University of California, Davis
Autoregressive Models for Distributional Time Series
Distributional time series consist of sequences of distributions indexed by time and are frequently encountered in modern data analysis. We propose two classes of autoregressive models for such time series that are based on the Wasserstein geometry and the Fisher-Rao geometry respectively, where the former is an intrinsic model that operates in the space of optimal transport maps and the latter utilizes rotation operators that map distributional regressions to geodesics on the (infiinite-dimensional) Hilbert sphere. While the Wasserstein geometry is popular in the literature due to its statistical utility and connections to optimal transport, its application for distributional time series has been limited to the case of univariate distributions, as optimal transport is unwieldy for multidimensional distributions in statistical applications. On the other hand, the Fisher-Rao geometry is not affected by the dimension of the distributions and we show that it can be utilized not only for multidimensional distributional time series but also for compositional time series, both giving rise to a new class of spherical time series. We derive theoretical properties of the ensuing autoregressive models and showcase these approaches with time series of yearly observations of uni/bivariate distributions of the minimum/maximum temperatures for a period of 120 days during each summer for the years 1990-2018 and with U.S. energy mix time series.