## 2 Research Methodology
### Machine Learning Modeling  
Before constructing the model, the dataset needs to be divided into a training set and a testing set. Among them, the training set is used to train the model, i.e., to learn the relationship between the inputs and the output. The testing set is treated as the independent \"new\" data to evaluate the generalization capability of the trained ML model [16]. In this study, we used a random split to divide 70% of the data into a training set, and the remaining 30% was used as a testing set [17]. DT and PSO were used to construct a hybrid model. Notably, the random splitting was repeated 10 times, and the average result was regarded as the representative performance during comparison. The DT is a graphical method of intuitive use of probability analysis that has been widely used for regression and classification problems [18]. The DT technique employs a tree structure to support the decision-making process. In DT, each internal node represents a test on an attribute, each branch represents a test output, and each leaf node represents a category. The decision rules are updated during the DT training process to minimize the discrepancy between the actual and estimated outputs, such as the mean squared error (MSE) for regression problems and the Gini index for classification problems [19]. For the prediction of each new data instance, the model will start with the root node and move downward until reaching the leaf node. The output of the leaf node will be the prediction for this data instance. The DT has been used widely in geotechnical engineering for its flexibility and interpretability [20]. Figure 4 shows a typical DT structure. The PSO is a population-based optimization algorithm proposed by [21]. PSO has been widely used in geotechnical engineering due to its simple operation, easy implementation, and rapid convergence to the optimal solution. PSO imitates the swarm behavior of insects, birds, fish, etc., when they are cooperatively searching for food. Each member of the group is constantly changing its search mode by learning from its own experience and the experience of other members [22]. The solution space is efficiently searched by a particle swarm or population. The first population of particles or individuals is initialized randomly in the solution space. Specifically, the velocity \(v_{i}\) and position \(x_{i}\) of each particle is randomly initialized [23]. The selected fitness function is used to evaluate the performance of each particle, and then the \(v_{i}\) and \(x_{i}\) of  
Figure 4: A typical DT structure. each particle will be updated. After multiple generations, the optimal solution is obtained (Figure 5). The update strategy is shown as follows:  
\[v_{i}^{t+1}=w\times v_{i}^{t}+c_{1}\times r_{1}(p_{best,i}^{t}-x_{i}^{t})+c_{2} \times r_{2}(g_{best}^{t}-x_{i}^{t}) \tag{2}\]  
\[x_{i}^{t+1}=x_{i}^{t}+v_{i}^{t+1} \tag{3}\]  
where \(v_{i}^{t}\) and \(v_{i}^{t+1}\) represent the velocity of the \(i\) individual at \(t\)-th and \((t+1)\)-th generation, respectively; \(x_{i}^{t}\) and \(x_{i}^{t+1}\) represent positions of \(i\)-th individual; \(w\), \(c_{1}\), and \(c_{2}\) are the inertia parameter, the cognitive influence parameter, and the social influence parameter, respectively; \(r_{1}\) and \(r_{2}\) are random values between 0 and 1; \(p_{best,i}^{t}\) and \(g_{best}^{t}\) represent the best position of an individual and the group's best position at \(t\)-th generation. The performance of the DT model on a specific dataset is controlled by hyper-parameters. Thus, we used PSO to tune the hyper-parameters of the DT model. Table 2 summarizes the PSO parameters employed in this study [24; 25].