## 2 Materials and Methods
### Methods
#### 2.3.4 Optimal Parameters XGBoost Model  
The XGBoost model is an ensemble learning algorithm based on gradient boosting, designed to reduce bias and underfitting by iteratively constructing multiple estimators ([PERSON] et al., 2017). It is widely recognized for its strong performance, computational efficiency, and scalability in regression, classification, and ranking tasks ([PERSON], 2018). In this study, to enhance the accuracy of the XGBoost model, optimal hyperparameters were determined using a combination of 10-fold cross-validation and random search optimization. The model was trained and evaluated 10,000 times, with the best-performing hyperparameters selected for training the final model. The structure and workflow of the optimized XGBoost model are illustrated in Figure 2. Additionally, to further improve training efficiency, the model utilized GPU acceleration and parallel processing.  
The XGBoost model was developed using seven datasets, where the dependent variable was one of the CWUE metrics for irrigated cropland, and the independent variables included precipitation, temperature, LAI, GDP, radiation, and sunshine. To determine the optimal hyperparameters, a combination of 10-fold cross-validation and randomized search was employed. The hyperparameter optimization process consisted of the following steps: Step 1: The dataset was equally divided into 10 subsets of similar size, each representative of the overall data characteristics. The XGBoost model was trained 10 times, with each iteration using the \(i\)th subset as the validation set and the remaining nine subsets combined as the training set. Step 2: After completing the 10 training iterations, the average model predictions were computed and outputted. Step 3: Model tuning was performed using a random search hyperparameter optimization method. This method randomly sampled parameter combinations within the parameter space, evaluating model performance via Root Mean Square Error (RMSE) and Coefficient of Determination (R\({}^{2}\)). Among 10,000 trained XGBoost models, the parameter set with the best performance was selected. The optimal hyperparameters for the three CWUE indicators in Henan Province are listed in Table 2, with all models achieving R\({}^{2}\) values above 0.81. These results demonstrate that the optimized XGBoost model accurately and reliably identifies the spatiotemporal heterogeneity of key drivers underlying CWUE dynamics in Henan Province.  
We provide the exact range of hyperparameters in the article as follows: param_distributions = {'n_estimators': randint (100, 500), 'learning_rate': uniform (0.01, 0.2),'max_depth': randint (3, 10),'subsample': uniform (0.7, 0.3), 'colsample_bytree': uniform (0.7, 0.3), lambda: [0.1, 10], and alpha: [0, 5]). The computer environment information used in this study shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**CWUE** & \begin{tabular}{c} **N** \\ **Estimators** \\ \end{tabular} & \
\begin{tabular}{c} **Learning** \\ **Rate** \\ \end{tabular} &
\begin{tabular}{c} **Max** \\ **Depth** \\ \end{tabular} & **Subsample** & **R\({}^{2}\)** & **RMSE** \\ \hline CUE & 454 & 0.1163 & 9 & 0.8912 & 0.8787 & 0.0250 \\ WUE\({}_{\text{NPP}}\) & 423 & 0.1206 & 9 & 0.7243 & 0.8547 & 0.1042 \\ WUE\({}_{\text{GPP}}\) & 423 & 0.1206 & 9 & 0.7243 & 0.8122 & 0.1869 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Optimal hyperparameter details for optimal parameter XGBoost.  
Figure 2: The working principle of the optimal parameter XGBoost model.