple random densities are observed for each subject, we propose the Wasserstein covariance matrix, yielding a scalar summary measure of covariance for each pair of random densities. Using the fMRI connectivity distributions as an example, we find that the Wasserstein covariance matrix provides an interpretable summary of dependence across regions that also reflects key distinguishing features between normal and Alzheimer's subjects.