# Gaussian Process simulation and circulant embeddings

### I might shoehorn Whittle likelihoods in here too

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 RVs with the desired covariance structure

[Following the introduction in (DiNe93)]

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?)