Gaussian Processes on Directed Metric Graphs
Abstract
We introduce a statistical framework for Gaussian fields indexed at arbitrary edge locations on general compact directed metric graphs. The construction is based on a stochastic differential equation with a first-order operator and conditions at the vertices. We characterise well-posedness and identify the covariance reproducing kernel Hilbert space. We also connect the proposed framework to earlier stream-network models, showing that these arise from the same system under particular boundary conditions, and introduce new boundary conditions that yield more physically realistic processes. The differential-equation representation enables computationally efficient inference and prediction. This makes the method applicable to large data sets without approximation. Applications to temperature modelling on river networks and traffic speeds on road networks illustrate the framework, including the computational efficiency and improved performance under physically informed vertex conditions.
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BibTeX
@misc{bolin2026gaussian,
title = {{Gaussian Processes on Directed Metric Graphs}},
author = {David Bolin and Alexandre de Bustamante Simas and Erik Karlsson Strandh and Jonas Wallin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01435},
eprint = {2609.01435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01435},
}
RIS
TY - UNPB
TI - Gaussian Processes on Directed Metric Graphs
AU - David Bolin
AU - Alexandre de Bustamante Simas
AU - Erik Karlsson Strandh
AU - Jonas Wallin
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01435
UR - https://arxiv.org/abs/2609.01435
AB - We introduce a statistical framework for Gaussian fields indexed at arbitrary edge locations on general compact directed metric graphs. The construction is based on a stochastic differential equation with a first-order operator and conditions at the vertices. We characterise well-posedness and identify the covariance reproducing kernel Hilbert space. We also connect the proposed framework to earlier stream-network models, showing that these arise from the same system under particular boundary conditions, and introduce new boundary conditions that yield more physically realistic processes. The differential-equation representation enables computationally efficient inference and prediction. This makes the method applicable to large data sets without approximation. Applications to temperature modelling on river networks and traffic speeds on road networks illustrate the framework, including the computational efficiency and improved performance under physically informed vertex conditions.
ER -