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A New Construction of Covariance Functions for Gaussian Random Fields

Author

Listed:
  • Weichao Wu

    (Sun Yat-Sen University)

  • Athanasios C. Micheas

    (University of Missouri)

Abstract

We develop a new approach to creating covariance functions for Gaussian random fields via point processes on the complex plane. We present two approaches to construct valid covariance functions by exploiting Bochner’s theorem and then modeling the characteristic function of a covariance function. In particular, we use a complex point process (CPP) to model the Fourier coefficients and illustrate how to estimate the covariance function of a Gaussian random field model from data. We further illustrate our construction approaches and compare several algorithms via simulations. The methods are exemplified via applications to real-life research data in wheat yields and earthquake studies.

Suggested Citation

  • Weichao Wu & Athanasios C. Micheas, 2024. "A New Construction of Covariance Functions for Gaussian Random Fields," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(1), pages 530-574, February.
    • Handle: RePEc:spr:sankha:v:86:y:2024:i:1:d:10.1007_s13171-023-00336-4
    DOI: 10.1007/s13171-023-00336-4
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    References listed on IDEAS

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    1. Athanasios Christou Micheas, 2014. "Hierarchical Bayesian modeling of marked non-homogeneous Poisson processes with finite mixtures and inclusion of covariate information," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(12), pages 2596-2615, December.
    2. G. Avlogiaris & A. C. Micheas & K. Zografos, 2019. "A Criterion for Local Model Selection," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(2), pages 406-444, December.
    3. Michael L. Stein, 2005. "Space-Time Covariance Functions," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 310-321, March.
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    5. Finn Lindgren & Håvard Rue & Johan Lindström, 2011. "An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(4), pages 423-498, September.
    6. Athanasios C. Micheas & Jiaxun Chen, 2018. "sppmix: Poisson point process modeling using normal mixture models," Computational Statistics, Springer, vol. 33(4), pages 1767-1798, December.
    7. Luai Al Labadi & Mahmoud Zarepour, 2018. "On Approximations of the Beta Process in Latent Feature Models: Point Processes Approach," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 59-79, February.
    8. Edoardo M Airoldi, 2007. "Getting Started in Probabilistic Graphical Models," PLOS Computational Biology, Public Library of Science, vol. 3(12), pages 1-5, December.
    9. Jesper Møller & Kateřina Helisová, 2010. "Likelihood Inference for Unions of Interacting Discs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 365-381, September.
    10. Zoubin Ghahramani, 2015. "Probabilistic machine learning and artificial intelligence," Nature, Nature, vol. 521(7553), pages 452-459, May.
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    Cited by:

    1. A. C. Micheas, 2025. "Random mixture Cox point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 77(2), pages 289-330, April.

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