Wasserstein Gradients for the Temporal Evolution of Probability Distributions

TL;DR

This paper introduces a method to model and estimate the instantaneous evolution of time-varying probability distributions using Wasserstein gradients, with applications to income and mortality data.

Contribution

It develops a novel approach employing local Fréchet regression in Wasserstein space to estimate distribution derivatives over time.

Findings

Consistent estimation of distribution derivatives over time.

Application to income and mortality distribution data.

Derived convergence rates for the estimators.

Abstract

Many studies have been conducted on flows of probability measures, often in terms of gradient flows. We utilize a generalized notion of derivatives with respect to time to model the instantaneous evolution of empirically observed one-dimensional distributions that vary over time and develop consistent estimates for these derivatives. Employing local Fr\'echet regression and working in local tangent spaces with regard to the Wasserstein metric, we derive the rate of convergence of the proposed estimators. The resulting time dynamics are illustrated with time-varying distribution data that include yearly income distributions and the evolution of mortality over calendar years.