Whittle-Mat\'ern fields are a recently introduced class of Gaussian processes on metric graphs, which are specified as solutions to a fractional-order stochastic differential equation. Unlike earlier covariance-based approaches for specifying Gaussian fields on metric graphs, the Whittle-Mat\'ern fields are well-defined for any compact metric graph and can provide Gaussian processes with differentiable sample paths. We derive the main statistical properties of the model class, particularly the consistency and asymptotic normality of maximum likelihood estimators of model parameters and the necessary and sufficient conditions for asymptotic optimality properties of linear prediction based on the model with misspecified parameters. The covariance function of the Whittle-Mat\'ern fields is generally unavailable in closed form, and they have therefore been challenging to use for statistical inference. However, we show that for specific values of the fractional exponent, when the fields have Markov properties, likelihood-based inference and spatial prediction can be performed exactly and computationally efficiently. This facilitates using the Whittle-Mat\'ern fields in statistical applications involving big datasets without the need for any approximations. The methods are illustrated via an application to modeling of traffic data, where allowing for differentiable processes dramatically improves the results.

翻译：Whittle-Matérn场是近年来提出的一类定义在度量图上的高斯过程，其通过分数阶随机微分方程的解来刻画。与早期基于协方差函数在度量图上定义高斯场的方法不同，Whittle-Matérn场对任意紧度量图均有良好定义，并能生成具有可微样本路径的高斯过程。本文推导了该模型类的主要统计性质，特别包括模型参数极大似然估计的相合性与渐进正态性，以及基于错误指定参数模型进行线性预测时渐进最优性的充要条件。由于Whittle-Matérn场的协方差函数通常无法显式表达，其在统计推断中的应用面临挑战。然而我们证明，当分数阶指数取特定值使得场具有马尔可夫性时，基于似然的推断与空间预测可实现精确且高效计算。这使Whittle-Matérn场得以在大数据统计应用中无需近似方法即可直接使用。通过交通数据建模的应用实例表明，允许过程具有可微性可显著提升建模效果。