AI 中文总结
本研究针对度量图场景，提出引入潜在GRF的灵活协方差模型，经模拟与马德里交通流量数据验证，该模型在数据充足时协方差结构估计与预测能力优于较不灵活模型。
AI 中文摘要
Whittle-Matérn（WM）高斯随机场（GRF）被定义为随机偏微分方程（SPDE）的解，它是Matérn GRF在非欧几里得几何上的自然类似物，在非欧几里得几何中Matérn协方差函数不再适用。特别地，受道路和河流网络的启发，度量图上的WM GRF成为研究热点，这类网络中空间相关性用网络内的内在距离描述比欧几里得距离更自然。该GRF族由边缘方差、空间范围和平滑度三个参数控制，可通过SPDE中的空间变化系数扩展为广义WM GRF。近期研究已考虑使用空间变化协变量，但尚未挖掘其全部灵活性。本研究引入描述SPDE空间变化系数的潜在GRF，在模拟研究中将该灵活模型与灵活性较低的模型对比，评估两者估计协方差结构和预测能力。研究重点是可靠恢复协方差结构所需的观测值和重复次数。结果表明，在数据充足时，灵活模型比灵活性较低的模型表现更优；还在马德里某区域的交通流量数据上验证了该模型的实际适用性，观察到对比模型的样本内和样本外预测能力存在显著差异。
英文摘要
Whittle-Matérn (WM) Gaussian random fields (GRFs) are defined as solutions of stochastic partial differential equations (SPDEs) and provide a natural analog of Matérn GRFs on non-Euclidean geometry where the Matérn covariance function is not valid. In particular, WM GRFs on metric graphs have been an active area of research motivated by road and river networks where spatial dependence is more naturally described by intrinsic distances in the network than by Euclidean distances. This family of GRFs is controlled by three parameters relating to marginal variance, spatial range, and smoothness, but can be extended to so-called generalized WM GRFs through spatially varying coefficients in the SPDE. Recent work has considered the use of spatially varying covariates, but the full possibilities of flexibility have not been considered. In this work, we introduce latent GRFs that describe the spatially varying coefficients of the SPDE. This flexible model is compared to less flexible models in a simulation study evaluating both the ability to estimate the covariance structure and predictive ability. An important focus is the number of observations and replications necessary to reliably recover the covariance structure. We find that the flexible model improves over less flexible models in the presence of sufficient data. We also demonstrate practical applicability on traffic counts in a part of Madrid, and observe major differences between in-sample and out-of-sample predictive abilities of the models compared.
