  In spatial statistics, a common objective is to predict values of a spatial
process at unobserved locations by exploiting spatial dependence. Kriging
provides the best linear unbiased predictor using covariance functions and is
often associated with Gaussian processes. However, when considering non-linear
prediction for non-Gaussian and categorical data, the Kriging prediction is no
longer optimal, and the associated variance is often overly optimistic.
Although deep neural networks (DNNs) are widely used for general classification
and prediction, they have not been studied thoroughly for data with spatial
dependence. In this work, we propose a novel DNN structure for spatial
prediction, where the spatial dependence is captured by adding an embedding
layer of spatial coordinates with basis functions. We show in theory and
simulation studies that the proposed DeepKriging method has a direct link to
Kriging in the Gaussian case, and it has multiple advantages over Kriging for
non-Gaussian and non-stationary data, i.e., it provides non-linear predictions
and thus has smaller approximation errors, it does not require operations on
covariance matrices and thus is scalable for large datasets, and with
sufficiently many hidden neurons, it provides the optimal prediction in terms
of model capacity. We further explore the possibility of quantifying prediction
uncertainties based on density prediction without assuming any data
distribution. Finally, we apply the method to predicting PM2.5 concentrations
across the continental United States.
