Spatial prediction of environmental processes using random forests: How best to account for spatial dependence?

Technical Summary: Spatial Prediction of Environmental Processes Using Random Forests

Problem Statement
Spatial prediction of environmental processes, such as air pollution concentrations, is critical for regulatory compliance and epidemiological research. While Gaussian process models (e.g., Kriging) are the standard for spatial prediction due to their ability to handle spatial autocorrelation, machine learning (ML) algorithms like Random Forests (RF) are state-of-the-art for non-spatial prediction due to their capacity to model complex nonlinear covariate-response relationships. A significant gap exists in the literature regarding the optimal method for fusing these paradigms. While numerous approaches have been proposed to imbue ML algorithms with spatial capabilities—such as fusing with Gaussian processes, utilizing observation-driven correlation structures, or employing local geographical fitting—there has been no numerical comparison of their relative predictive performances. Consequently, environmental scientists lack guidance on which approach to select, particularly when the underlying cause of spatial autocorrelation (e.g., unmeasured confounding, spatial spillover, or heterogeneous effects) is unknown.

Methodology
This study focuses on Random Forests as the primary ML algorithm due to their computational efficiency and conceptual straightforwardness compared to deep learning alternatives. The authors evaluate five distinct frameworks for adapting RF to spatial data:

1. Standard RF: A non-spatial baseline implementation.
2. Spatially Smoothed Features: Augmenting the covariate set with spatially smoothed versions of covariates and/or the response variable using exponential kernel weights (Soltani et al., 2022).
3. Spatial Basis Functions (RF-Basis): Incorporating multi-resolution radial basis functions (constructed with Wendland compactly supported correlation functions) as additional covariates (Chen et al., 2024).
4. Local Data Fitting (RF-Local): A Geographically Random Forest approach that fits a local RF model using only nearby data points for each prediction location, combined with a global model (Georganos et al., 2021).
5. Spatially Smoothed Gaussian Process (RF-GP): A residual learning strategy where a Gaussian Process (GP) is fitted to the residuals of an RF model. This includes both a one-step approach (Hengl et al., 2015) and a full iterative approach (MacBride et al., 2025) where the GP and RF are sequentially re-fitted.

The evaluation was conducted through two studies:

• Simulation Study: A controlled experiment generating 50 datasets across six scenarios varying by noise levels (low/high) and the cause of spatial autocorrelation (unmeasured confounding, spatial spillover, and spatially heterogeneous effects). Performance was measured by Root Mean Square Prediction Error (RMSPE) and bias.
• Real Data Study: An application to the SCALE study in Blantyre, Malawi, predicting indoor PM2.5 concentrations for households lacking measurements. The dataset included 7,173 households with various covariates (meteorology, socio-economic status, dwelling characteristics). The study utilized an 80/20 train-test split repeated five times to assess out-of-sample performance.

Key Results

• Simulation Findings:

  • The standard non-spatial RF consistently yielded the highest RMSPE across all scenarios.
  • No single spatial adaptation method was universally superior across all types of spatial autocorrelation.
  • RF-Local performed exceptionally well when spatial autocorrelation was driven by spatially heterogeneous covariate-response relationships but performed poorly in other scenarios due to overfitting on small local subsets, particularly in high-noise settings.
  • RF-GP models (one-step and iterative) excelled when autocorrelation stemmed from unmeasured confounding or spatial spillover but failed when relationships were spatially heterogeneous.
  • RF-Basis (spatial basis functions) and RF-Smooth (smoothed features) demonstrated the most consistent performance, outperforming the non-spatial baseline in all scenarios and remaining close to the best-performing model in specific contexts. Among these, RF-Basis showed the most robust consistency.
  • Computational Efficiency: RF-Local was significantly slower (approx. 1 hour per dataset) due to fitting local models for every point. RF-Basis and RF-Smooth models were computationally feasible (minutes), while RF-GP models were intermediate.

• SCALE Study Findings:

  • In the real-world application, RF-Local achieved the lowest RMSPE (14.03 µg/m³) and coverage probabilities closest to the nominal 95%.
  • RF-Basis was a close second in predictive accuracy but required significantly less computational time.
  • All spatially adapted models outperformed the standard non-spatial RF.
  • Variable importance analysis indicated that meteorological variables (pressure, humidity, temperature) and temporal indicators (month, hour) were the dominant predictors, while household-specific characteristics (e.g., cooking fuel, poverty level) were less influential.

Significance and Claims
The paper claims to provide the first numerical comparison of spatially adapted Random Forest approaches across a range of real and simulated datasets. The authors conclude that while no single method is optimal for all spatial autocorrelation mechanisms, utilizing spatial basis functions (RF-Basis) appears to be the most reliable "safe bet" for researchers. This approach offers a favorable balance, performing consistently well across diverse autocorrelation types in both simulation and real data, while remaining computationally efficient compared to local fitting or iterative Gaussian process methods.

The study highlights that extending Random Forests to account for spatial autocorrelation consistently improves predictive performance and uncertainty quantification. However, the authors modestly note that the optimal model for a specific dataset can only be determined through comparative out-of-sample experiments. They also observe that in the SCALE study, the sampling design (single-time sampling of nearby clusters) made it impossible to disentangle spatial and temporal effects, suggesting that future studies would benefit from repeated monitoring to accurately model spatio-temporal fields.