Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series

TL;DR

This paper introduces a novel class of autoregressive models for spherical time series, including distributional and compositional data, using rotation operators to handle the non-linearity of spheres, with applications to temperature distribution data.

Contribution

It develops a new autoregressive modeling framework for spherical and Hilbert sphere-valued time series using rotation operators and skew-symmetric operators, addressing non-linearity challenges.

Findings

Successfully modeled temperature distribution time series.

Demonstrated the effectiveness of rotation-based autoregressive models.

Applied methods to real-world temperature data from Los Angeles and JFK.

Abstract

We introduce a new class of autoregressive models for spherical time series, where the dimension of the spheres on which the observations of the time series are situated may be finite-dimensional or infinite-dimensional as in the case of a general Hilbert sphere. Spherical time series arise in various settings. We focus here on distributional and compositional time series. Applying a square root transformation to the densities of the observations of a distributional time series maps the distributional observations to the Hilbert sphere, equipped with the Fisher-Rao metric. Likewise, applying a square root transformation to the components of the observations of a compositional time series maps the compositional observations to a finite-dimensional sphere, equipped with the geodesic metric on spheres. The challenge in modeling such time series lies in the intrinsic non-linearity of spheres and Hilbert spheres, where conventional arithmetic operations such as addition or scalar multiplication are no longer available. To address this difficulty, we consider rotation operators to map observations on the sphere. Specifically, we introduce a class of skew-symmetric operator such that the associated exponential operators are rotation operators that for each given pair of points on the sphere map one of the points to the other one. We exploit the fact that the space of skew-symmetric operators is Hilbertian to develop autoregressive modeling of geometric differences that correspond to rotations of spherical and distributional time series. Motivating data for our methods include a time series of yearly observations of bivariate distributions of the minimum/maximum temperatures for a period of 120 days during each summer for the years 1990-2018 at Los Angeles (LAX) and John F. Kennedy (JFK) international airports.