Covariance mapping

Discrete mathematics

Statistics

Multivariate statistics

Covariance intersection

Applied mathematics

Economics

Data mining

Computer science

Covariance

Wasserstein metric

Covariance function

Metric space

Mathematics

Estimator

Econometrics

Mathematical analysis

Operations management

Measure (data warehouse)

Probability density function

Density estimation

Metric (unit)

Consistency (knowledge bases)

Univariate

Biometrika

A common feature of methods for analysing 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 Fréchet mean is typically used to quantify and visualize the average density of 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–Fréchet mean or barycentre 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.

Wasserstein covariance for multiple random densities