## 4 Results, Discussion, and Concluding Remarks
### Relative Importance of Influencing Variables  
The importance of influencing variables was analyzed for a deeper understanding of \(Q_{\text{u}}\) and \(u_{\text{p}}\). As indicated before, partial dependence plots and the relative importance score were choose as the methods for interpreting the importance of influencing variables [58; 59].  
In Figure 10, partial dependence plots of four influencing variables of \(Q_{\text{u}}\) and \(u_{\text{p}}\) are shown. Normally, the significance of influencing variable is reflected by the output response when an influencing variable is changed. As shown, there was an almost linear growth relation between \(Q_{\text{u}}\) and \(H/D\). The finding is quite understandable as the self-weight stress of soil increases with the increase in \(H/D\) when \(D\) remains unchanged. An almost linear initial growth relation is also shown between \(u_{\text{p}}\) and \(H/D\); however, the growth rate gradually decreases with increasing \(H/D\) and tends toward a stable value when \(H/D\) reaches a certain size. The output response to variation in the inputs indicates that \(H/D\) was the most significant variable for both \(Q_{\text{u}}\) and \(u_{\text{p}}\). The importance of the remaining three input variables was similar, although both \(Q_{\text{u}}\) and \(u_{\text{p}}\) exhibit slow growth increases with increasing \(n\). The authors also note here the relatively small influence of \(Dr\) (%) in the current study was due to the limited change in \(Dr\). If the data for loose and medium-dense sand were also included, their influence on \(Q_{\text{u}}\) and \(u_{\text{p}}\) would be significantly greater.  
Figure 10: _Cont._  
Figure 9: Repeated evaluation of model performance based on cumulative frequency of error: (**a**) for \(u_{\text{p}}\), (**b**) for \(Q_{\text{u}}\).  
Figure 11 shows the relative importance score of four influencing variables with the summation of all importance scores being scaled to one; \(H/D\) was the most sensitive variable for both \(Q_{\text{u}}\) and \(u_{\text{p}}\). The importance scores of \(H/D\) were 0.614 and 0.622 for \(Q_{\text{u}}\) and \(u_{\text{p}}\), respectively. The influence of \(H/D\) on \(Q_{\text{u}}\) is consistent with results from previous studies [24; 26; 60] and the influence of \(H/D\) on \(u_{\text{p}}\) also agrees with the findings of previous research [26; 61]. The other three variables show almost equal importance scores for \(Q_{\text{u}}\). The above results agree well with the partial dependence plots. For \(u_{\text{p}}\), the \(n\) parameter had an importance score of 0.058, indicating it is also a non-negligible influencing variable.  
Figure 11: Cont.  
Figure 10: Partial dependence plots of the influencing variables in the optimal GBDT model for predicting: (**a-1**–**a-**4**) \(Q_{\text{u}}\), (**b-1**–**b-**4**) \(u_{\text{p}}\).  
In general, the partial dependence plots and relative importance scores of influencing variables in this study highlight some important outcomes and indicate future potential experimental studies concerning \(Q_{\text{u}}\) and \(u_{\text{p}}\) in dense sand. These findings have particular importance for optimizing \(Q_{\text{u}}\) and \(u_{\text{p}}\) and represent a useful tool for helical anchor design.