DeepKriging: Spatially Dependent Deep Neural Networks for Spatial Prediction

• Activity Number: 217 - High-Fidelity Gaussian Process Surrogate Modeling: Deep and Shallow
• Type: Invited
• Date/Time: Wednesday, August 11, 2021 : 10:00 AM to 11:50 AM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): Yuxiao Li and Ying Sun* and Brian Reich
• Companies: King Abdullah University of Science and Technology (KAUST) and King Abdullah University of Science and Technology (KAUST) and North Carolina State University
• Keywords: Deep learning; Gaussian processes; Radial basis function; Spatial Regression; Feature embedding

Abstract

In spatial statistics, a common objective is to predict the values of a spatial process at unobserved locations by exploiting spatial dependence. In geostatistics, 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 not necessarily optimal, and the associated variance is often overly optimistic. We propose to use deep neural networks (DNNs) for spatial prediction. Although 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 neural network structure for spatial prediction by adding an embedding layer of spatial coordinates with basis functions. We show in theory that the proposed DeepKriging method has multiple advantages over Kriging and classical DNNs only with spatial coordinates as features. We also provide density prediction for uncertainty quantification without any distributional assumption and apply the method to PM2.5 concentrations across the continental United States.