Title: Wasserstein covariance for vectors of random densities
Authors: Hans-Georg Mueller - University of California Davis (United States) [presenting]
Alexander Petersen - Brigham Young University (United States)
Abstract: Samples of data that consist of probability densities or distributions are encountered in various applications. Once a metric for densities is specified, the Frechet mean or barycenter is typically used to determine the average density. The Wasserstein metric is popular due to its good practical performance and interpretation as an optimal transport metric. Motivated by applications in neuroimaging, we consider data that consist of a $p$-vector of univariate random densities for each sampling unit. We introduce Wasserstein covariance to quantify the dependency of the component densities and provide corresponding estimators for fixed and diverging $p$, where the latter corresponds to continuously-indexed densities. Consistency and asymptotic normality are established, while accounting for errors introduced in the unavoidable preparatory density estimation step. The utility of the Wasserstein covariance matrix is demonstrated in applications that include functional connectivity in the brain and the secular evolution of mortality.