Whittle–Matérn metrics and isotropic Gaussian processes on compact metric graphs

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 $d_κ$, $κ\ge0$, 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 $κ\ge0$, 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.