Whittle--Matérn metrics and isotropic Gaussian processes on compact metric graphs
Abstract
Isotropic Gaussian processes on networks have a covariance that is a function of a distance between locations, but valid constructions have so far required graphs with Euclidean edges and the resistance metric. We introduce a one-parameter family of metrics dκ, κ≥0, on arbitrary compact metric graphs, including those with loops and parallel edges, defined as the variograms of first-order Whittle--Matérn fields, where κ=0 corresponds to an intrinsic limiting field. We show that these Whittle--Matérn metrics are invariant under insertion and removal of vertices of degree two, that they decrease monotonically in κ, and that they can be evaluated by sparse linear solves. The metric d0 coincides with the resistance metric on graphs with Euclidean edges, and thus extends it to general compact metric graphs without any modification of the graph. For every completely monotone function r and every κ≥0, we show that r(dκ) is a positive semidefinite function. This yields isotropic Gaussian processes on general compact metric graphs. For κ>0 the metric levels off beyond the screening length κ−1, which can be used to induce long-range dependence in short-range covariance models, and κ can be estimated from data together with the parameters of r. The models are compared on daily traffic maps from the street network of Paris and the motorway network around London. For both, κ is estimated to be positive, dκ increases the maximised log-likelihood relative to d0 for every covariance function, and it gives the best predictions at randomly held-out locations.
Related
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BibTeX
@misc{bolin2026whittlematérn,
title = {{Whittle--Matérn metrics and isotropic Gaussian processes on compact metric graphs}},
author = {David Bolin and Alexandre B. Simas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35732},
eprint = {2610.35732},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35732},
}
RIS
TY - UNPB
TI - Whittle--Matérn metrics and isotropic Gaussian processes on compact metric graphs
AU - David Bolin
AU - Alexandre B. Simas
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35732
UR - https://arxiv.org/abs/2610.35732
AB - Isotropic Gaussian processes on networks have a covariance that is a function of a distance between locations, but valid constructions have so far required graphs with Euclidean edges and the resistance metric. We introduce a one-parameter family of metrics d_κ, κ≥0, on arbitrary compact metric graphs, including those with loops and parallel edges, defined as the variograms of first-order Whittle--Matérn fields, where κ=0 corresponds to an intrinsic limiting field. We show that these Whittle--Matérn metrics are invariant under insertion and removal of vertices of degree two, that they decrease monotonically in κ, and that they can be evaluated by sparse linear solves. The metric d_0 coincides with the resistance metric on graphs with Euclidean edges, and thus extends it to general compact metric graphs without any modification of the graph. For every completely monotone function r and every κ≥0, we show that r(d_κ) is a positive semidefinite function. This yields isotropic Gaussian processes on general compact metric graphs. For κ>0 the metric levels off beyond the screening length κ^−1, which can be used to induce long-range dependence in short-range covariance models, and κ can be estimated from data together with the parameters of r. The models are compared on daily traffic maps from the street network of Paris and the motorway network around London. For both, κ is estimated to be positive, d_κ increases the maximised log-likelihood relative to d_0 for every covariance function, and it gives the best predictions at randomly held-out locations.
ER -