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. 🏗
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