Gaussian processes (GP) and Kriging are widely used in traditional spatio-temporal modelling and prediction. These techniques typically presuppose that the data are observed from a stationary GP with a 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 assumptions and is computationally efficient. Therefore, it is suitable for large-scale prediction of complex spatio-temporal processes. We apply our method to monthly $P{M}_{2.5}$ 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.

高斯过程（GP）和克里金法广泛应用于传统的时空建模和预测。这些技术通常假定数据是从具有参数协方差结构的固定 GP观察到的。然而，现实应用中的过程通常表现出非高斯性和非平稳性。此外，基于似然的 GP 推理的计算成本很高，因此对于大型数据集来说是望而却步的。在本文中，我们提出了一种深度神经网络（DNN）基于时空插值和预测的两阶段模型。第一步执行插值，利用依赖 DNN 和由时空基函数构建的嵌入层。对于第二阶段，我们使用长短期记忆 (LSTM) 和卷积 LSTM 来预测给定位置的未来观测结果。我们在 DNN 中采用基于分位数的损失函数来提供概率预测。与克里金法相比，所提出的方法不需要指定协方差函数或进行平稳性假设，并且计算效率高。因此，它适用于复杂时空过程的大规模预测。我们将我们的方法应用于每月$磷{\mathrm{中号}}_{2。5}$1999 年 1 月至 2022 年 12 月超过 200,000 个时空位置的数据，用于快速估算缺失值和具有不确定性的预测。