Neural networks for geospatial data

TL;DR

This work advances geostatistical modeling by integrating neural networks into Gaussian process mean functions while preserving explicit spatial covariance via generalized least squares (GLS) loss. The NN-GLS algorithm is shown to be realizable as a two-layer graph neural network using nearest-neighbor Gaussian process (NNGP) structure, enabling scalable mini-batching, backpropagation, and kriging. The authors provide rigorous theory for existence, consistency under irregular spatial designs, and finite-sample error rates that quantify the advantage of incorporating accurate spatial covariance over vanilla NN with independence assumptions. Simulations and a PM$_{2.5}$ case study demonstrate that NN-GLS yields superior estimation and prediction performance, reliable uncertainty quantification via spatial bootstrap, and interpretable covariate effects through the nonlinear mean function. Overall, NN-GLS offers a principled, scalable framework for nonparametric mean modeling in spatial settings, with broad applicability to irregular geospatial data and complex covariate interactions.

Abstract

Analysis of geospatial data has traditionally been model-based, with a mean model, customarily specified as a linear regression on the covariates, and a covariance model, encoding the spatial dependence. We relax the strong assumption of linearity and propose embedding neural networks directly within the traditional geostatistical models to accommodate non-linear mean functions while retaining all other advantages including use of Gaussian Processes to explicitly model the spatial covariance, enabling inference on the covariate effect through the mean and on the spatial dependence through the covariance, and offering predictions at new locations via kriging. We propose NN-GLS, a new neural network estimation algorithm for the non-linear mean in GP models that explicitly accounts for the spatial covariance through generalized least squares (GLS), the same loss used in the linear case. We show that NN-GLS admits a representation as a special type of graph neural network (GNN). This connection facilitates use of standard neural network computational techniques for irregular geospatial data, enabling novel and scalable mini-batching, backpropagation, and kriging schemes. 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 simulated and real datasets.

Figures (32)

• Figure 1: NN-GLS as a graph neural network with two graph convolution layers
• Figure 2: (a): The estimation performance comparison with $f_0 = f_2$; (b): (Section \ref{['sec-sim-friedman']}) Estimation performance comparison among GAM, GAMGLS and NN-GLS against $rho$, i.e. the interaction strength; (c): The prediction performance comparison with $f_0 = f_2$; (d): (Section \ref{['sec-sim-NN']}) Prediction performance comparison among non-spatial NN, NN-GLS and NN-splines against sample size; (e): (Section \ref{['sec-sim-large']}) The consistency of estimation. (f): (Section \ref{['sec-sim-large']}) The running time for estimation.
• Figure 3: PM$_{2.5}$ data analysis.
• Figure 4: Partial dependency plots showing the marginal effects of temperature and west wind (U-wind).
• Figure S5: Parameter estimations in NN-GLS. We denote NN-nonspatial by NN here to save space for axis text.

Theorems & Definitions (20)

• Theorem 1: Existence of seive estimator
• Theorem 2: Consistency
• Proposition 1
• Proposition 2
• Theorem 3: Convergence rate
• Theorem S1: Theorem 2.2 in white1991some
• Lemma S1
• Lemma S2
• Definition S4.1: Orlicz norm
• Proposition S1