Abstract:

While a variety of methods exist for analyzing samples of probability density functions, a common element in the majority of these is the incorporation of the constraints inherent to the space of densities. In the setting of one dimensional densities, the Wasserstein metric is popular due to its theoretical appeal and interpretive value, leading to the WassersteinFréchet mean or barycenter. We extend the existing methodology for samples of densities in two important ways. First, motivated by applications in neuroimaging, we consider multivariate density data, where a vector of univariate random densities is observed for each sampling unit. Second, we define a novel population object, the Wasserstein covariance matrix. Intuitive estimators and their asymptotic performance are investigated while accounting for errors introduced by preliminary density estimation. The utility of the Wasserstein covariance matrix is demonstrated through the analysis of local functional connectivity distributions arising from fMRI data. In particular, Wasserstein covariance matrices at the individual level are shown to be intimately related to higher cognitive scores of episodic memory.
