Spherical Spatial Autoregressive Model for Spherically Embedded Spatial Data

TL;DR

This paper introduces a new spherical spatial autoregressive model for analyzing data on the sphere, addressing non-Euclidean challenges with novel inference, covariate integration, and uncertainty quantification methods, validated through simulations and real-world applications.

Contribution

It presents a unified framework with a novel model, asymptotic theory, a distribution-free test, and conformal prediction for spherical spatial data analysis.

Findings

Model effectively captures spatial dependence on the sphere.

Method performs well in simulations and real data applications.

Provides reliable uncertainty quantification tailored to spherical data.

Abstract

Spherically embedded spatial data are spatially indexed observations whose values naturally reside on or can be equivalently mapped to the unit sphere. Such data are increasingly ubiquitous in fields ranging from geochemistry to demography. However, analysing such data presents unique difficulties due to the intrinsic non-Euclidean nature of the sphere, and rigorous methodologies for statistical modelling, inference, and uncertainty quantification remain limited. This paper introduces a unified framework to address these three limitations for spherically embedded spatial data. We first propose a novel spherical spatial autoregressive model that leverages optimal transport geometry and then extend it to accommodate exogenous covariates. Second, for either scenario with or without covariates, we establish the asymptotic properties of the estimators and derive a distribution-free Wald test for spatial dependence, complemented by a bootstrap procedure to enhance finite-sample performance. Third, we contribute a novel approach to uncertainty quantification by developing a conformal prediction procedure specifically tailored to spherically embedded spatial data. The practical utility of these methodological advances is illustrated through extensive simulations and applications to Spanish geochemical compositions and Japanese age-at-death mortality distributions.