Wasserstein Regression

By Yaqing Chen, Zhenhua Lin, H. Muller. Journal of the American Statistical Association, 2020

142 citations recorded by the source corpus.

Abstract The analysis of samples of random objects that do not lie in a vector space is gaining increasing attention in statistics. 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 use the geometry of tangent bundles of the space of random measures endowed with the Wasserstein metric for mapping distributions to tangent spaces. 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. In simulations, it performs better than an alternative transformation approach where one maps distributions to a Hilbert space through the log quantile density transformation and then applies traditional functional regression. We derive asymptotic rates of convergence for the estimator of the regression operator and for predicted distributions and also study an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data on human mortality and distributional time series of house prices.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/99d884a6f8dbe34c9d19aaf0ea0da41c46efa9bc

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Distribution-on-Distribution Regression via Optimal Transport Maps (2021, methodology): “By comparison, Chen, Lin and M¨uller [2] introduce (linear) regression in Wasserstein space by means of a geometric approach, that is in a sense a linear model between tangent spaces.”
Wasserstein autoregressive models for density time series (2020, methodology): “Thus, the key difference between the WAR(p) model proposed in this article and that of Chen et al. (2020) is in how the quantities Tt−j − id from previous timepoints are mapped to the tangent space prior to adding the innovations.”