## 4 Results, Discussion, and Concluding Remarks
### Results of the Optimum GBDT Model  
In the existing literature, machine learning predictions for \(Q_{\text{u}}\) and \(u_{\text{p}}\) have not been attempted. Thus, the prediction accuracy achieved in this study cannot be compared with previous results. However, GBDT modeling has been widely verified to display better performance than commonly used machine learning techniques in many problems [36; 37; 55; 56; 57]. Furthermore, the performance of GBDT was further increased using PSO in this study, improving the representativeness of the current work. To verify the efficiency of the proposed optimization model, we applied it to the testing set for evaluation (Figure 7). For \(u_{\text{p}}\), the R value achieved on the testing set was above 0.8, while the R on the testing set for \(Q_{\text{u}}\) is as high as 0.92. Furthermore, the MAE and MSE values of the two datasets were small, implying the high generalization ability of the model. However, there were clear differences in the MSE and MAE of \(Q_{\text{u}}\) on the training and testing sets, which will lead to weak predictive ability for \(Q_{\text{u}}\). In addition, the EVS of \(u_{\text{p}}\) was between 0.6 and 0.7, and the predicted and sample values were more dispersed than those of \(Q_{\text{u}}\) (0.85). Figure 6: Comparison of model performance before and after hyperparameter tunning. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Hyperparameters** & \(Q_{\text{u}}\)** **Dataset** & \(u_{\text{p}}\)** **Dataset** \\ \hline Max\_depth & 4 & 15 \\ Min\_samples\_split & 7 & 7 \\ Min\_samples\_leaf & 1 & 1 \\ Max\_RT & 1319 & 873 \\ Learning rate & 0.061 & 0.890 \\ Max\_features & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimized hyperparameters for both datasets. The experimental and predicted \(Q_{\text{u}}\) and \(u_{\text{p}}\) values were further compared, as shown in Figure 8. The variation trends and the numerical value of \(Q_{\text{u}}\) and \(u_{\text{p}}\) were predicted well except for some special cases, such as T33 for \(u_{\text{p}}\) and T32 for \(Q_{\text{u}}\). The R value was 0.941 and 0.913 on the whole datasets of \(Q_{\text{u}}\) and \(u_{\text{p}}\), respectively. Furthermore, Figure 8 shows that there are three different parts to the results: (1) T1-T5 consist of five tests concerning single-plate anchors embedded under different depths with similar \(D_{\text{r}}\)--in this part, \(u_{\text{p}}\) and \(Q_{\text{u}}\) increase with the embedment depth; (2) T6-T21 consist of ten different embedment depths of single-helix anchors in sand with similar \(D_{\text{r}}\)--in this part, \(u_{\text{p}}\) and \(Q_{\text{u}}\) also increase with the embedment depth and the relevant size is close to that for single-plate anchors in part (1); (3) T22-T33 comprise multiple-helix anchors with different \(H\)/\(D\), \(S\)/\(D\), and \(n\), with double-helix anchors with different \(H\)/\(D\) and \(S\)/\(D\) to those analyzed in T22-T29. Adjacent helices were found to have no influence on each other for \(Q_{\text{u}}\) and \(u_{\text{p}}\) when \(S\)/\(D>6\) in dense fine silica sand; this outcome differs from the threshold value result of \(S\)/\(D>9\) in [PERSON] et al. [23]. Combined with the finding by [PERSON] et al. [16] (i.e., the relevant influence of circular-plate anchors in loose sand is smaller than that in dense sand), the findings of this study have potentially important implications for optimization, \(Q_{\text{u}}\), for multiple-helix anchors if they can be verified by other experimental studies in the future. Figure 7: Evaluation of optimized GBDT models on testing and training sets: (**a**) for \(u_{\text{p}}\), (**b**) for \(Q_{\text{u}}\). With the increasing number of helices, the differences between the predicted and actual values fluctuate, which may be caused by the relatively small available data volume for multiple-helix anchors.