Gaussian processes (GP) and Kriging are widely used in traditional spatio-temporal mod-elling and prediction. These techniques typically presuppose that the data are observed from a stationary GP with parametric covariance structure. However, processes in real-world applications often exhibit non-Gaussianity and nonstationarity. Moreover, likelihood-based inference for GPs is computationally expensive and thus prohibitive for large datasets. In this paper we propose a deep neural network (DNN) based two-stage model for spatio-temporal interpolation and forecasting. Interpolation is performed in the first step, which utilizes a dependent DNN with the embedding layer constructed with spatio-temporal basis functions. For the second stage, we use Long-Short Term Memory (LSTM) and convolutional LSTM to forecast future observations at a given location. We adopt the quantile-based loss function in the DNN to provide probabilistic forecasting. Compared to Kriging, the proposed method does not require specifying covariance functions or making stationarity assumption, and is computationally efficient. Therefore, it is suitable for large-scale prediction of complex spatio-temporal processes. We apply our method to monthly $PM_{2.5}$ data at more than $200,000$ space-time locations from January 1999 to December 2022 for fast imputation of missing values and forecasts with uncertainties.

翻译：高斯过程（GP）与克里金法广泛应用于传统时空建模与预测。这些技术通常假设观测数据源自具有参数化协方差结构的平稳高斯过程。然而，现实世界中的过程往往呈现非高斯性与非平稳性。此外，基于似然的GP推断计算成本高昂，难以应用于大规模数据集。本文提出一种基于深度神经网络（DNN）的两阶段时空插值与预测模型。第一阶段执行插值，利用依赖型DNN，其嵌入层由时空基函数构建；第二阶段采用长短期记忆网络（LSTM）与卷积LSTM预测给定位置的未来观测值。我们在DNN中采用基于分位数的损失函数以实现概率预测。与克里金法相比，所提方法无需指定协方差函数或假设平稳性，且计算高效，适用于复杂时空过程的大规模预测。我们将该方法应用于1999年1月至2022年12月期间超过20万个时空位置的月均PM₂.₅数据，实现缺失值的快速填补及含不确定性的预测。