Multivariate Modelling of the Trace Element Chemistry of Arsenopyrite from Gold Deposits Using Higher-Dimensional Algebras

Abstract

In geochemistry, the elevated concentrations of certain elements in rock or mineral samples are used for the assessment of a mineralising system’s important geological characteristics. The relationship between the elements and system characteristics may take one of two forms, regression or classification. The collection and analysis of geochemical samples typically leads to the development of sparse and fuzzy multivariate datasets, which often have complex interrelationships. As a result, it is difficult to quantitatively and qualitatively interpret many of these geochemical datasets and relate the resulting data to relevant geological characteristics. There are no governing expressions in the current literature that can relate mineral-based trace element assemblages to geological characteristics. In this work, a generalised formulation for multivariate systems modelling is presented and it is applied to the discovery of the manifested system characteristics of arsenopyrite samples collected from Australian gold (Au) deposits, using fuzzy arsenopyrite trace element datasets. Arsenopyrite trace element assemblages, collected using laser ablation-inductively coupled plasma-mass spectrometry, are represented as spatially distributed networks embedded in ${\mathbf{R}}^{4,1}$ conformal geometric algebraic space. The interactions of elements within these networks generate multivectors which are referred to as hyperfields. The functions and parameters used to generate these hyperfields are optimised to create multiple unique models, which are used to discover relationships between system characteristics and the multivariate dataset. The models generated allow a specialist to discover qualitative relations, using higher-dimensional representations of multivariate data. In this work, the generalised hyperfield formulation is applied to the problem of predicting the endowment, lithology and camp of Au deposits, using an arsenopyrite trace element dataset. The hyperfield formulation is compared against the state-of-the-art random forest decision tree algorithm ‘XGBoost’ and artificial neural networks (ANNs).

Appendices

Appendix 1

1.1 Sample Node Inputs ($\alpha_{qk}$)

See Table 23, 24, 25 and 26.

Table 23 The analyte responses ($\alpha_{qk}$) of all Au deposits for $G_{1}$

Table 24 The analyte responses ($\alpha_{qk}$) of all Au deposits for $G_{2}$

Table 25 The analyte responses ($\alpha_{qk}$) of all Au deposits for $G_{3}$

Table 26 The analyte responses ($\alpha_{qk}$) of all Au deposits for $G_{4}$

Appendix 2

2.1 Prediction Results

See Tables 27, 28 and 29.

Table 27 The real and predicted $E_{E}$, using Eq. (61)

Table 28 The real and predicted lithology class, using the model $\left. M \right|_{5,17}$

Table 29 The real and predicted camp class, using the model $\left. M \right|_{1,13}$

Appendix 3: MLP Loss Functions

3.1 Endowment Loss Function

See Fig. 16.

Fig. 16

Loss function for MLP A, for the endowment regression case study (Sect. 3.1)

3.2 Lithology Loss Function

See Fig. 17.

Fig. 17

Loss function for MLP B, for the lithology classification case study (Sect. 3.2)

3.3 Camp Loss Function

See Fig. 18.

Fig. 18

Loss function for MLP B, for the camp classification case study (Sect. 3.3)