Gaussian Whittle-Mat\'ern fields on metric graphs

TL;DR

This paper introduces a new class of Gaussian processes called Whittle–Matérn fields on compact metric graphs, extending Gaussian fields with Matérn covariance to non-Euclidean graph domains, with properties, expansions, and numerical comparisons.

Contribution

The paper constructs the first differentiable Gaussian processes on general compact metric graphs using fractional stochastic differential equations.

Findings

Processes are well-defined with sample path regularity.

The models are invariant under addition/removal of degree-two vertices.

Karhunen–Loève expansions and numerical comparisons are provided.

Abstract

We define a new class of Gaussian processes on compact metric graphs such as street or river networks. The proposed models, the Whittle--Mat\'ern fields, are defined via a fractional stochastic differential equation on the compact metric graph and are a natural extension of Gaussian fields with Mat\'ern covariance functions on Euclidean domains to the non-Euclidean metric graph setting. Existence of the processes, as well as some of their main properties, such as sample path regularity are derived. The model class in particular contains differentiable processes. To the best of our knowledge, this is the first construction of a differentiable Gaussian process on general compact metric graphs. Further, we prove an intrinsic property of these processes: that they do not change upon addition or removal of vertices with degree two. Finally, we obtain Karhunen--Lo\`eve expansions of the processes, provide numerical experiments, and compare them to Gaussian processes with isotropic covariance functions.