Statistical inference for Gaussian Whittle–Matérn fields on metric graphs

Abstract Whittle–Matérn fields are a recently introduced class of Gaussian processes on metric graphs, specified as solutions to a fractional-order stochastic differential equation. Unlike previous covariance-based methods, these fields are well-defined for any compact metric graph and can provide Gaussian processes with differentiable sample paths. We derive the main statistical properties, including the consistency and asymptotic normality of maximum likelihood estimators and the necessary and sufficient conditions for optimal prediction with misspecified parameters. The covariance function is generally unavailable in closed form, which makes statistical inference challenging. However, we show that for specific values of the fractional exponent where the fields exhibit Markov properties, likelihood-based inference and spatial prediction can be performed exactly and efficiently. This enables the use of Whittle–Matérn fields in large datasets without approximations. The results and methods are illustrated through simulation studies and through an application to traffic data modelling, where allowing for differentiable processes significantly improves results.