Comparison of Trivariate Copula-Based Conditional Quantile Regression Versus Machine Learning Methods for Estimating Copper Recovery

Heber Hernández, Martín Alberto Díaz-Viera, Elisabete Alberdi and Aitor Goti ()
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Heber Hernández: Facultad de Ingeniería, Universidad Santo Tomás, Ejército Libertador 146, Santiago 8370003, Chile
Martín Alberto Díaz-Viera: Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas No. 152, Ciudad de Mexico 07730, Mexico
Elisabete Alberdi: Department of Applied Mathematics, University of the Basque Country UPV/EHU, 48013 Bilbao, Spain
Aitor Goti: Department of Mechanics, Design and Organization, University of Deusto, 48007 Bilbao, Spain

Mathematics, 2025, vol. 13, issue 4, 1-22

Abstract: In this study, an innovative methodology using trivariate copula-based conditional quantile regression (CBQR) is proposed for estimating copper recovery. This approach is compared with six supervised machine learning regression methods, namely, Decision Tree, Extra Tree, Support Vector Regression (linear and epsilon), Multilayer Perceptron, and Random Forest. For comparison purposes, an open access database representative of a porphyry copper deposit is used. The database contains geochemical information on minerals, mineral zoning data, and metallurgical test results related to copper recovery by flotation. To simulate a high undersampling scenario, only 5% of the copper recovery information was used for training and validation, while the remaining 95% was used for prediction, applying in all these stages error metrics, such as R 2 , MaxRE, MAE, MSE, MedAE, and MAPE. The results demonstrate that trivariate CBQR outperforms machine learning methods in accuracy and flexibility, offering a robust alternative solution to model complex relationships between variables under limited data conditions. This approach not only avoids the need for intensive tuning of multiple hyperparameters, but also effectively addresses estimation challenges in scenarios where traditional methods are insufficient. Finally, the feasibility of applying this methodology to different data scales is evaluated, integrating the error associated with the change in scale as an inherent part of the estimation of conditioning variables in the geostatistical context.

Keywords: metallurgical copper recovery; copula; quantile regression; kernel smoothing; machine learning (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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