Bayesian Decomposition Modelling: An Interpretable Nonlinear Approach for Mineral Prospectivity Mapping

Abstract

Prospectivity models that quantify spatial associations between predictor variables and mineralization are critical in data-driven mineral prospectivity mapping (MPM). The current prospectivity modelling approaches, however, are limited in terms of meeting both nonlinearity and interpretability, which are two essential factors in evaluating the performance of a model. Along these lines, this work proposed a novel Bayesian decomposition modelling (BDM) method with nonlinear ability and without loss of interpretability. Following the “decomposition-integration” strategy, the BDM method first decomposed the predictor variables by dimension and then transformed them through nonlinear mapping functions (NMFs). All nonlinear mappings of the predictor variables were then linearly integrated to generate the prospectivity model. Throughout the process, the interpretability of the model was maintained via the decomposition of the predictor variables, and the introduction of NMFs ensured better predictive performance compared to methods that only use linear transformations, such as logistic regression and weights of evidence. The BDM method was applied to the Dayingezhuang deposit, a structurally controlled hydrothermal gold deposit in the Jiaodong Peninsula. The results demonstrate that the area under the receiver operating characteristic curve for the classification model was 0.97, and the coefficient of determination of the gold grade predicted by the regression model was 0.91, clearly indicating that the BDM method possesses excellent predictive ability compared to that of several well-known approaches. Moreover, the classification and regression coefficients of the BDM reveal favourable conditions for gold mineralization in Dayingezhuang, including the negative distance (footwall of fault), gentle dip angle, negative divergence of fluid (mineral fluid convergence) and positive volumetric strains (dilation space). Therefore, the proposed BDM method, which combines robust predictive capability and interpretability, is an effective new method for MPM.

Appendix

Here, the product of powers of two univariate Gaussian probability density functions (PDFs) is discussed based on Bromiley (2003). Let $f(x)$ and $g(x)$ be Gaussian PDFs with arbitrary means ${\mu }_{f}$ and ${\mu }_{g}$ and standard deviations ${\sigma }_{f}$ and ${\sigma }_{g}$, respectively

$$\begin{aligned}f\left(x\right)= \frac{1}{\sqrt{2\pi }{\sigma }_{f}}{e}^{-\frac{{\left(x-{\mu }_{f}\right)}^{2}}{2{{\sigma }_{f}}^{2}}} \mathrm{and}\, g\left(x\right)= \frac{1}{\sqrt{2\pi }{\sigma }_{g}}{e}^{-\frac{{\left(x-{\mu }_{g}\right)}^{2}}{2{{\sigma }_{g}}^{2}}}.\end{aligned}$$

(40)

By supposing that $a$ and $b$ are power exponents of $f(x)$ and $g(x)$, respectively, their product is

$$\begin{aligned}{f\left(x\right)}^{a}{g\left(x\right)}^{b}= \frac{1}{{\sqrt{2\pi }}^{(a+b)}{{\sigma }_{f}}^{a}{{\sigma }_{g}}^{b}}{e}^{-\left(\frac{{a\left(x-{\mu }_{f}\right)}^{2}}{2{{\sigma }_{f}}^{2}}+\frac{b{\left(x-{\mu }_{g}\right)}^{2}}{2{{\sigma }_{g}}^{2}}\right)}.\end{aligned}$$

(41)

Next, the term in the exponent is examined

$$\begin{aligned}\beta &= \frac{{a\left(x-{\mu }_{f}\right)}^{2}}{2{{\sigma }_{f}}^{2}}+\frac{b{\left(x-{\mu }_{g}\right)}^{2}}{2{{\sigma }_{g}}^{2}}\\ &=\frac{{a{{\sigma }_{g}}^{2}x}^{2}-2a{{\sigma }_{g}}^{2}{\mu }_{f}x+a{{\sigma }_{g}}^{2}{{\mu }_{f}}^{2}+b{{\sigma }_{f}}^{2}{x}^{2}-2b{{\sigma }_{f}}^{2}{\mu }_{g}x+b{{\sigma }_{f}}^{2}{{\mu }_{g}}^{2}}{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}\\ &=\frac{\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right){x}^{2}-2\left(a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}\right)x+a{{\sigma }_{g}}^{2}{{\mu }_{f}}^{2}+b{{\sigma }_{f}}^{2}{{\mu }_{g}}^{2}}{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}\\ &=\frac{{x}^{2}-2\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}x+\frac{a{{\sigma }_{g}}^{2}{{\mu }_{f}}^{2}+b{{\sigma }_{f}}^{2}{{\mu }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}}{\frac{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}}.\end{aligned}$$

