# Optimized Weighted Ensemble Approach for Enhancing Gold Mineralization Prediction
## 2 Method
### Gaussian Process Regression (GPR)  
Gaussian processes are non-parametric techniques utilized in the Bayesian method of machine learning. They can solve both supervised and unsupervised learning problems, such as regression and classification. The GPR-based system offers various practical benefits over other supervised learning algorithms, like flexibility, abilities to generate uncertainty predictions, and learning noisy and smooth parameters through training data. Other supervised learning methods are limited in their ability to learn these parameters. In addition, GPR performs well on small datasets and can provide ([PERSON] et al., 2017). Therefore, it can be employed in continuous variable forecasting, modeling purposes, mapping, and interpolating ([PERSON] et al., 2017),and it can be guided from training data to represent the intrinsic nonlinear relationship in any dataset. Due to the presence of this characteristic, it is suitable for evaluating the grade of the ore.  
\[\text{F(x)}\sim\text{GP}\left(\text{m(x),k(x,x^{\prime})}\right. \tag{4}\]  
where m(x) = E[f(x)] is the mean function at input x, and k(x; x^{\prime}) = E[(f(x) - m(x))(f(x^{\prime})-m(x^{\prime}))^{\text{T}}]\) is the covariance function that represents the dependency between the function values for various input points x and x^{\prime}\). The squared exponential covariance function is a popular option for the kernel in the statistical technique known as Gaussian process regression. This kernel assumes that the function being modeled is continuous and smooth and that data points that are close together strongly correlate with one another. Gaussian process regression uses zero mean and squared exponential covariance functions. This is seen in Equation (5). \[\text{K}\left(\text{x,x^{\prime}}\right)=\sigma_{\text{f}}{}^{2}\exp\left( \frac{-r}{2}\right) \tag{5}\]  
where \(r\) is equal to \((\,|\,\text{x}-x^{\prime}\,|\,^{2}/l^{2})\) and \(\sigma_{\text{f}}\) and 1 are hyperparameters that have a major impact on the performance of the GP algorithm. Model noise is \(\sigma_{\text{f}}\), and length scale is l. Input parameters have a high covariance regardless of their proximity, but this value decreases exponentially with distance. Several covariance and kernel functions can be used to optimize the GPR rational quadratic. The model of the distribution of predictions is conditioned on the training data. This is made possible via  
\[p(f^{*}|x^{*},D)=GP\Big{(}f^{*}\Big{|}\mu^{*},\sigma^{*2}\Big{)} \tag{6}\]  
The mean prediction \(\mu^{*}\) may be calculated using the formula below. \[\mu^{*}=k(x^{*})^{T}\Big{(}K(X,X)+\sigma_{N}^{2}I_{N}\Big{)}^{-1}*t \tag{7}\]  
Furthermore, the variance prediction \(\sigma^{*}\) may be computed as  
\[\sigma^{*2}=\sigma_{N}^{2}-k(x^{*})^{T}\Big{(}K(X,X)+\sigma_{N}^{2}I_{N}\Big{)} ^{-1}k(x^{*})+k(x^{*},x^{*}) \tag{8}\]  
The training dataset's covariance matrix is represented by the matrices K(X, X), and the testing dataset's covariance matrix is represented by the matrices K(X*, X*). The previous equations make it possible to demonstrate that the linear combination of the observed target may be proven to be the mean forecast. The observed target did not influence the variance; instead, it depended entirely on the observed inputs. Gaussian distributions have this attribute. GPR is a robust technique that can be used to make predictions and estimate uncertainty. It assumes the function to be a random sampling from a _Gaussian process (GP)_ and returns a probability density for the function's value at each given input position. It is flexible enough to include information from multiple sources and models and can deal with non-linear and non-stationary functions. The choice of kernel function can be challenging and requires prior information, and it can be computationally expensive for massive datasets or high-dimensional input spaces. In mineral exploration, GPR can predict the _geographical distribution of minerals_ using information gathered from _geological_, _geophysical_, _and geochemical studies_.