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Series ridge regression for spatial data on $\mathbb{R}^d$

Daisuke Kurisu, Yasumasa Matsuda

TL;DR

The paper develops a general theory for series ridge estimators with an $L^2$ penalty applied to irregularly spaced spatial data in $\mathbb{R}^d$, covering spatial trend and spatial regression with covariates. It establishes uniform and $L^2$ convergence rates, multivariate CLTs, and consistent long-run variance estimation, proving that spline and wavelet bases achieve optimal rates under Hölder smoothness and enabling valid confidence intervals. The framework accommodates a broad dependence structure, including Lévy-driven moving average fields, and demonstrates practical utility with an application to Tokyo population data, yielding interpretable trend surfaces and uncertainty quantification. The results provide building blocks for global nonparametric spatial regression under irregular sampling and pave the way for extensions to more general penalized sieve estimators in spatio-temporal settings.

Abstract

This paper develops a general asymptotic theory of series estimators for spatial data collected at irregularly spaced locations within a sampling region $R_n \subset \mathbb{R}^d$. We employ a stochastic sampling design that can flexibly generate irregularly spaced sampling sites, encompassing both pure increasing and mixed increasing domain frameworks. Specifically, we focus on a spatial trend regression model and a nonparametric regression model with spatially dependent covariates. For these models, we investigate $L^2$-penalized series estimation of the trend and regression functions. We establish uniform and $L^2$ convergence rates and multivariate central limit theorems for general series estimators as main results. Additionally, we show that spline and wavelet series estimators achieve optimal uniform and $L^2$ convergence rates and propose methods for constructing confidence intervals for these estimators. Finally, we demonstrate that our dependence structure conditions on the underlying spatial processes cover a broad class of random fields, including Lévy-driven continuous autoregressive and moving average random fields.

Series ridge regression for spatial data on $\mathbb{R}^d$

TL;DR

The paper develops a general theory for series ridge estimators with an penalty applied to irregularly spaced spatial data in , covering spatial trend and spatial regression with covariates. It establishes uniform and convergence rates, multivariate CLTs, and consistent long-run variance estimation, proving that spline and wavelet bases achieve optimal rates under Hölder smoothness and enabling valid confidence intervals. The framework accommodates a broad dependence structure, including Lévy-driven moving average fields, and demonstrates practical utility with an application to Tokyo population data, yielding interpretable trend surfaces and uncertainty quantification. The results provide building blocks for global nonparametric spatial regression under irregular sampling and pave the way for extensions to more general penalized sieve estimators in spatio-temporal settings.

Abstract

This paper develops a general asymptotic theory of series estimators for spatial data collected at irregularly spaced locations within a sampling region . We employ a stochastic sampling design that can flexibly generate irregularly spaced sampling sites, encompassing both pure increasing and mixed increasing domain frameworks. Specifically, we focus on a spatial trend regression model and a nonparametric regression model with spatially dependent covariates. For these models, we investigate -penalized series estimation of the trend and regression functions. We establish uniform and convergence rates and multivariate central limit theorems for general series estimators as main results. Additionally, we show that spline and wavelet series estimators achieve optimal uniform and convergence rates and propose methods for constructing confidence intervals for these estimators. Finally, we demonstrate that our dependence structure conditions on the underlying spatial processes cover a broad class of random fields, including Lévy-driven continuous autoregressive and moving average random fields.
Paper Structure (41 sections, 24 theorems, 211 equations, 1 figure)

This paper contains 41 sections, 24 theorems, 211 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Series trend estimators
  4. Spatial trend regression model
  5. Sampling design
  6. Dependence structure
  7. Uniform convergence rates
  8. Optimal uniform rates for spline and wavelet estimators
  9. $L^2$ convergence rates
  10. Asymptotic normality
  11. Examples
  12. Lévy-driven MA random fields
  13. Application to population data
  14. Conclusion
  15. Proofs for Section \ref{['Sec:reg-trend']}
  16. ...and 26 more sections

Key Result

Proposition 2.1

Suppose that Assumptions Ass:sample, Ass:model, Ass:sieve1, and Ass:ze-lam hold. Additionally, assume that $\|m_0\|_\infty<\infty$, $\varsigma_{J,n} \lesssim \zeta_J\lambda_J^{-1}\sqrt{\log n/n}$, and $\zeta_J^2 \lambda_J^{-1}(\zeta_J + \lambda_J^{-1})\sqrt{\log n/n} \to 0$ as $n,J \to \infty$. The

Figures (1)

  • Figure 1: Point estimator (upper left), standard error (upper right), 2.5% lower bound (lower left) and 2.5% upper bound (lower right), for the trend surface of human population in the Tokyo 23 central districts at 4:00 AM in May.

Theorems & Definitions (29)

  • Proposition 2.1
  • Corollary 2.1
  • Theorem 2.1
  • Proposition 2.2
  • Corollary 2.2
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition B.1
  • Corollary B.1
  • ...and 19 more