## 4 Development of the Strength Model  
### Selection of the Base Model  
The selection of base learners directly influences final performance in establishing an ensemble learning framework for predicting compressive strength of MICP-improved sand [22; 27]. Therefore, this study employs diverse machine learning methods for comparative analysis to identify optimal model combinations. Through a comprehensive consideration of theoretical foundations, learning mechanisms, and applicable scenarios of different approaches, this study selected eight representative machine learning methods, including SVM, XGBoost, categorical boosting  
Figure 3: Correlation matrix of the input parameters used in this study. (CatBoost), random forest (RF), light gradient boosting machine (LGBM), decision tree (DT), MLP, and k-nearest neighbours (KNN). These models encompass different ML paradigms, including tree-based methods, kernel function methods, and neural network approaches, establishing a solid foundation for subsequent ensemble development. To evaluate model performance, this study adopts multiple cross-validation approaches to systematically assess each model's performance across different datasets. Specifically, to ensure evaluation result stability and reliability, multiple repeated experiments were conducted to perform statistical analysis of error performance for each model across multiple data subsets, with confidence intervals of error distributions serving as the primary basis for performance evaluation. Based on the performance comparison results shown in Figure 4a, various models demonstrate significant differences in accuracy. From an overall effectiveness perspective, XGBoost exhibits the most outstanding performance among multiple tree models, thereby being selected as the representative tree-based method. Additionally, SVM effectively handles nonlinear problems in high-dimensional spaces and demonstrates strong resistance to interference. Its error range remains relatively concentrated, with overall performance superior to most comparative models. MLP similarly demonstrates favourable error distribution performance. Although its error levels are slightly higher than XGBoost, fluctuations remain minimal with strong consistency, making it particularly suitable as a meta-learner to capture interactive features among different base model outputs. In comparison, KNN and DT exhibit significantly larger error fluctuations with relatively poor prediction consistency, therefore excluding them from the final selection. Given the objective existence of performance differences among models, this section further employs non-parametric statistical methods to quantitatively evaluate significant differences between models. Specifically, the study utilises Friedman testing to assess whether significant differences exist among multiple models, followed by Nemenyi testing for pairwise comparison analysis. Nemenyi testing determines whether performance differences between any two models reach statistical significance levels by calculating critical difference values [39; 40]. The calculation formula for critical difference values is  
\[CD=q_{\mathrm{a}}\times\sqrt{\frac{\lambda(\lambda+1)}{6N_{l}}} \tag{13}\]  
Figure 4: Performance evaluation and statistical comparison of ML models: (**a**) Model performance comparison; (**b**) Statistical ranking of models. where \(q_{a}\) represents the critical value based on the significance level, \(\lambda\) denotes the number of learners participating in comparison, and \(N_{t}\) represents the number of datasets. Figure 4b shows that XGBoost achieves the best average ranking, while the ranking differences among XGBoost, SVM, and MLP remain below the critical difference threshold, forming a homogeneous performance group. In contrast, models such as DT, LGBM, and KNN exhibit significantly lower average rankings, particularly DT, positioned at the bottom, with ranking differences from XGBoost far exceeding critical difference values, indicating highly significant statistical differences in performance.