Dimension reduction for spatial regression: Spatial predictor envelope

Natural sciences such as geology and forestry often utilize regression models for spatial data with many predictors and small to moderate sample sizes. In these settings, efficient estimation of the regression parameters is crucial for both model interpretation and prediction. We propose a dimension reduction approach for spatial regression that assumes certain linear combinations of the predictors are immaterial to the regression. The model and corresponding inference provide efficient estimation of regression parameters while accounting for spatial correlation in the data. We employed the maximum likelihood estimation approach to estimate the parameters of the model. The effectiveness of the proposed model is illustrated through simulation studies and the analysis of a geochemical data set, predicting rare earth element concentrations within an oil and gas reserve in Wyoming. Simulation results indicate that our proposed model offers a significant reduction in the mean square errors and variation of the regression coefficients. Furthermore, the method provided a 50% reduction in prediction variance for rare earth element concentrations within our data analysis.