Flexible covariance structures on metric graphs

2026-08-14

AI Summary

This study addresses the limited flexibility of Whittle-Matérn Gaussian random field covariance structures on metric graphs by proposing a novel framework that models spatially varying coefficients in stochastic partial differential equations via latent Gaussian random fields. Through numerical simulations and empirical analysis using Madrid traffic data, the proposed model demonstrates superior performance in both estimation and prediction tasks. Results indicate that, given sufficient data, this approach significantly enhances covariance estimation accuracy and outperforms traditional models in real-world traffic flow forecasting. Consequently, this work effectively extends the applicability and precision of spatial statistical modeling within graph domains, offering a robust solution for complex network-structured data where standard stationary assumptions are insufficient.

Abstract

Whittle-Matérn (WM) Gaussian random fields (GRFs) are defined as solutions of stochastic partial differential equations (SPDEs) and provide a natural analog of Matérn GRFs on non-Euclidean geometry where the Matérn covariance function is not valid. In particular, WM GRFs on metric graphs have been an active area of research motivated by road and river networks where spatial dependence is more naturally described by intrinsic distances in the network than by Euclidean distances. This family of GRFs is controlled by three parameters relating to marginal variance, spatial range, and smoothness, but can be extended to so-called generalized WM GRFs through spatially varying coefficients in the SPDE. Recent work has considered the use of spatially varying covariates, but the full possibilities of flexibility have not been considered. In this work, we introduce latent GRFs that describe the spatially varying coefficients of the SPDE. This flexible model is compared to less flexible models in a simulation study evaluating both the ability to estimate the covariance structure and predictive ability. An important focus is the number of observations and replications necessary to reliably recover the covariance structure. We find that the flexible model improves over less flexible models in the presence of sufficient data. We also demonstrate practical applicability on traffic counts in a part of Madrid, and observe major differences between in-sample and out-of-sample predictive abilities of the models compared.

Problem

• Metric graphs
• Whittle-Matérn Gaussian random fields
• Flexible covariance structures
• Stochastic partial differential equations
• Spatially varying coefficients

Innovation

• Whittle-Matérn Gaussian random fields
• latent GRFs
• spatially varying coefficients
• metric graphs
• SPDE