Abstract
In this study, machine learning methods such as neural networks, random forests, and Gaussian processes are applied to the estimation of copper grade in a mineral deposit. The performance of these methods is compared to geostatistical techniques, such as ordinary kriging and indicator kriging. To ensure that these comparisons are realistic and relevant, the predictive accuracy is estimated on test instances located in drill holes that are different from the training data. The results of an extensive empirical study in the Sarcheshmeh porphyry copper deposit in Southeastern Iran illustrate that specially designed Gaussian processes with a symmetric standardization of the spatial location inputs and an anisotropic kernel yield the most accurate predictions. Furthermore, significant improvements are obtained when, besides location, information on the rock type is included in the set of predictor variables. This observation highlights the importance of carrying out detailed studies of the geological composition of the deposit to obtain more accurate ore grade predictions.
Similar content being viewed by others
References
Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995)
Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001)
Criminisi, A., Shotton, J., Konukoglu, E.: Decision forests: a unified framework for classification, regression, density estimation, manifold learning and semi-supervised learning. Foundations and Trends®, in Computer Graphics and Vision 7(2?3), 81–227 (2012)
Trehan, S., Carlberg, K., Durlofsky, L.J.: Error modeling for surrogates of dynamical systems using machine learning. International Journal for Numerical Methods in Engineering. https://doi.org/10.1002/nme.5583 (2017)
Ramussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press, Boston (2006)
Kapageridis, I., Denby, B.: Ore grade estimation with modular neural network systems. A case study. In: Panagiotou, G.N., Michalakopoulos, T.N. (eds.) Information Technology in the Mi- neral Industry, p 52. AA Balkema Publishers, Rotterdam (1998)
Koike, K., Matsuda, S., Suzuki, T., Ohmi, M.: Neural network-based estimation of principal metal contents in the Hokuroku district, northern Japan, for exploring Kuroko-type deposits. Nat. Resour. Res. 11(2), 135–156 (2002)
Matias, J.M., Vaamonde, A., Taboada, J., González-Manteiga, W.: Comparison of kriging and neural networks with application to the exploitation of a slate mine. Math. Geol. 36(4), 463–486 (2004)
Samanta, B., Bandopadhyay, S.: Construction of a radial basis function network using an evolutionary algorithm for grade estimation in a placer gold deposit. Comput. Geosci. 35(8), 1592–602 (2009)
Chatterjee, S, Bandopadhyay, S, Machuca, D.: Ore grade prediction using a genetic algorithm and clustering based ensemble neural network model. Math. Geosci. 42(3), 309–326 (2010)
Jafrasteh, B., Fathianpour, N.: A hybrid simultaneous perturbation artificial bee colony and back-propagation algorithm for training a local linear radial basis neural network on ore grade estimation. Neurocomputing 235, 217–227 (2017)
Samanta, B, Bandopadhyay, S, Ganguli, R.: Data segmentation and genetic algorithms for sparse data division in Nome placer gold grade estimation using neural network and geostatistics. Explor. Min. Geol. 11(1-4), 69–76 (2002)
Chatterjee, S., Samanta, A., Bhattacherjee, B., Pal, S.K.: Ore grade estimation of a limestone deposit in India using an artificial neural network. Applied GIS 2(1), 1–2 (2006)
Samanta, B., Bandopadhyay, S.: Construction of a radial basis function network using an evolutionary algorithm for grade estimation in a placer gold deposit. Comput. Geosci. 35(8), 1592–1602 (2009)
Jafrasteh, B., Fathianpour, N., Suárez, A.: Advanced machine learning methods for copper ore grade estimation. In: Near Surface Geoscience 2016-22nd European Meeting of Environmental and Engineering Geophysics (2016)
Pishbin, M., Fathianpour, N., Mokhtari, A.R.: Uniaxial compressive strength spatial estimation using different interpolation techniques. Int. J. Rock Mech. Min. Sci. 89, 136–150 (2016)
Webster, R., Oliver, M.A.: Geostatistics for Environmental Scientists. Wiley, Hoboken (2007)
