Gaussian processes on lattices

Gaussian Processes with a stationary kernel are faster if you are working on a grid of points. The main tricks here seem to be circulant embeddings and circulant approximations, which enable one to leverage fast Fourier transforms. This complements, perhaps, the trick of filtering Gaussian processes.


Chan, G., and A. T. A. Wood. 1999. “Simulation of Stationary Gaussian Vector Fields.” Statistics and Computing 9 (4): 265–68.
Choromanski, Krzysztof, and Vikas Sindhwani. 2016. “Recycling Randomness with Structure for Sublinear Time Kernel Expansions.” May 29, 2016.
Davies, Tilman M., and David Bryant. 2013. “On Circulant Embedding for Gaussian Random Fields in R.” Journal of Statistical Software 55 (9).
Dietrich, C. R., and G. N. Newsam. 1993. “A Fast and Exact Method for Multidimensional Gaussian Stochastic Simulations.” Water Resources Research 29 (8): 2861–69.
Durrande, Nicolas, Vincent Adam, Lucas Bordeaux, Stefanos Eleftheriadis, and James Hensman. 2019. “Banded Matrix Operators for Gaussian Markov Models in the Automatic Differentiation Era.” February 26, 2019.
Graham, Ivan G., Frances Y. Kuo, Dirk Nuyens, Rob Scheichl, and Ian H. Sloan. 2017a. “Analysis of Circulant Embedding Methods for Sampling Stationary Random Fields.” October 2, 2017.
———. 2017b. “Circulant Embedding with QMC – Analysis for Elliptic PDE with Lognormal Coefficients.” October 25, 2017.
Gray, Robert M. 2006. Toeplitz and Circulant Matrices: A Review. Vol. 2.
Guinness, Joseph, and Montserrat Fuentes. 2016. “Circulant Embedding of Approximate Covariances for Inference From Gaussian Data on Large Lattices.” Journal of Computational and Graphical Statistics 26 (1): 88–97.
Kroese, Dirk P., and Zdravko I. Botev. 2013. “Spatial Process Generation.” August 1, 2013.
Powell, Catherine E. 2014. “Generating Realisations of Stationary Gaussian Random Fields by Circulant Embedding.” Matrix 2 (2): 1.
Sigrist, Fabio, Hans R. Künsch, and Werner A. Stahel. 2015a. “Spate : An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process.” Journal of Statistical Software 63 (14).
———. 2015b. “Stochastic Partial Differential Equation Based Modelling of Large Space-Time Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 (1): 3–33.
Stroud, Jonathan R., Michael L. Stein, and Shaun Lysen. 2017. “Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice.” Journal of Computational and Graphical Statistics 26 (1): 108–20.
Whittle, P. 1952. “Tests of Fit in Time Series.” Biometrika 39 (3-4): 309–18.
———. 1953a. “The Analysis of Multiple Stationary Time Series.” Journal of the Royal Statistical Society: Series B (Methodological) 15 (1): 125–39.
———. 1953b. “Estimation and Information in Stationary Time Series.” Arkiv För Matematik 2 (5): 423–34.
Whittle, P. 1954. “On Stationary Processes in the Plane.” Biometrika 41 (3/4): 434–49.
Whittle, Peter. 1952. “Some Results in Time Series Analysis.” Scandinavian Actuarial Journal 1952 (1-2): 48–60.

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