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Comparison of machine learning methods for copper ore grade estimation

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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.

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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).

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Correspondence to Nader Fathianpour.

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

$$ \mathbf{K_{ij}} = \sigma^{2} \exp \left( - \sqrt{\mathbf{ \mathbf{D}^{2}(\mathbf{x_{ir}}, \mathbf{x_{jr} }) }}\right), $$
(20)

where

$$ \mathbf{D}^{2}(\mathbf{x_{ir}}, \mathbf{x_{jr}}) = (\mathbf{x_{ir}}- \mathbf{x_{jr}})^{T} \cdot (\mathbf{x_{ir}}- \mathbf{x_{jr}} ) $$
(21)

is the squared Mahalanobis distance between xir and xjr. These quantities are rotations of the original location vectors

$$ \mathbf{x_{ir}} = \mathbf{x_{i}} \mathbf{\Gamma}, \quad \mathbf{x_{jr}} = \mathbf{x_{j}} \mathbf{\Gamma}, $$
(22)

where

$$ \mathbf{\Gamma} = \mathbf{R^{T}} \mathbf{{\Lambda} } \mathbf{R} $$
(23)

and R is the rotation matrix

$$\begin{array}{@{}rcl@{}} \mathbf{R} = \left( \begin{array}{lll} cos(\alpha) cos(\theta) + sin(\alpha) sin(\beta) sin(\theta)\\ sin(\alpha) cos(\beta) \\ cos(\alpha) sin(\theta) - sin(\alpha) sin(\beta) cos(\theta) \\ \end{array}\right. \\ \left. \begin{array}{lll} -sin(\alpha) cos(\theta) + cos(\alpha) sin(\beta) sin(\theta)\\ cos(\alpha) cos(\beta) \\ -sin(\alpha) sin(\theta) - cos(\alpha) sin(\beta) cos(\theta) \\ \end{array}\right. \\ \left. \begin{array}{lll} -cos(\beta) sin(\theta)\\ sin(\beta) \\ cos(\beta) cos(\theta) \\ \end{array}\right) \end{array} $$
(24)

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

$$ \mathbf{{\Lambda} } =\left[\begin{array}{lll} \frac{1}{S_{x}}&0 & 0\\ 0&\frac{1}{S_{y}}&0\\ 0&0&\frac{1}{S_{z}} \end{array}\right] $$
(25)

Figure 13 shows the scale parameters and the rotated coordinate axis.

Fig. 13
figure 13

Anisotropy ellipsoid, its scale parameter, and the rotated coordinate axis

When the rock type is used for prediction, the matrices Λ and R are modified as follows:

(26)
(27)

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].

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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

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