Abstract: 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, for nonlinear predictions for nonGaussian 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 we capture the spatial dependence 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 has multiple advantages over Kriging for nonGaussian and nonstationary data. That is, it provides nonlinear predictions, and thus has smaller approximation errors. Furthermore, it does not require operations on covariance matrices, and thus is scalable for large data sets. With sufficiently many hidden neurons, the proposed method provides an optimal prediction in terms of model capacity. In addition, we quantify prediction uncertainties based on density prediction, without assuming a data distribution. Finally, we apply the method to PM_2.5 concentrations across the continental United States.

Key words and phrases: Basis function, deep learning, feature embedding, Gaussian process, spatial prediction.