Wasserstein Gradients for the Temporal Evolution of Probability Distributions

• Activity Number: 120 - SPEED: Nonparametric Statistics: Estimation, Testing, and Modeling
• Type: Contributed
• Date/Time: Monday, July 30, 2018 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Yaqing Chen* and Hans Mueller
• Companies: University of California, Davis and UC Davis
• Keywords: Estimation; Derivatives; Density Functions; Income Distributions; Evolution of Mortality; Height Growth

Abstract

While many studies have been conducted on flows of probability measures, often in terms of gradient flows, modeling of the instantaneous evolution of observed distribution flows over time has not yet been explored. Our goal is to develop statistical models to reflect the observed flow of distributions in one-dimensional Euclidean space over time, based on the Wasserstein distance and corresponding optimal transport maps. For this purpose, we introduce Wasserstein temporal gradients, the notion of derivatives of optimal transport maps with respect to time. An implementation for empirical data is presented and it has been shown that it provides a consistent estimator of the derivative. These time dynamics of optimal transport maps are illustrated with time-varying distribution data that include yearly income distributions, the evolution of mortality over calendar years, and data on age-dependent height distributions of children from the longitudinal Zürich growth study.