The increasing availability of network data has driven the development of advanced statistical models specifically designed for metric graphs, where Gaussian processes play a pivotal role. While models such as Whittle-Matérn fields have been introduced, there remains a lack of practically applicable options that accommodate flexible non-stationary covariance structures or general smoothness. To address this gap, we propose a novel class of generalized Whittle-Matérn fields, which are rigorously defined on general compact metric graphs and permit both non-stationarity and arbitrary smoothness. We establish new regularity results for these fields, which extend even to the standard Whittle-Matérn case. Furthermore, we introduce a method to approximate the covariance operator of these processes by combining the finite element method with a rational approximation of the operator's fractional power, enabling computationally efficient Bayesian inference for large datasets. Theoretical guarantees are provided by deriving explicit convergence rates for the covariance approximation error, and the practical utility of our approach is demonstrated through simulation studies and an application to traffic speed data, highlighting the flexibility and effectiveness of the proposed model class.

翻译：随着网络数据的日益普及，专门针对度量图设计的先进统计模型得到了快速发展，其中高斯过程发挥着关键作用。尽管Whittle-Matérn场等模型已被提出，但仍缺乏能够适应灵活非平稳协方差结构或一般光滑性的实际可用模型。为填补这一空白，我们提出了一类新颖的广义Whittle-Matérn场，该模型严格定义在一般紧致度量图上，同时允许非平稳性和任意光滑性。我们为这些场建立了新的正则性结果，这些结果甚至可推广至标准Whittle-Matérn场情形。此外，我们提出了一种通过结合有限元方法与算子分数阶幂的有理逼近来近似这些过程协方差算子的方法，从而实现了对大计算数据集的高效贝叶斯推断。通过推导协方差近似误差的显式收敛速率，我们提供了理论保证，并通过模拟研究和交通速度数据的应用展示了所提方法的实用价值，凸显了该模型类别的灵活性与有效性。