Non-Stationary Covariance Functions for Spatial Data on Linear Networks

We introduce a novel class of non-stationary covariance functions for random fields on linear networks that allows both the variance and the correlation range of the random field to vary spatially. The proposed covariance functions are useful to model random fields with a spatial dependence that is locally isotropic with respect to the resistance metric, a distance that reflects the topology of the network. The framework admits explicit stochastic representations of the associated random fields and can be naturally extended to matrix-valued covariance functions for vector-valued random fields. We assess the statistical and computational performance of a weighted local likelihood estimator for the proposed models using synthetic data generated on the street network of the University of Chicago neighborhood.

💡 Research Summary

The paper introduces a new class of non‑stationary covariance functions for Gaussian random fields defined on linear networks such as road, river, or electrical grids. Classical spatial statistics on networks have largely relied on isotropic models based on Euclidean or geodesic distances, which ignore the underlying graph topology and often lead to unrealistic dependence structures, especially when the network contains cycles. To overcome these limitations, the authors adopt the resistance metric—a distance derived from electrical network theory that reflects the connectivity of the graph and can be defined continuously along edges.

The core theoretical contribution is Theorem 3.1, which shows that for any two strictly positive functions a(·) and b(·) on the network and any finite positive measure F on (0,∞), the kernel

K(s,t)=∫₀^∞ b(s)b(t)