Simulating Gaussian processes on a lattice



Assumed audience:

ML people

How can I simulate a Gaussian Processes on a lattice with a given covariance?

The general (non-lattice) case is given in historical overview in Liu et al. (2019), but in this notebook we are interested in specialising a little. Following the introduction in Dietrich and Newsam (1993), let’s say we wish to generate a stationary Gaussian process \(Y(x)\) on a points \(\Omega\). \(\Omega=(x_0, x_1,\dots, x_m)\).

Stationary in this context means that the covariance function \(r\) is translation-invariance and depend only on distance, so that it may be given \(r(|x|)\). Without loss of generality, we assume that \(\bb E[Y(x)]=0\) and \(\var[Y(x)]=1\).

The problem then reduces to generating a vector \(\vv y=(Y(x_0), Y(x_1), \dots, Y(x_m) )\sim \mathcal{N}(0, R)\) where \(R\) has entries \(R[p,q]=r(|x_p-x_q|).\)

Note that if \(\bb \varepsilon\sim\mathcal{N}(0, I)\) is an \(m+1\)-dimensional normal random variable, and \(AA^T=R\), then \(\vv y=\mm A \vv \varepsilon\) has the required distribution.

The circulant embedding trick

If we have additional structure, we can work more efficiently.

Suppose further that our points form a grid, \(\Omega=(x_0, x_0+h,\dots, x_0+mh)\); specifically, equally-spaced-points on a line.

We know that \(R\) has a Toeplitz structure. Moreover it is non-negative definite, with \(\vv x^t\mm R \vv x \geq 0\forall \vv x.\) (Why?) πŸ—

Wilson et al. (2021) credits the following authors:

Well-known examples of this trend include banded and sparse matrices in the context of one-dimensional Gaussian processes and Gauss–Markov random fields [;Durrande et al. (2019)], as well as Kronecker and Toeplitz matrices when working with regularly-spaced grids (Dietrich and Newsam 1997; Grace Chan and Wood 1997).

References

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Chan, Grace, and Andrew T.A. Wood. 1997. β€œAlgorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 46 (1): 171–81.
Chan, G., and A. T. A. Wood. 1999. β€œSimulation of Stationary Gaussian Vector Fields.” Statistics and Computing 9 (4): 265–68.
Davies, Tilman M., and David Bryant. 2013. β€œOn Circulant Embedding for Gaussian Random Fields in R.” Journal of Statistical Software 55 (9).
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Durrande, Nicolas, Vincent Adam, Lucas Bordeaux, Stefanos Eleftheriadis, and James Hensman. 2019. β€œBanded Matrix Operators for Gaussian Markov Models in the Automatic Differentiation Era.” In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, 2780–89. PMLR.
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