## 5 Conclusions  
In this study, based on relevant centrifuge test data, a GBDT-PSO prediction model was constructed to explore the nonlinear relationship between four input variables (e.g., soil relative density) and the ultimate monotonic uplift resistance, \(Q_{\text{u}}\), and the anchor mobilization distance, \(u_{\text{p}}\). A 30-fold cross-validation approach was applied, and six external indicators (R, EVS, MSE, MAE, a20-index, and P30) were used to verify the performance of the optimal GBDT model. In addition, partial dependence plots and importance scores were selected to investigate the parameters' relative importance. The specific conclusions of the work are as follows:  
1. PSO was efficient in the hyperparameter tuning of GBDT models with maximum R values of 0.987 on the \(Q_{\text{u}}\) dataset and 0.957 on the \(u_{\text{p}}\) dataset; these results were achieved in the first 10 PSO iterations.
2. The optimal GBDT-PSO model has a high generalization ability. For \(Q_{\text{u}}\) and \(u_{\text{p}}\), the R values on the testing set were up to 0.93 and 0.82, respectively, and the a20-index values were 0.70 and 0.724, indicating that the model has predictive ability for the lifting behavior of spiral anchors in sand.  
3. The embedment ratio, \(H/D\), was found to be the most important variable; however, the helix spacing ratio, \(S/D\), was found to have less influence on the capacity of adjacent helices when \(S/D>6\). The influence of other input variable is not obvious.  
The prediction results for multiple-helix anchors have non-negligible differences with experimental results, which may be a result of insufficient data. With the filling of more centrifuge test results of in situ test results, the accuracy of prediction could be markedly improved, and the present model can be used for more robust and economic designs of helical anchors in sand.  
Conceptualization, L.W. and C.Q.; methodology, C.Q. and M.W.; validation, L.W., D.H. and C.Q.; formal analysis, M.W. and H.C. investigation, L.W. data curation, D.H. writing--original draft preparation, L.W. and C.Q. writing--review and editing, all authors; supervision, Y.T. funding acquisition, L.W., D.H. and C.Q.. All authors have read and agreed to the published version of the manuscript.  
This research was funded by [State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University] grant number [HESS-2120], [State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology], grant number [LP2120], and [National Natural Science Foundation of China], grant number [52078108].  
Not applicable, this research not involving humans or animals.  
Not applicable.  
The first three authors acknowledge the support by State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University (HESS-2120), State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (LP2120) and National Natural Science Foundation of China (52078108).  
The author declares no conflict of interest.  
The first three authors acknowledge the support by State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University (HESS-2120), State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (LP2120) and National Natural Science Foundation of China (52078108).  
The second author declares no conflict of interest.