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Frequency Domain Resampling for Gridded Spatial Data

Souvick Bera, Daniel J. Nordman, Soutir Bandyopadhyay

TL;DR

This paper tackles the challenge of making nonparametric inferences for spectral mean statistics in gridded spatial data, where the sampling distribution involves a complex variance that includes a fourth-order cumulant component. It introduces the Hybrid Frequency Domain Bootstrap (HFDB), which couples spatial subsampling (to estimate variance components) with a bootstrap scheme (to recover distributional shape), yielding valid inference under mild spatial conditions and without requiring Gaussianity. The authors establish consistency results for the variance estimators, derive theoretical guarantees for HFDB, and demonstrate through extensive simulations that HFDB outperforms prior spatial bootstraps, especially for non-Gaussian processes, and is effective in isotropy testing. The approach extends the scope of frequency-domain resampling in spatial statistics and provides practical tools for uncertainty quantification in spectral analyses of gridded spatial data.

Abstract

In frequency domain analysis for spatial data, spectral averages based on the periodogram often play an important role in understanding spatial covariance structure, but also have complicated sampling distributions due to complex variances from aggregated periodograms. In order to nonparametrically approximate these sampling distributions for purposes of inference, resampling can be useful, but previous developments in spatial bootstrap have faced challenges in the scope of their validity, specifically due to issues in capturing the complex variances of spatial spectral averages. As a consequence, existing frequency domain bootstraps for spatial data are highly restricted in application to only special processes (e.g. Gaussian) or certain spatial statistics. To address this limitation and to approximate a wide range of spatial spectral averages, we propose a practical hybrid-resampling approach that combines two different resampling techniques in the forms of spatial subsampling and spatial bootstrap. Subsampling helps to capture the variance of spectral averages while bootstrap captures the distributional shape. The hybrid resampling procedure can then accurately quantify uncertainty in spectral inference under mild spatial assumptions. Moreover, compared to the more studied time series setting, this work fills a gap in the theory of subsampling/bootstrap for spatial data regarding spectral average statistics.

Frequency Domain Resampling for Gridded Spatial Data

TL;DR

This paper tackles the challenge of making nonparametric inferences for spectral mean statistics in gridded spatial data, where the sampling distribution involves a complex variance that includes a fourth-order cumulant component. It introduces the Hybrid Frequency Domain Bootstrap (HFDB), which couples spatial subsampling (to estimate variance components) with a bootstrap scheme (to recover distributional shape), yielding valid inference under mild spatial conditions and without requiring Gaussianity. The authors establish consistency results for the variance estimators, derive theoretical guarantees for HFDB, and demonstrate through extensive simulations that HFDB outperforms prior spatial bootstraps, especially for non-Gaussian processes, and is effective in isotropy testing. The approach extends the scope of frequency-domain resampling in spatial statistics and provides practical tools for uncertainty quantification in spectral analyses of gridded spatial data.

Abstract

In frequency domain analysis for spatial data, spectral averages based on the periodogram often play an important role in understanding spatial covariance structure, but also have complicated sampling distributions due to complex variances from aggregated periodograms. In order to nonparametrically approximate these sampling distributions for purposes of inference, resampling can be useful, but previous developments in spatial bootstrap have faced challenges in the scope of their validity, specifically due to issues in capturing the complex variances of spatial spectral averages. As a consequence, existing frequency domain bootstraps for spatial data are highly restricted in application to only special processes (e.g. Gaussian) or certain spatial statistics. To address this limitation and to approximate a wide range of spatial spectral averages, we propose a practical hybrid-resampling approach that combines two different resampling techniques in the forms of spatial subsampling and spatial bootstrap. Subsampling helps to capture the variance of spectral averages while bootstrap captures the distributional shape. The hybrid resampling procedure can then accurately quantify uncertainty in spectral inference under mild spatial assumptions. Moreover, compared to the more studied time series setting, this work fills a gap in the theory of subsampling/bootstrap for spatial data regarding spectral average statistics.
Paper Structure (14 sections, 5 theorems, 62 equations, 3 figures, 2 tables)

This paper contains 14 sections, 5 theorems, 62 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Distributions of Spectral Mean Statistics
  3. Spatial process and sampling design
  4. Spectral mean parameters
  5. Spectral mean statistics and sampling distributions
  6. Spatial Subsampling in Frequency Domain
  7. Spatial subsampling variance estimators
  8. Consistency of spatial subsampling estimators
  9. Spatial Hybrid Frequency Domain Bootstrap (HFDB)
  10. Numerical Studies of Accuracy
  11. Results for Gaussian processes
  12. Results for non-Gaussian processes
  13. Application to Calibrating Spatial Isotropy Tests
  14. Concluding Remarks

Key Result

Theorem 3.1

Suppose Assumptions assumption:assump1-assumption:assump4 hold and the subsample size $b_n \equiv n_1^{(b)} \times n_2^{(b)}$ satisfies $1/n_k^{(b)} + n_k^{-1} n_k^{(b)} \rightarrow 0$ for $k=1,2$ as $n\equiv n_1\times n_2\rightarrow \infty$. Then, the spatial subsampling estimators of variance comp

Figures (3)

  • Figure 1: Coverages of 90% HFDB intervals for the covariance parameter $\gamma(\mathbf{h}), \mathbf{h} = (1,0)^{T}$, based on either FDWB ($Q^{*}_{FDWB,n}(\psi)$) (blue line) or corrected ($H^{*}_{HFDB,n}(\psi)$) (green line) HFDB versions with different subsample sizes $b_n$, range, and sample sizes $n$.
  • Figure 2: Coverages of 90% HFDB intervals for the covariance parameter $\gamma(\mathbf{h}), \mathbf{h} = (1,0)^{^{\rm T}}$, based on either non-corrected ($Q^{*}_{FDWB,n}(\psi)$) (blue line) or corrected ($H^{*}_{HFDB,n}(\psi)$) (green line) HFDB versions with different subsample sizes $b_n$, innovations, and sample sizes $n$.
  • Figure 3: Coverages of 90% HFDB intervals for the covariance parameter $\gamma(\mathbf{h}), \mathbf{h} = (1,0)^{^{\rm T}}$, based on either non-corrected ($Q^{*}_{FDWB,n}(\psi)$) (blue line) or corrected ($H^{*}_{HFDB,n}(\psi)$) (green line) HFDB versions with different subsample sizes $b_n$, and sample sizes $n$.

Theorems & Definitions (7)

  • Theorem 3.1
  • Theorem 4.1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Proposition 3