Exact Conditioning of Regression Random Forest for Spatial Prediction

Abstract

Regression random forest is becoming a widely-used machine learning technique for spatial prediction that shows competitive prediction performance in various geoscience fields. Like other popular machine learning methods for spatial prediction, regression random forest does not exactly honor the response variable’s measured values at sampled locations. However, competitor methods such as regression-kriging perfectly fit the response variable’s observed values at sampled locations by construction. Exactly matching the response variable’s measured values at sampled locations is often desirable in many geoscience applications. This paper presents a new approach ensuring that regression random forest perfectly matches the response variable’s observed values at sampled locations. The main idea consists of using the principal component analysis to create an orthogonal representation of the ensemble of regression tree predictors resulting from the traditional regression random forest. Then, the exact conditioning problem is reformulated as a Bayes-linear-Gauss problem on principal component scores. This problem has an analytical solution making it easy to perform Monte Carlo sampling of new principal component scores and then reconstruct regression tree predictors that perfectly match the response variable’s observed values at sampled locations. The reconstructed regression tree predictors’ average also precisely matches the response variable’s measured values at sampled locations by construction. The proposed method’s effectiveness is illustrated on the one hand using a synthetic dataset where the ground-truth is available everywhere within the study region, and on the other hand, using a real dataset comprising southwest England’s geochemical concentration data. It is compared with the regression-kriging and the traditional regression random forest. It appears that the proposed method can perfectly fit the response variable’s measured values at sampled locations while achieving good out of sample predictive performance comparatively to regression-kriging and traditional regression random forest.