Ensemble Machine Learning Geostatistical Hybrid Models for Grade Control

Samples collected from densely drilled grade control boreholes are used to create spatial models for ore sorting, classifying material as ore or waste prior to extraction. Geostatistics (typically ordinary kriging) is used to spatially estimate mineral grade at unknown locations; however, hybrid techniques combine geostatistical and machine learning models to take advantage of available dense data and improve overall model performance. There are many different machine learning models; using an ensemble learning-based approach that combines individual models improves estimation accuracy. Two-layer stacked, global, and local weighted ensemble models are proposed. In the two-layer stacking ensemble (SE), the first layer combines n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n $$\end{document} individual models; this work considers four individual models, elliptical radial basis neural network (ERBFN), locally weighted support vector regression (LWSVR), kernel density estimated trend (KDET), and a novel convolutional neural network (CNN). In the second layer, either random forest (RF) or support vector regression (SVR) is trained on outputs of the first layer to generate the final model, which is incorporated into intrinsic collocated cokriging (ICCK) as a secondary variable. The global and local weighting-based ensemble models combine ICCK estimates in which each individual model is considered a secondary variable whose performance is evaluated with cross-validation error. The performance of the ensemble models is compared to inverse distance, ordinary kriging, and hybrid models assessed on 10 blast areas at Teck Resources Limited's Carmen de Andacollo copper mine in Chile. Considering these 10 blasts, ordinary kriging obtains an R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<^>{2}$$\end{document} of 0.39, inverse distance obtains an R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<^>{2}$$\end{document} of 0.38, and the proposed ensemble approach obtains an R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<^>{2}$$\end{document} of 0.67, demonstrating a clear improvement over traditional spatial estimation workflows. The proposed method is fully automated and requires the same amount of professional time as implementing ordinary kriging.