Wasserstein Regression for Distributions, with Application to Distributional Time Series

• Activity Number: 10 - Advances in Functional and Geometric Data Analysis
• Type: Invited
• Date/Time: Sunday, August 8, 2021 : 1:30 PM to 3:20 PM
• Sponsor: IMS
• Author(s): Hans-Georg Müller* and Yaqing Chen and Zhenhua Lin
• Companies: University of California, Davis and University of California, Davis and National University of Singapore
• Keywords: Distributions; Wasserstein Metric; Time Series ; Object Data ; Random Objects ; Prediction

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 special case is the class of univariate probability measures defined on the real line. Adopting the Wasserstein-2 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. We study asymptotic rates of convergence for the estimator of the regression coefficient function and for predicted distributions and an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data on human mortality and distributions of house prices.