Wasserstein Regression

• Activity Number: 443 - Student Paper Competition Presentations
• Type: Topic Contributed
• Date/Time: Thursday, August 6, 2020 : 10:00 AM to 11:50 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Yaqing Chen* and Zhenhua Lin and Hans-Georg Müller
• Companies: University of California, Davis and National University of Singapore and University of California, Davis
• Keywords: distribution regression; Wasserstein geometry; random measures; parallel transport; tangent bundles; functional data analysis

Abstract

The analysis of samples of random objects that do not lie in a vector space has found increasing attention in statistics in recent years. An important class of such object data is univariate probability measures defined on the real line. Adopting the Wasserstein metric, we develop a class of regression models for such data, where random distributions serve as predictors and the responses are either also distributions or scalars. To define this regression model, we utilize the geometry of tangent bundles of the metric space of random measures with the Wasserstein metric. The proposed distribution-to-distribution regression model provides an extension of multivariate linear regression for Euclidean data and function-to-function regression for Hilbert space valued data in functional data analysis. We derive asymptotic rates of convergence for the estimator of the regression coefficient function and for predicted distributions and illustrate the proposed methods with human mortality data.