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Bayesian Smoothing with Gaussian Processes Using Fourier Basis Functions in the spectralGP Package

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  • Paciorek, Christopher J.

Abstract

The spectral representation of stationary Gaussian processes via the Fourier basis provides a computationally efficient specification of spatial surfaces and nonparametric regression functions for use in various statistical models. I describe the representation in detail and introduce the spectralGP package in R for computations. Because of the large number of basis coefficients, some form of shrinkage is necessary; I focus on a natural Bayesian approach via a particular parameterized prior structure that approximates stationary Gaussian processes on a regular grid. I review several models from the literature for data that do not lie on a grid, suggest a simple model modification, and provide example code demonstrating MCMC sampling using the spectralGP package. I describe reasons that mixing can be slow in certain situations and provide some suggestions for MCMC techniques to improve mixing, also with example code, and some general recommendations grounded in experience.

Suggested Citation

  • Paciorek, Christopher J., 2007. "Bayesian Smoothing with Gaussian Processes Using Fourier Basis Functions in the spectralGP Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 19(i02).
    • Handle: RePEc:jss:jstsof:v:019:i02
    DOI: http://hdl.handle.net/10.18637/jss.v019.i02
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    1. Berger J.O. & De Oliveira V. & Sanso B., 2001. "Objective Bayesian Analysis of Spatially Correlated Data," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1361-1374, December.
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    Cited by:

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    2. Silas Bergen & Lianne Sheppard & Joel D. Kaufman & Adam A. Szpiro, 2016. "Multipollutant measurement error in air pollution epidemiology studies arising from predicting exposures with penalized regression splines," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 65(5), pages 731-753, November.
    3. Sigrist, Fabio & Künsch, Hans R. & Stahel, Werner A., 2015. "spate: An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 63(i14).
    4. Thomas Kneib & Nadja Klein & Stefan Lang & Nikolaus Umlauf, 2019. "Modular regression - a Lego system for building structured additive distributional regression models with tensor product interactions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(1), pages 1-39, March.
    5. Anyosa, Susan & Eidsvik, Jo & Pizarro, Oscar, 2023. "Adaptive spatial designs minimizing the integrated Bernoulli variance in spatial logistic regression models - with an application to benthic habitat mapping," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).
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    7. Maitreyee Bose & James S. Hodges & Sudipto Banerjee, 2018. "Toward a diagnostic toolkit for linear models with Gaussian‐process distributed random effects," Biometrics, The International Biometric Society, vol. 74(3), pages 863-873, September.

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