Neural networks for geospatial data

Analysis of geospatial data has traditionally been model-based, with a mean model, customarily specified as a linear regression on the covariates, and a Gaussian process covariance model, encoding the spatial dependence. While nonlinear machine learning algorithms like neural networks are increasingly being used for spatial analysis, current approaches depart from the model-based setup and cannot explicitly incorporate spatial covariance. We propose NN-GLS, embedding neural networks directly within the traditional Gaussian process (GP) geostatistical model to accommodate non-linear mean functions while retaining all other advantages of GP, like explicit modeling of the spatial covariance and predicting at new locations via kriging. In NN-GLS, estimation of the neural network parameters for the non-linear mean of the Gaussian Process explicitly accounts for the spatial covariance through use of the generalized least squares (GLS) loss, thus extending the linear case. We show that NN-GLS admits a representation as a special type of graph neural network (GNN). This connection facilitates the use of standard neural network computational techniques for irregular geospatial data, enabling novel and scalable mini-batching, backpropagation, and kriging schemes. We provide methodology to obtain uncertainty bounds for estimation and predictions from NN-GLS. Theoretically, we show that NN-GLS will be consistent for irregularly observed spatially correlated data processes. We also provide a finite sample concentration rate, which quantifies the need to accurately model the spatial covariance in neural networks for dependent data. To our knowledge, these are the first large-sample results for any neural network algorithm for irregular spatial data. We demonstrate the methodology through numerous simulations and an application to air pollution modeling. We develop a software implementation of NN-GLS in the Python package geospaNN.