## 3 Materials and Methods
### Gaussian Processes Regression  
Gaussian processes regression (GPR) is a suitable and recently proposed method that has been used in a variety of machine learning applications [39]. The GPR model's probabilistic solution leads to the identification of generic regression problems using kernels. The applied regressor's training process can be categorized as Bayesian framing, and the model relations are presumed to follow a Gaussian distribution to encode the previous information about the output function [40]. The Gaussian process is defined \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Dataset** & **Statistical Parameter** & \(B\) **(m)** & \(D\) **(m)** & \(L\)/B & \(\gamma\) **(kN/m3)** & \(\phi\) (\({}^{\circ}\)) & \(q_{u}\) **(kPa)** \\ \hline \multirow{4}{*}{Training} & Minimum & 0.0585 & 0 & 1 & 9.85 & 32 & 58.5 \\  & Maximum & 3.016 & 0.889 & 6 & 17.1 & 44.8 & 2847 \\  & Mean & 0.398 & 0.156 & 3.126 & 14.279 & 38.492 & 407.899 \\  & Standard deviation & 0.543 & 0.195 & 2.192 & 2.642 & 3.308 & 511.870 \\ \hline \multirow{4}{*}{Testing} & Minimum & 0.0585 & 0 & 1 & 10.2 & 32 & 91.5 \\  & Maximum & 1.492 & 0.762 & 6 & 17.1 & 44.8 & 2033 \\ \cline{1-1}  & Mean & 0.420 & 0.204 & 2.995 & 13.777 & 38.800 & 569.826 \\ \cline{1-1}  & Standard deviation & 0.361 & 0.221 & 1.964 & 2.573 & 3.225 & 600.020 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical aspects of the dataset. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(B\) & \(D\) & \(L\)/B & \(\gamma\) & \(\phi\) & \(q_{u}\) \\ \hline \(B\) & 1.000 & & & & & \\ \(D\) & 0.710 & 1.000 & & & & \\ \(L/B\) & \(-\)0.351 & \(-\)0.249 & 1.000 & & & \\ \(\gamma\) & \(-\)0.269 & \(-\)0.125 & 0.340 & 1.000 & & \\ \(\phi\) & \(-\)0.378 & \(-\)0.286 & 0.124 & 0.076 & 1.000 & \\ \(q_{u}\) & 0.452 & 0.671 & \(-\)0.248 & \(-\)0.238 & 0.258 & 1.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Correlation coefficients between various factors. by a set of variables for which each one has a joint Gaussian allocation [41]. The overall structure of the Gaussian process is described by the following equation:  
\[g(x)\sim GP\big{(}w(x),k\big{(}x,x^{\prime}\big{)}\big{)} \tag{2}\]  
The mean function of the Gaussian process is denoted by \(w(x)\), and the kernel function is denoted by \(k(x,x^{\prime})\). The mean function is typically constant, either zero or the mean of the training dataset. In this study, the mean of the training dataset was used. Consider a learning dataset with \(N\) pairs in the form of \(S=\{(x_{i},y_{i})\,|\,i=1,\,2,\,\ldots,\,N\}\) where \(x\) is an \(N\)-dimensional input vector, and \(y\) is the corresponding target. Using \(x_{j}\) as a test sample, the GPR model utilizes the below formulation in determining the relationship between the provided inputs and targets [41]. \[w_{j}=k_{j}^{\top}\Big{[}K(X,X)+{\sigma^{2}}_{n}I\Big{]}y \tag{3}\]  
\[\sigma_{j}^{2}=k\big{(}x_{j},x_{j}\big{)}-k_{j}^{\top}\Big{[}K(X,X)+\sigma_{n} ^{2}I\Big{]}^{-1}k_{j} \tag{4}\]  
where \(w_{j}\) denotes the mean value of the most compatible predicted outputs for the test input vector \((x_{j})\). Additionally, \(K(X,X)\), \(k_{j}\), \(\sigma_{n}^{2}\) and \(y\) represent covariance matrix, the kernel distance between training and testing data, the noise variance, and the training observation, respectively. Furthermore, the produced variance \(\left(\sigma_{j}^{2}\right)\) by Equation (4) indicates a confidence measure for the acquired results. It is worth noting that this variance is inversely proportional to the confidence associated with the \(w_{j}\)[42]. The formulas above can be combined in the form of a linear combination of the kernel function and the mean estimation \(\overline{f}\big{(}x_{j}\big{)}\) as follows:  
\[\overline{f}\big{(}x_{j}\big{)}=\sum\limits_{i=1}^{s}\Big{(}K(X,X)+{\sigma^{2 }}_{n}I\Big{)}^{-1}yk\big{(}x_{j},x_{j}\big{)} \tag{5}\]  
GPR utilizes a number of kernel functions. GPR has a limitation in terms of selecting a suitable kernel function.