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Spatial linear mixed-models, consisting of a linear covariate effect and a Gaussian process (GP) distributed spatial random effect, are widely used for analyses of geospatial data. We consider the setting where the covariate effect is nonlinear. Random forests (RF) are popular for estimating nonlinear functions but applications of RF for spatial data have often ignored the spatial correlation. We show that this impacts the performance of RF adversely. We propose RF-GLS, a novel and well-principled extension of RF, for estimating nonlinear covariate effects in spatial mixed models where the spatial correlation is modeled using GP. RF-GLS extends RF in the same way generalized least squares (GLS) fundamentally extends ordinary least squares (OLS) to accommodate for dependence in linear models. RF becomes a special case of RF-GLS, and is substantially outperformed by RF-GLS for both estimation and prediction across extensive numerical experiments with spatially correlated data. RF-GLS can be used for functional estimation in other types of dependent data like time series. We prove consistency of RF-GLS for  β -mixing dependent error processes that include the popular spatial Matérn GP. As a byproduct, we also establish, to our knowledge, the first consistency result for RF under dependence. We establish results of independent importance, including a general consistency result of GLS optimizers of data-driven function classes, and a uniform law of large number under  β -mixing dependence with weaker assumptions. These new tools can be potentially useful for asymptotic analysis of other GLS-style estimators in nonparametric regression with dependent data.

Random Forests for Spatially Dependent Data