AI 中文总结
该研究提出适用于有向度量图的高斯过程统计框架，关联流网络模型并引入新边界条件，可高效处理大型数据集，在河流温度、道路交通速度建模中表现出性能优势。
AI 中文摘要
我们提出了一个统计框架，用于处理一般紧致有向度量图上任意边位置索引的高斯场。该构造基于带有一阶算子和顶点处条件的随机微分方程，我们刻画了其适定性并确定了协方差再生核希尔伯特空间。我们还将所提框架与早期的流网络模型关联，表明这些模型在特定边界条件下源于同一系统，并引入了能产生更符合物理实际过程的新边界条件。该微分方程表示可实现计算高效的推断与预测，这使得该方法无需近似即可应用于大型数据集。将其应用于河流网络的温度建模和道路网络的交通速度建模，验证了该框架的计算效率以及在物理信息顶点条件下的性能提升。
英文摘要
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.
Comments29 pages, 5 figures; includes supplementary material
