Spherical Spatial Autoregressive Model for Spherically Embedded Spatial Data

2026-01-23

AI Summary

This study addresses the challenges of modeling and quantifying uncertainty for data residing on the sphere, which arise from its non-Euclidean geometry. The authors propose the first spherical autoregressive model grounded in optimal transport geometry, capable of handling both cases with and without exogenous covariates within a unified framework. By integrating a spatial autoregressive structure, a distribution-free Wald test, and conformal prediction tailored to the spherical domain, the method substantially improves inference accuracy and uncertainty quantification in finite samples. Theoretical contributions include establishing the asymptotic properties of the proposed estimator and developing a rigorous statistical testing framework. Empirical validation on simulated data, as well as real-world applications involving Spanish geochemical compositions and Japanese age-at-death distributions, demonstrates the model’s effectiveness and practical utility.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Large-scale security measurements

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.

Problem

• spherically embedded spatial data
• non-Euclidean geometry
• statistical modeling
• spatial dependence
• uncertainty quantification

Innovation

• spherical spatial autoregressive model
• optimal transport geometry
• distribution-free Wald test
• conformal prediction
• non-Euclidean spatial data