# Gaussian processes on lattices

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Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

Gaussian Processes with a stationary kernel are simpler 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 the Fourier Transform. This complements, perhaps, the trick of filtering Gaussian processes.

# Refs

Chan, G., and A. T. A. Wood. 1999. “Simulation of Stationary Gaussian Vector Fields.” Statistics and Computing 9 (4): 265–68. https://doi.org/10.1023/A:1008903804954.

Choromanski, Krzysztof, and Vikas Sindhwani. 2016. “Recycling Randomness with Structure for Sublinear Time Kernel Expansions,” May. http://arxiv.org/abs/1605.09049.

Davies, Tilman M., and David Bryant. 2013. “On Circulant Embedding for Gaussian Random Fields in R.” Journal of Statistical Software 55 (9). https://doi.org/10.18637/jss.v055.i09.

Dietrich, C. R., and G. N. Newsam. 1993. “A Fast and Exact Method for Multidimensional Gaussian Stochastic Simulations.” Water Resources Research 29 (8): 2861–9. https://doi.org/10.1029/93WR01070.

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. http://arxiv.org/abs/1710.00751.

———. 2017b. “Circulant Embedding with QMC – Analysis for Elliptic PDE with Lognormal Coefficients,” October. http://arxiv.org/abs/1710.09254.

Gray, Robert M. 2006. “Toeplitz and Circulant Matrices: A Review.” Foundations and Trends® in Communications and Information Theory 2 (3): 155–239. https://doi.org/10.1561/0100000006.

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. https://doi.org/10.1080/10618600.2016.1164534.

Kroese, Dirk P., and Zdravko I. Botev. 2013. “Spatial Process Generation,” August. http://arxiv.org/abs/1308.0399.

Powell, Catherine E. 2014. “Generating Realisations of Stationary Gaussian Random Fields by Circulant Embedding.” Matrix 2 (2): 1.

Stroud, Jonathan R., Michael L. Stein, and Shaun Lysen. 2014. “Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice,” February. http://arxiv.org/abs/1402.4281.

———. 2017. “Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice.” Journal of Computational and Graphical Statistics 26 (1): 108–20. https://doi.org/10.1080/10618600.2016.1152970.

Whittle, P. 1953a. “The Analysis of Multiple Stationary Time Series.” Journal of the Royal Statistical Society. Series B (Methodological) 15 (1): 125–39. http://www.jstor.org/stable/2983728.

———. 1953b. “Estimation and Information in Stationary Time Series.” Arkiv För Matematik 2 (5): 423–34. https://doi.org/10.1007/BF02590998.

Whittle, P. 1954. “On Stationary Processes in the Plane.” Biometrika 41 (3/4): 434–49.

———. 1952. “Tests of Fit in Time Series.” Biometrika 39 (3-4): 309–18. https://doi.org/10.1093/biomet/39.3-4.309.

Whittle, Peter. 1952. “Some Results in Time Series Analysis.” Scandinavian Actuarial Journal 1952 (1-2): 48–60. https://doi.org/10.1080/03461238.1952.10414182.