## 3 Results and Discussion
### Feature Importance of CatBoost-Bayesian Hybrid Model  
The CatBoost algorithm (CatBoost-Bayesian hybrid model) was optimized using the Bayesian optimization algorithm under the five-fold cross-validation, and the optimal hyperparameters of the CatBoost algorithm were obtained (Table 2); this study was carried out under the optimal parameters to avoid the excessive model fit, and was beneficial to the CatBoost-Bayesian hybrid model's adaptively coupled modified theoretical equations for stable estimation of the undrained shear strength of clay. Through the CatBoost-Bayesian hybrid model, an importance ranking of soil parameters that affect the undrained shear strength of clay was obtained.  
When the CatBoost-Bayesian hybrid model was used to estimate the undrained shear strength of clay, the feature importance of the model input parameters to the hybrid model under five-fold cross-validation was obtained, and it was explained whether the input parameters contribute positively (positive correlation) or negatively (negative correlation). Each point in the graph represents a data point from the training set. The color represents the value of the feature parameter; red represents the sample with a higher value of the feature parameter, and blue represents the sample with a lower value of the feature parameter. The length of the horizontal line represents the importance of the estimation of the undrained shear strength of the clay.  
It can be seen from the whole that the overconsolidation ratio (\(OCR\)) is the most important parameter for estimating the undrained shear strength of clay, followed by preconsolidation pressure (\(\sigma^{\prime}_{v0}\)), effective overburden pressure (\(\sigma^{\prime}_{vv}\)), etc. The effective overburden pressure (\(\sigma^{\prime}_{vv}\)) increases and the undrained shear strength depends on the consolidation stress before shearing, that is, the research status of the preconsolidation pressure (\(\sigma^{\prime}_{v0}\)) [52; 53; 54]. The void ratio (\(e\)) and clay content (\(CI\)) features are the least important. At the same time, the characteristic variables significantly show that the compression index (\(C_{\rm c}\)) makes a negative contribution to the computed value of the drainage shear strength, while the preconsolidation pressure (\(\sigma^{\prime}_{v0}\)) and the overconsolidation ratio (\(OCR\)) make a positive contribution to the computed value of the drainage shear strength (Figure 6). The model's interpretability method is expected to help geotechnical engineers in the selection of soil parameters in practical engineering. However, the CatBoost-Bayesian hybrid model is still a black-box model, which finds it difficult to explain the internal mechanism of the model and the feature importance; therefore, the combination of the feature importance results with the theoretical equations derived from the constitutive model can provide a reference for geotechnical analysis.  
\begin{table}
\begin{tabular}{c c} \hline
**Optuna_Parameters of CatBoost** & **Description** \\ \hline loss function & MAE \\ n\_estimators & 1000 \\ learning rate & 0.153 \\ random state & 2019 \\ l2\_leaf\_reg & 0.030 \\ colsample\_bylevel & 0.098 \\ depth & 1 \\ boosting type & Plain \\ bootstrap type & MVS \\ min\_data\_in\_leaf & 4 \\ one\_hot\_max\_size & 3 \\ early\_stopping\_rounds & 100 \\ \hline \end{tabular}
\end{table}
Table 2: CatBoost optimal hyperparameters.  
Figure 6: Clay parameter importance ranking of CatBoost–Bayesian. (**a**) Feature importance analysis of K = 1. (**b**) Feature importance analysis of K = 2. (**c**) Feature importance analysis of K = 3. (**d**) Feature importance analysis of K = 4. (**e**) Feature importance analysis of K = 5. (**f**) Feature importance ranking.