## 2 Materials and Methods
### Data Analysis
The calculation formula is as follows:  
\[\frac{R_{\tau}}{S_{\tau}}=(c\tau)^{H} \tag{8}\]  
\[R_{\tau}=\text{max}_{1\leq t\leq\tau}X_{\text{t},\tau}-\text{min}_{1\leq t\leq \tau}X_{\text{t},\tau} \tag{9}\]  
\[S_{\tau}= \begin{bmatrix}\frac{1}{\tau}\sum_{t=1}^{\tau}\big{(}WUIE_{\text{t}} -\overline{WUE_{\tau}}\big{)}^{2}\end{bmatrix} ^{\frac{1}{2}}(\tau=1,2,\dots,n) \tag{10}\]  
\[X_{\text{t},\tau}=\sum_{t=1}^{\tau}\big{(}WUIE_{\text{t}}-\overline{WUE_{ \tau}}\big{)}(1\leq\text{t}\leq\tau) \tag{11}\]  
\[\overline{WUEC_{\tau}}=\frac{1}{\tau}\sum_{t=1}^{\tau}(WUE_{\text{t}})(\tau=1,2,\dots,n) \tag{12}\]  
where \(H\) is the Hurst index and \(c\) is a constant. (5) Stability Analysis  
The coefficient of variation (CV) is a statistical parameter used to measure the degree of variation in variables over time. It objectively reflects the degree of difference in the spatial values of WUEc data in Xinjiang, serving as a means of evaluating the stability of WUEc [50]. The calculation formula is as follows:  
\[CV=\frac{\sigma}{\mu} \tag{13}\]  
where \(\sigma\) and \(\mu\) are the standard deviation and mean of WUEc in Xinjiang from 2002 to 2022, respectively; the larger the \(CV\), the greater the data fluctuation. (6) Visualizing Machine Learning Models (XGBoost-SHAP)  
The XGBoost model constructs multiple decision trees sequentially and combines them into a strong classifier. By aggregating the prediction results of samples passing through each decision tree, the final predictions are obtained. The XGBoost model processes data more efficiently than other machine learning models, such as the random forest and support vector machine, and also achieves higher accuracy in predicting the final results [51; 52; 53]. The coefficient of determination (R2), root mean square error (RMSE), and mean absolute error (MAE) are used as evaluation metrics for the model. The optimal objective function (Obj) formula for generating the decision tree by training the model with the selected WUEc influencing factors is as follows:  
\[Ob_{j}=-\frac{1}{2}{\sum_{j=1}^{T}\frac{G_{j}^{2}}{H_{j}+\lambda}+\gamma^{T}} \tag{14}\]  
where \(G_{j}=\sum i\in I_{j}g_{i}\) is the sum of all samples \(g_{i}\) belonging to the jth leaf node and \(g_{i}\) is the first order derivative of the loss function; \(H_{i}=\sum i\in I_{j}h_{i}+\lambda\) is the sum of all samples \(h_{i}\) belonging to the jth leaf node and \(h_{i}\) is the second order derivative of the loss function; T is the number of leaf nodes; and \(\lambda\) & \(\gamma\) are the hyperparameters. An XGBoost machine learning model was developed using Python (version 3.1.2); 80% of the data were used for training, and 20% for testing. Spatial 10-fold cross-validation was used to tune the model parameters. The final full-sample MAE was 0.11, RMSE was 0.15, and R2 was 0.96. The key parameters were set as follows: 60 decision trees, a tree depth of 5, and a learning rate of 0.03. The evaluation results are shown in Figure 2.  
SHAP is an additive feature attribution method based on the Shapley value principle from game theory, commonly used to explain complex models. It quantifies the importance  
Figure 2: Evaluation diagram of the XGBoost model. The evaluation results for R2, MAE, and RMSE metrics for the overall data, training set, and test set are displayed in the top-left corner. The red line represents the predicted values fitted by the model. of each influencing factor and converts it into the corresponding SHAP value [54]. The formula for calculating the SHAP value is as follows:  
\[SHAP(f,x,S)=\sum_{T\subseteq S}\frac{|T|!(n-|T|)!}{n!}[f(x)-f(x_{S/T})] \tag{15}\]  
\[SHAP(f,x,S/\{i\})=f(x)-SHAP(f,x,S) \tag{16}\]  
where \(f\) is the predictive function of the model; \(x\) is the observed feature vector; \(S\) is the set of all features; \(n\) is the number of features; and \(x_{s/i}\) is the observed values of all features except \(T\).