Random fields as stochastic differential equations

On ordering space in time

The representation of certain random fields, especially Gaussian random fields as stochastic differential equations. This is the engine that makes filtering Gaussian processes go, and is also a natural framing for probabilistic spectral analysis.

I do not have much to say right now about this, but I am using it so watch this space.

Creating an SDE with desired covariance

Warning: I'm taking crib notes for myself here, so I lazily switch between signal processing filter terminology. I assume Bochner’s and Yaglom’s Theorems as comprehensible methods for analysing covariance kernels.

Let’s start with stationry kernels. We consider an SDE \(f: \mathbb{R}\to\mathbb{R}\) at stationarity. We will let its driving noise to be some Wiener process. We care concerned with deriving the parameters of the SDE such that it has a given stationary covariance function \(k\).

If there are no zeros in the spectral density, then there are no poles in the inverse transfer function, and we can model it with an all-pole SDE. This includes all the classic Matérn functions. This is covered in Hartikainen and Särkkä (2010), and Lindgren, Rue, and Lindström (2011). Worked examples starting from a discrete time formulation are given in a tutorial introduction Grigorievskiy and Karhunen (2016).

More generally, (quasi-)periodic covariances have zeros and we need to find a full rational function approximation. Särkkä, Solin, and Hartikainen (2013) introduces one such method.

Solin and Särkkä (2014) has a fancier method employing resonators a.k.a. filter banks, to address a concern of Steven Reece et al. (2014) that atomic spectral peaks in the Fourier transform are not well approximated by rational functions.


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