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Fixed-Domain Asymptotics Under Vecchia's Approximation of Spatial Process Likelihoods.

Lu Zhang1, Wenpin Tang1, Sudipto Banerjee1

  • 1University of Southern California, Los Angeles, Columbia University, New York, University of California, Los Angeles.

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Summary

This study validates microergodic spatial covariance parameter estimation using Vecchia's approximation for large spatial datasets. It confirms the method's effectiveness under fixed-domain asymptotics, crucial for scalable Gaussian process modeling.

Keywords:
Fixed-domain asymptoticsGaussian processesMatérn covariance functionMicroergodic parametersSpatial statistics

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Area of Science:

  • Statistics
  • Spatial Analysis
  • Computational Statistics

Background:

  • Massive spatial data necessitates scalable statistical models.
  • Vecchia's approximation offers efficient Gaussian process (GP) likelihood evaluation by limiting dependencies to neighboring locations.
  • Understanding the inferential properties of spatial covariance parameters is vital for accurate modeling.

Purpose of the Study:

  • To establish the inferential properties of microergodic spatial covariance parameters when estimated with Vecchia's approximation.
  • To investigate the theoretical and empirical conditions required for these properties.
  • To corroborate the effectiveness of Vecchia's approximation within fixed-domain asymptotic frameworks.

Main Methods:

  • Utilizing Vecchia's approximation for Gaussian process models.
  • Applying fixed-domain asymptotic theory.
  • Theoretical analysis and empirical validation of parameter estimation properties.

Main Results:

  • Formal establishment of inferential properties for microergodic spatial covariance parameters under Vecchia's approximation.
  • Identification of conditions supporting these properties.
  • Empirical corroboration of Vecchia's approximation's effectiveness in fixed-domain asymptotics.

Conclusions:

  • Vecchia's approximation is a statistically sound method for estimating spatial covariance parameters in large datasets.
  • The approach is validated under fixed-domain asymptotics, enhancing its reliability for scalable spatial statistics.
  • This work strengthens the theoretical foundation for using Vecchia's approximation in massive spatial data analysis.