## 2 Materials and Methods
### Algorithm Principle for the XGBoost Model  
XGBoost, firstly proposed by Washington computer Ph.D. [PERSON] and [PERSON] [45], uses CART as a sub-model to achieve the ensemble learning of multiple CART trees by gradient tree boosting. The integration follows the typical additive model and forward distribution algorithm. The basic structure is shown in Figure 1.  
The XGBoost algorithm, including the objective function, regularization function, and loss function, enabled it to learn the weights of each feature autonomously with excellent accuracy while being trained. We assumed that the sample dataset used for training was \((x_{i},y_{i})\), where \(x_{i}\in R^{m},y_{i}\in R\), and where \(x_{i}\) is a feature vector with \(m\) dimensions, \(y_{i}\) denotes the sample label, and the model contains K trees. The XGBoost model was defined as shown in Equation (1):  
\[\hat{y}_{i}^{(t)}=\sum_{k=1}^{t}f_{k}(x_{i})=\hat{y}_{i}^{(t-1)}+f_{i}(x_{i}), \tag{1}\]  
where \(\hat{y}_{i}^{(t)}\) is the model prediction value of a sample \(x_{i}\) at the \(t\)th iteration, \(\hat{y}_{i}^{(t-1)}\) is the model prediction value of sample \(x_{i}\) at the previous iteration, and \(f_{i}(x_{i})\) is the new sub-model trained for the \(t\)th time. The objective function (_Obj_) of the XGBoost model was based on the loss function (\(L\)) with the addition of the regularization term (\(\Omega\)), measuring the complexity of model, as in Equation (2):  
\[Obj^{(t)}=\sum_{i=1}^{n}L(y_{i},\hat{y}_{i}^{(t)})+\sum_{i=1}^{t}\Omega(f_{i}). \tag{2}\]  
The loss function (\(L\)) was used for evaluating the deviation between the predicted and measured values of the model. In addition, the regularization term (\(\Omega\)) was introduced to account for the overall complexity of all trees in the model, effectively mitigating the risk of overfitting [45]. The expression for the regularization term (\(\Omega\)) is presented in Equation (3):  
\[\Omega(f)=\gamma T+\frac{1}{2}\lambda\|\omega\|^{2}, \tag{3}\]  
Figure 1: Basic structure of the XGBoost tree model. where \(\gamma\) and \(\lambda\) are both regularization term coefficients, \(T\) is the number of leaf nodes in the decision tree, and \(\omega\) is a vector of the weight scores controlling the leaf nodes. \(\gamma T\) affects the complexity of the model by controlling the tree depth. The optimization objective of the model was to find the optimal loss function satisfying the minimum objective function, and this task was completed by introducing the Taylor formula into the XGBoost model. After expanding Equation (1) using [PERSON]'s formula and substituting Equation (2) into Equation (1), the objective function was transformed into Equation (4):  
\[Obj^{(t)}=\sum\limits_{i=1}^{n}{[g_{i}f_{s}(x_{i})+\frac{1}{2}h_{i}f_{t}^{2}(x _{i})]}+\gamma T+\frac{1}{2}\lambda\sum\limits_{j=1}^{T}{\omega_{j}^{2}}. \tag{4}\]  
During the model training process, a crucial task is to determine the optimal cut points for constructing the \(t\)th tree's leaf nodes. In the XGBoost model, an exact greedy algorithm was adopted to iterate through all the split leaf nodes that were eligible for splitting. Among these nodes, the one with the highest gain in the splitting objective function was chosen as the optimal splitting point. The gain of the splitting point was calculated by Equation (5). Notably, the gain provided a basis for the feature importance output. The steps of the optimal cut-point selection process are outlined in Algorithm 1. \[Gain=\frac{1}{2}[\frac{G_{L}^{2}}{H_{L}+\lambda}+\frac{G_{R}^{2}}{H_{R}+\lambda }-\frac{\left(G_{L}+G_{R}\right)^{2}}{H_{L}+H_{R}+\lambda}]-\lambda, \tag{5}\]  
where \(G_{L}\) and \(G_{R}\) denote the first-order gradient statistical sums of the sample sets of the left and right child nodes, respectively, that were split from the current node, and \(H_{L}\) and \(H_{R}\) are the second-order gradient statistical sums of the sample sets of the left and right child nodes, respectively.