# Wasserstein Covariance for Multiple Random Densities

## TL;DR

This paper introduces a Wasserstein covariance measure for dependent density data, extending optimal transport-based analysis to multivariate and continuously-indexed densities, with theoretical guarantees and practical applications.

## Contribution

It proposes a novel Wasserstein covariance framework for dependent densities, including estimators and asymptotic theory, applicable to neuroimaging and demographic data.

## Key findings

- Wasserstein covariance effectively captures dependencies in density data.
- Theoretical results establish estimator consistency and asymptotic normality.
- Applications demonstrate utility in brain connectivity and mortality trends.

## Abstract

A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fr\'echet mean is typically used to quantify and visualize the average density from the sample. For one-dimensional densities, the Wasserstein metric is popular due to its theoretical appeal and interpretive value as an optimal transport metric, leading to the Wasserstein-Fr\'echet mean or barycenter as the mean density. We extend the existing methodology for samples of densities in two key directions. First, motivated by applications in neuroimaging, we consider dependent density data, where a $p$-vector of univariate random densities is observed for each sampling unit. Second, we introduce a Wasserstein covariance measure and propose intuitively appealing estimators for both fixed and diverging $p$, where the latter corresponds to continuously-indexed densities. We also give theory demonstrating consistency and asymptotic normality, while accounting for errors introduced in the unavoidable preparatory density estimation step. The utility of the Wasserstein covariance matrix is demonstrated through applications to functional connectivity in the brain using functional magnetic resonance imaging data and to the secular evolution of mortality for various countries.