## 4 Construction of Prediction Models
### Model Evaluation Indexes  
The results of the proposed models are evaluated using \(R^{2}\), \(MAE\), \(RMSE\), \(MAE\), \(NSE\) and \(RSR\), as more commonly used criteria in the literature. The following equations are used to calculate these metrics:  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2}}{\sum_{i=1}^{n}(x_{i}-\hat{ x})^{2}} \tag{13}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Hyperparameters** & **Meanings** & **Optimal Values** \\ \hline \multirow{3}{*}{XGBoost} & n estimators & Number of trees & 133 \\ \cline{2-4}  & Learning rate & Shrinkage coefficient of tree & 0.03 \\ \cline{2-4}  & Maximum depth & Maximum depth of a tree & 4 \\ \hline \multirow{3}{*}{RF} & n estimators & Number of trees in forest & 500 \\ \cline{2-4}  & Minimum split & Minimum samples of split for nodes & 5 \\ \cline{2-4}  & Maximum depth & Maximum depth of a tree & 5 \\ \cline{2-4}  & Minimum leaf & Minimum samples of nodes for leaf & 8 \\ \hline \multirow{3}{*}{AdaBoost} & n estimators & Number of trees & 500 \\ \cline{2-4}  & Learning rate & Shrinkage coefficient of tree & 1 \\ \hline \multirow{3}{*}{SVM} & C\({}_{2}\) & Regularization parameter & 2.5 \\ \cline{2-4}  & Minimum split & Minimum samples of split for nodes & 4 \\ \cline{2-4}  & Maximum depth & Maximum depth of a tree & 100 \\ \cline{2-4}  & Minimum leaf & Minimum samples of nodes for leaf & 7 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Hyperparameters optimal results.  
Figure 7: The flowchart for applying a data-driven technique to anticipate pile bearing capacity.  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\hat{x}_{i}) \tag{14}\] \[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2}}\] (15) \[MARE=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{x_{i}-\hat{x}_{i}}{x_{i}} \times 100\right|\] (16) \[NSE=1-\frac{\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2}}{\sum_{i=1}^{n}(x_{i}- \overline{x})^{2}}\] (17) \[RSR=\frac{\sqrt{\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2}}}{\sqrt{ \sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}} \tag{18}\]  
where \(n\) denotes the number of points, \(x_{i}\) and \(\hat{x}_{i}\) denotes the actual and expected outputs of i-th sample, respectively; \(\overline{x}\) is data averaged actual output. \(R^{2}\) is a number that ranges from 0 to 1, a higher \(R^{2}\) value indicates a more efficient model. The model is considered effective when \(R^{2}\) is more than 0.8 and close to 1 [22]. The mean squared difference between projected outputs and targets is the criterion \(RMSE\), and the mean magnitude of errors is the criterion \(MAE\), \(RMSE\) and \(MAE\) are similar in that the closer these criterion values of these errors are to 0, the better the model's performance. In circumstances where the \(MAE\) and \(RMSE\) are minimal, the model's accuracy is greater. Models yielded the lowest \(MARE\) value, indicating that the model has superior predictive power. The \(RSR\) ranges from 0 to a considerable positive number. Lower \(RSR\) indicates lower \(RMSE\), indicating that the model is more productive. \(RSR\) and \(NSE\) categorization ranges as very good, good, satisfactory, and unsatisfactory with ranges \(0\ \leq RSR\ \leq\ 0.5\), \(0.5<RSR\ \leq\ 0.6\), \(0.6<RSR\ \leq\ 0.7\), \(RSR>0.7\) and \(0.75<NSE\ \leq\ 1\), \(0.65<NSE\ \leq\ 0.75\), \(0.5<NSE\ \leq\ 0.65\), and \(NSE\ \leq\ 0.5\), respectively [65].  
In addition, the Taylor diagram was used to compare the model's performance visually [66]. Taylor diagram shows how similar patterns are and how closely a model pattern relates to reference. The standard deviation (\(\sigma\)), \(R^{2}\), and the \(RMSE\) are three equivalent model performance statistics that can be shown on a two-dimensional plot using the law of cosines. Taylor diagram is the best method for comparing the performance of various models in particular.