# Gaussian Process simulation and circulant embeddings

### I might shoehorn Whittle likelihoods in here too

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

An converse problem to covariance estimation. Related: phase retrieval. Gaussian process regression.

## TODO

Discuss in the context of

• deterministic covariance
• expected autocorrelation in covariance

## Simulating Gaussian fields with the desired covariance structure

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 $$\vv y=\mm A\bb \varepsilon$$ has the required distribution.

### The circulant embedding trick

But what if that’s too slow? If we have additional structure, we can do better.

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?) 🚧

## Simulating point processes with the desired covariance structure

For now, see spatial point 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.

Ellis, Robert L., and David C. Lay. 1992. “Factorization of Finite Rank Hankel and Toeplitz Matrices.” Linear Algebra and Its Applications 173 (August): 19–38. https://doi.org/10.1016/0024-3795(92)90420-F.

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.

Heinig, Georg, and Karla Rost. 2011. “Fast Algorithms for Toeplitz and Hankel Matrices.” Linear Algebra and Its Applications 435 (1): 1–59. https://doi.org/10.1016/j.laa.2010.12.001.

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

Ye, Ke, and Lek-Heng Lim. 2016. “Every Matrix Is a Product of Toeplitz Matrices.” Foundations of Computational Mathematics 16 (3): 577–98. https://doi.org/10.1007/s10208-015-9254-z.