(42)

Similar to Bromiley (2003), we denote the variance ${{\sigma }_{fg}}^{2}$ and mean ${\mu }_{fg}$ by

$$\begin{aligned}{\sigma }_{fg}&=\sqrt{\frac{{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}},\\ {\mu }_{fg}&=\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}.\end{aligned}$$

(43)

By supposing that $\epsilon$ is the term required to complete the square in $\beta$

$$\begin{aligned}\epsilon =\frac{{\left(\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}\right)}^{2}-{\left(\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}\right)}^{2}}{\frac{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}}=0.\end{aligned}$$

(44)

Adding this term to $\beta$ gives

$$\begin{aligned}\beta &=\frac{{x}^{2}-2\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}x+{\left(\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}\right)}^{2}}{\frac{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}}\\ &+\frac{\frac{a{{\sigma }_{g}}^{2}{{\mu }_{f}}^{2}+b{{\sigma }_{f}}^{2}{{\mu }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}-{\left(\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}\right)}^{2}}{\frac{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}}\\ &={\beta }_{1}+{\beta }_{2}.\end{aligned}$$

(45)

Simplifying, the above formulas are

$$\begin{aligned}{\beta }_{1}&=\frac{{\left(x-\frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}\right)}^{2}}{\frac{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}}=\frac{{\left({x-\mu }_{fg}\right)}^{2}}{2{{\sigma }_{fg}}^{2}},\\ {\beta }_{2}&=\frac{\left(a{{\sigma }_{g}}^{2}{{\mu }_{f}}^{2}+b{{\sigma }_{f}}^{2}{{\mu }_{g}}^{2}\right)\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)-{\left(a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}\right)}^{2}}{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)}\\ &=\frac{{a}^{2}{{\sigma }_{g}}^{4}{{\mu }_{f}}^{2}+ab{{\sigma }_{g}}^{2}{{\mu }_{f}}^{2}{{\sigma }_{f}}^{2}+ab{{\sigma }_{f}}^{2}{{\mu }_{g}}^{2}{{\sigma }_{g}}^{2}+{b}^{2}{{\sigma }_{f}}^{4}{{\mu }_{g}}^{2}}{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)}\\ & \quad -\frac{{a}^{2}{{\sigma }_{g}}^{4}{{\mu }_{f}}^{2}+2ab{{\sigma }_{g}}^{2}{\mu }_{f}{{\sigma }_{f}}^{2}{\mu }_{g}+{b}^{2}{{\sigma }_{f}}^{4}{{\mu }_{g}}^{2}}{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)}\\ &=\frac{ab{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}\left({{\mu }_{f}}^{2}{{-{2\mu }_{f}{\mu }_{g}+\mu }_{g}}^{2}\right)}{2{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)}\\ &=\frac{ab{\left({\mu }_{f}-{\mu }_{g}\right)}^{2}}{2\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)},\\ \beta &=\frac{{\left({x-\mu }_{fg}\right)}^{2}}{2{{\sigma }_{fg}}^{2}}+\frac{ab{\left({\mu }_{f}-{\mu }_{g}\right)}^{2}}{2\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)}.\end{aligned}$$

(46)

Substituting back into Eq. 41 gives

$$\begin{aligned}{f\left(x\right)}^{a}{g\left(x\right)}^{b}= \frac{1}{{\sqrt{2\pi }}^{(a+b)}{{\sigma }_{f}}^{a}{{\sigma }_{g}}^{b}}{\exp}\left[-\frac{{\left({x-\mu }_{fg}\right)}^{2}}{2{{\sigma }_{fg}}^{2}}\right]{\exp}\left[-\frac{ab{\left({\mu }_{f}-{\mu }_{g}\right)}^{2}}{2\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)}\right].\end{aligned}$$

(47)

Multiplying by ${\sigma }_{fg}/{\sigma }_{fg}$ and rearranging gives

$$\begin{aligned}{f\left(x\right)}^{a}{g\left(x\right)}^{b}&=\frac{1}{\sqrt{2\pi }{\sigma }_{fg}}{\exp}\left[-\frac{{\left({x-\mu }_{fg}\right)}^{2}}{2{{\sigma }_{fg}}^{2}}\right]\cdot \frac{{\exp}\left[-\frac{ab{\left({\mu }_{f}-{\mu }_{g}\right)}^{2}}{2\left(a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}\right)}\right]}{{\sqrt{2\pi }}^{\left(a+b\right)-1}{{\sigma }_{f}}^{a-1}{{\sigma }_{g}}^{b-1}\sqrt{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}}\\ &={S}_{fg}\cdot \frac{1}{\sqrt{2\pi }{\sigma }_{fg}}{\exp}\left[-\frac{{\left({x-\mu }_{fg}\right)}^{2}}{2{{\sigma }_{fg}}^{2}}\right].\end{aligned}$$

