Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spherical Spatial Autoregressive Model for Spherically Embedded Spatial Data

Jiazhen Xu, Han Lin Shang

TL;DR

This work addresses the analysis of spherically embedded spatial data, which reside on the unit sphere and exhibit intrinsic non-Euclidean geometry. It develops the Pure Spherical Spatial Autoregressive (PSSAR) model and its covariate-augmented extension (SRMSAR) using optimal-transport geometry on the sphere, paired with generalised method of moments estimation, asymptotic theory, and Wald-type tests, complemented by bootstrap refinements. A distribution-free conformal prediction framework is introduced to quantify predictive uncertainty under spatial dependence. The authors validate the methods via extensive simulations and real-data applications to geochemical compositions in Spain and life-table death counts for Japan, demonstrating accurate prediction and valid uncertainty quantification without distributional assumptions. Overall, the framework enables robust, geometry-respecting modelling and inference for high-dimensional, non-Euclidean spatial data across diverse scientific domains.

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.

Spherical Spatial Autoregressive Model for Spherically Embedded Spatial Data

TL;DR

This work addresses the analysis of spherically embedded spatial data, which reside on the unit sphere and exhibit intrinsic non-Euclidean geometry. It develops the Pure Spherical Spatial Autoregressive (PSSAR) model and its covariate-augmented extension (SRMSAR) using optimal-transport geometry on the sphere, paired with generalised method of moments estimation, asymptotic theory, and Wald-type tests, complemented by bootstrap refinements. A distribution-free conformal prediction framework is introduced to quantify predictive uncertainty under spatial dependence. The authors validate the methods via extensive simulations and real-data applications to geochemical compositions in Spain and life-table death counts for Japan, demonstrating accurate prediction and valid uncertainty quantification without distributional assumptions. Overall, the framework enables robust, geometry-respecting modelling and inference for high-dimensional, non-Euclidean spatial data across diverse scientific domains.

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.
Paper Structure (15 sections, 10 theorems, 31 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 15 sections, 10 theorems, 31 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Spherically Embedded Spatial Data
  3. Pure Spherical Spatial Autoregressive Model
  4. Estimation and Hypothesis Testing
  5. Prediction and Conformal Prediction Set
  6. Spherical Regression Model with Spatial Autoregressive Disturbances
  7. Estimation and Hypothesis Testing
  8. Prediction and Conformal Prediction Set
  9. Simulation Studies
  10. Simulations on Estimation and Inference
  11. Simulations on Prediction
  12. Empirical Data Analysis
  13. Geochemical Data in Spain
  14. Life-Table Death Counts for Japanese Prefectures
  15. Conclusion

Key Result

Proposition 1

When $S_n(\rho_0)=I_n-\rho_0 W_n$ is invertible, for any constant $P_n=(P_{ij})_{n\times n}\in\mathbb{R}^{n\times n}$ with ${\rm tr}(P_n)=0$, $\mathbb{E} \left[ \langle\left\{(P_nS_n(\rho_0))\otimes {\rm id} \right\} Q_n , \Gamma_n \rangle_{\mathcal{C}^n(\mathcal{H})} \right] = 0$, where $\langle(h_

Figures (8)

  • Figure 1: Plots of the age distribution of deaths in 2020, by single-year age group, for (a) males and (b) females across six countries.
  • Figure 2: The empirical test power of the Wald test based on the PSSAR model using the PCA-based procedure and the bootstrap-based procedure for observations located within $\mathcal{S}^5$ (the first row) or $\mathcal{S}^{110}$ (the second row), with fixed neighbour numbers being 10, signal strengths $\{0,1,2,3,4,5\}$ corresponding to $\rho_0$ taking values in $\{0,0.1,-0.3,0.4,-0.7,0.9\}$, respectively, and sample size varying in (a) $n=200$, (b) $n=500$ and (c) $n=1000$. The blue dashed line represents the PCA-based procedure, and the red solid line represents the bootstrap-based procedure. The horizontal grey dotted line represents the $5\%$ rejection rate.
  • Figure 3: Violin plots for the prediction errors produced by the PSSAR model and the MSAR model, with prediction error calculated via angles (radian) between the true observation and the prediction. The simulated data locate in $\mathcal{S}^3$ with sample size $n$ varying in $\{200,400,600,800,1000\}$.
  • Figure 4: Plots for the PSSAR model and the MSAR model with measures produced by (a) angles (radian) between the true observation and the prediction; (b) MSE between the true observation and the prediction using the GEMAS data in Spain.
  • Figure 5: Comparison between the predictions produced by the PSSAR model and the MSAR model and true observations of six randomly selected locations using the GEMAS data in Spain. The predictions produced by the PSSAR model are represented as red triangles, the predictions produced by the MSAR model are represented as blue squares, and the true observations are represented by pink circles.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 2
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 1 more