Spatial statistics is traditionally based on stationary models on $\mathbb{R^d}$ like Mat\'ern fields. The adaptation of traditional spatial statistical methods, originally designed for stationary models in Euclidean spaces, to effectively model phenomena on linear networks such as stream systems and urban road networks is challenging. The current study aims to analyze the incidence of traffic accidents on road networks using three different methodologies and compare the model performance for each methodology. Initially, we analyzed the application of spatial triangulation precisely on road networks instead of traditional continuous regions. However, this approach posed challenges in areas with complex boundaries, leading to the emergence of artificial spatial dependencies. To address this, we applied an alternative computational method to construct nonstationary barrier models. Finally, we explored a recently proposed class of Gaussian processes on compact metric graphs, the Whittle-Mat\'ern fields, defined by a fractional SPDE on the metric graph. The latter fields are a natural extension of Gaussian fields with Mat\'ern covariance functions on Euclidean domains to non-Euclidean metric graph settings. A ten-year period (2010-2019) of daily traffic-accident records from Barcelona, Spain have been used to evaluate the three models referred above. While comparing model performance we observed that the Whittle-Mat\'ern fields defined directly on the network outperformed the network triangulation and barrier models. Due to their flexibility, the Whittle-Mat\'ern fields can be applied to a wide range of environmental problems on linear networks such as spatio-temporal modeling of water contamination in stream networks or modeling air quality or accidents on urban road networks.

翻译：空间统计学传统上基于$\mathbb{R^d}$上的平稳模型（如Matérn场）。将最初为欧氏空间平稳模型设计的传统空间统计方法有效应用于流系、城市路网等线性网络上的现象建模具有挑战性。本研究旨在采用三种不同方法分析道路网络交通事故发生率，并比较各方法的模型性能。首先，我们分析了在道路网络上（而非传统连续区域）精确应用空间三角剖分的方法，但此方法在边界复杂区域面临挑战，导致出现人为空间依赖性。为解决该问题，我们采用替代性计算方法构建非平稳屏障模型。最终，我们探索了近期提出的紧致度量图上高斯过程新类别——Whittle-Matérn场，该场由度量图上的分数阶SPDE定义。后者是欧氏域上具有Matérn协方差函数的高斯场向非欧氏度量图环境的自然延伸。研究采用西班牙巴塞罗那2010-2019年十年间的每日交通事故记录评估上述三种模型。在模型性能比较中，我们观察到直接在网络上定义的Whittle-Matérn场优于网络三角剖分和屏障模型。由于其灵活性，Whittle-Matérn场可广泛应用于线性网络上的环境问题，如河流网络水污染时空建模、城市道路网络空气质量或事故建模。