Neural networks for geospatial data

Geostatistics, the analysis of geocoded data, is traditionally based on stochastic process models which offer a coherent way to model data at any finite collection of locations while ensuring the generalizability of inference to the entire region.Gaussian processes (GP) with a mean function capturing effects of covariates and the covariance function encoding the spatial dependence, is a staple for geostatistical analysis, offering theoretical guarantees and practical benefits. GP are flexible enough to model any smooth spatial surface, and can be specified parsimoniously with covariance functions using a very small set of parameters. The spatial covariance parameters offer insights into the smoothness and spatial properties of the response process (Stein, 1999). The finite dimensional realizations of a GP are multivariate Gaussian, thereby offering estimates of the mean and covariance parameters via convenient maximization of the Gaussian likelihood, and predictions at new locations by using conditional Gaussian distributions (see, e.g., Banerjee et al., 2014; Cressie and Wikle, 2015, for detailed exposition on GP models for spatial and spatio-temporal data). Also, computational roadblocks to using GP for large spatial data have been greatly mitigated by recent advances (see, Heaton et al., 2019, for a recent review of scalable GP approaches). The mean function of a Gaussian process is often modeled as a linear regression on the covariates. The growing popularity and accessibility of machine learning algorithms such as neural networks, random forests, gradient boosted trees, capable of modeling complex non-linear relationships has heralded a paradigm shift. Practitioners are increasingly shunning models with parametric assumptions like linearity in favor of these machine learning approaches that can capture non-linearity and high-order interactions in a data-driven manner. The field of spatial statistics has not been insulated from this machine learning revolution.