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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

Simulating Gaussian RVs with the desired covariance structure

[Following the introduction in (DiNe93)]

Let's say we wish to generate a stationary Gaussian process on a points . .

Stationary in this context means that the covariance function is translation-invariance and depend only on distance, so that it may be given . Without loss of generality, we assume that and .

The problem then reduces to generating a vector where has entries

Note that if is an -dimensional normal random variable, and , then 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, ; specifically, equally-spaced-points on a line.

We know that has a Toeplitz structure. Moreover it is non-negative definite, with (Why?)

Refs