  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.