Olea, R.A.: Geostatistics for Engineers and Earth Scientists. Springer Science & Business Media, Berlin (1999)
Hohn, M.: Geostatistics and Petroleum Geology. Springer Science & Business Media, Berlin (2013)
Irie, B., Miyake, S.: Capabilities of three-layered perceptrons. In: IEEE International Conference on Neural Networks, 1988, pp. 641–648 (1988)
Gomes, G.S., Ludermir, T.B., Lima, L.M.: Comparison of new activation functions in neural network for forecasting financial time series. Neural Comput. Applic. 20(3), 417–439 (2011)
Fernández-Delgado, M, Cernadas, E, Barro, S, Amorim, D.: Do we need hundreds of classifiers to solve real world classification problems? J. Mach. Learn. Res. 15, 3133–3181 (2014)
Breiman, L., Friedman, J., Stone, C.J., Olshen, R.A.: Classification and Regression Trees. Wadsworth and Brooks (1984)
Ho, T.K.: The random subspace method for constructing decision forests. IEEE Trans. Pattern Anal. Mach. Intell. 20(8), 832–44 (1998)
Breiman, L.: Bagging predictors. Mach. Learn. 24(2), 123–140 (1996)
Fletcher, R.: Practical Methods of Optimization. Wiley, Hoboken (1987)
Rasmussen, C.E.: Evaluation of Gaussian Processes and Other Methods for Non-Linear Regression. University of Toronto, Toronto (1996)
Atapour, H., Aftabi, A.: The geochemistry of gossans associated with Sarcheshmeh porphyry copper deposit, rafsanjan, Kerman, Iran: implications for exploration and the environment. J. Geochem. Explor. 93(1), 47–65 (2007)
Waterman, G.C., Hamilton, R.L.: The Sar Cheshmeh porphyry copper deposit. Econ. Geol. 70(3), 568–576 (1975)
Demuth, H., Beale, M.: MATLAB: Neural Network Toolbox: User’S Guide: Version 2 Math Works (1997)
Jalloh, A.B., Kyuro, S., Jalloh, Y., Barrie, A.K.: Integrating artificial neural networks and geostatistics for optimum 3D geological block modeling in mineral reserve estimation: a case study. Int. J. Min. Sci. Technol. 26 (4), 581–5 (2016)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference and Prediction. Springer, Berlin (2008)
Liaw, A, Wiener, M.: Classification and regression by randomForest. R news 2(3), 18–22 (2002)
Hernández-Lobato, D., Martínez-muñoz, G., Suárez, A.: How large should ensembles of classifiers be?. Pattern Recogn. 46(5), 1323–1336 (2013)
Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16(5), 1190–1208 (1995)
Gama, J., Žliobaitė, I., Bifet, A., Pechenizkiy, M., Bouchachia, A.: A survey on concept drift adaptation. ACM Computing Surveys (CSUR) 46(4), 44 (2014)
Sinclair, A.J., Blackwell, G.H.: Applied Mineral Inventory Estimation. Cambridge University Press, Cambridge (2002)
Funding
This research has been supported by the Spanish Ministry of Economy, Industry and Competitiveness, projects TIN2013-42351-P, TIN2015-70308-REDT, and TIN2016-76406-P, and of the Comunidad de Madrid, project CASI-CAM-CM (S2013/ICE-2845).
Author information
Authors and Affiliations
Corresponding author
Appendix: Anisotropic exponential kernel function
Appendix: Anisotropic exponential kernel function
An anisotropic exponential kernel function is introduced to capture actual anisotropies in the ore grade distribution. The kernel has the following form
where
is the squared Mahalanobis distance between xir and xjr. These quantities are rotations of the original location vectors
where
and R is the rotation matrix
and α, β, and γ are the azimuth, dip, and plunge angles, respectively. Λ is a diagonal matrix that is used to scale the components along the rotated coordinate axis
Figure 13 shows the scale parameters and the rotated coordinate axis.
When the rock type is used for prediction, the matrices Λ and R are modified as follows:
The hyperparameters of the anisotropic exponential kernel function H = (α, β, γ, Sx, Sy, Sz, Sr) are determined by optimization using the LBFGS algorithm [35].
Rights and permissions
About this article
Cite this article
Jafrasteh, B., Fathianpour, N. & Suárez, A. Comparison of machine learning methods for copper ore grade estimation. Comput Geosci 22, 1371–1388 (2018). https://doi.org/10.1007/s10596-018-9758-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-018-9758-0