(48)

Therefore, the product of the powers of two Gaussian PDFs ${f\left(x\right)}^{a}$ and ${g\left(x\right)}^{b}$ is also a scaled Gaussian PDF. The variance${{\sigma }_{fg}}^{2}$, the mean ${\mu }_{fg}$, and the scaling factor ${S}_{fg}$ can be written more conveniently as

$$\begin{aligned}\frac{1}{{{\sigma }_{fg}}^{2}}&=\frac{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}{{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}=\frac{a}{{{\sigma }_{f}}^{2}}+\frac{b}{{{\sigma }_{g}}^{2}}\\ {\mu }_{fg}&={{\sigma }_{fg}}^{2}\cdot \frac{a{{\sigma }_{g}}^{2}{\mu }_{f}+b{{\sigma }_{f}}^{2}{\mu }_{g}}{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}\cdot \frac{a{{\sigma }_{g}}^{2}+b{{\sigma }_{f}}^{2}}{{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}=\left(\frac{a{\mu }_{f}}{{{\sigma }_{f}}^{2}}+\frac{b{\mu }_{g}}{{{\sigma }_{g}}^{2}}\right){{\sigma }_{fg}}^{2}\\ {S}_{fg}&=\frac{{\exp}\left[-\frac{ab}{2}\frac{{\left({\mu }_{f}-{\mu }_{g}\right)}^{2}{{\sigma }_{fg}}^{2}}{{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}\right]}{{\sqrt{2\pi }}^{\left(a+b\right)-1}\sqrt{\frac{{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{{{\sigma }_{fg}}^{2}}}{{\sigma }_{f}}^{a-1}{{\sigma }_{g}}^{b-1}}=\frac{{\exp}\left[-\frac{1}{2}\left(\frac{a{{\mu }_{f}}^{2}}{{{\sigma }_{f}}^{2}}+\frac{b{{\mu }_{g}}^{2}}{{{\sigma }_{g}}^{2}}-\frac{{{\mu }_{fg}}^{2}}{{{\sigma }_{fg}}^{2}}\right)\right]}{{\sqrt{2\pi }}^{\left(a+b\right)-1}\sqrt{\frac{{{\sigma }_{f}}^{2}{{\sigma }_{g}}^{2}}{{{\sigma }_{fg}}^{2}}}{{\sigma }_{f}}^{a-1}{{\sigma }_{g}}^{b-1}}.\end{aligned}$$

(49)

Finally, Eq. 48 was extended to the product of powers of n univariate Gaussian PDFs according to the proof of Bromiley (2003)

$$\begin{aligned}\prod_{i=1}^{n}{\mathcal{N}\left({\mu }_{i},{\sigma }_{i}\right)}^{{\theta }_{i}}&={S}_{i=1\dots n}\mathcal{N}\left({\mu }_{i=1,...n},{\sigma }_{i=1,\dots n}\right)\\ \frac{1}{{\sigma }_{i=1,\dots n}^{2}}&=\sum_{i=1}^{n}\frac{{\theta }_{i}}{{{\sigma }_{i}}^{2}}\\ {\mu }_{i=1,\dots n}&=\left[\sum_{i=1}^{n}\frac{{\theta }_{i}{\mu }_{i}}{{{\sigma }_{i}}^{2}}\right]{\sigma }_{i=1,\dots n}^{2}\\ {S}_{i=1,\dots n}&=\frac{1}{{\sqrt{2\pi }}^{\sum_{i=1}^{n}{\theta }_{i}-1}\prod_{i=1}^{n}{\sigma }_{i}^{{\theta }_{i}-1}}\sqrt{\frac{{\sigma }_{i=1,\dots n}^{2}}{\prod_{i=1}^{n}{\sigma }_{i}^{2}}}\\ & {\exp}\left[-\frac{1}{2}\left(\sum_{i=1}^{n}\frac{{{{\theta }_{i}\mu }_{i}}^{2}}{{{\sigma }_{i}}^{2}}-\frac{{\mu }_{i=1,\dots n}^{2}}{{\sigma }_{i=1,\dots n}^{2}}\right)\right].\end{aligned}$$

(50)