2018-10-25 - Alex Petersen - Wasserstein Regression and Covariance for Random Densities

Samples of density functions appear in a variety of disciplines, including distributions of mortality across nations, CT density histograms of hematoma in post-stroke patients, and distributions of voxel-to-voxel correlations of fMRI signals across subjects. The nonlinear nature of density space 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. First, when densities appear as responses in a regression model, the utility of Fréchet regression, a general purpose methodology for response objects in a metric space, is demonstrated. Due to the manifold structure of the space, inferential methods are developed allowing for tests of global and partial effects, as well as simultaneous confidence bands for fitted densities. Second, a notion of Wasserstein covariance is proposed for multivariate density data (a vector of densities), where multiple densities are observed for each subject. This interpretable dependence measure is shown to reveal interesting differences in functional connectivity between a group of Alzheimer's subjects and a control group.
UC Santa Barbara
October 25, 2018