## 2 Materials and Methods
### Method
#### 2.2.4 Ensemble Learning  
Machine learning has strong capabilities in nonlinear modeling and adaptability to high-dimensional data. Vegetation's response to climate factors exhibits significant nonlinear characteristics, and machine learning models are adept at capturing such complex patterns. In machine learning, ensemble learning is a method that enhances predictive performance by combining multiple models. Stacking is an advanced ensemble learning technique that improves overall prediction performance by integrating predictions from multiple base models. This method not only leverages the unique strengths of each base model to improve prediction accuracy and robustness but also effectively reduces overfitting, enhancing the model's generalization ability. Through this multilevel learning mechanism, stacking models provide more stable and accurate predictions in complex data environments, which are widely applicable to various practical problems. In this study, to predict future NDVI, the authors first analyzed historical climate factor data from 2001 to 2023. To determine which meteorological factors should be prioritized as model inputs, this study first used the Random Forest method to analyze the relationships between multiple meteorological factors and NDVI. Through importance ranking, this study selected the top five meteorological factors with the strongest correlation as model inputs. In addition, although CO\({}_{2}\) does not rank highly in the importance order, this study specifically considered and included this variable in the final model inputs due to its key role in the carbon cycle, particularly its close relationship with emissions and plant respiration. Therefore, our final model is based on the following meteorological factors: relative humidity (RH), shortwave radiation (SWR), soil moisture (SWVL1), temperature at 2 m (T2m), precipitation (TP), and carbon dioxide concentration. This forms a comprehensive overlay model aimed at more accurately predicting future NDVI trends. The model integrated various machine learning algorithms, using Random Forest, XGBoost, and Support Vector Regression (SVR) as base models, and a linear regression model as the meta-model to determine the optimal weighting of these base models, optimizing the final prediction results and generating the final NDVI predictions. This study selects Random Forest, XGBoost, and SVR as the base models, primarily because of their unique advantages and complementarity. Random Forest provides strong anti-overfitting ability and good adaptability to nonlinear relationships by integrating multiple decision trees. XGBoost, with its efficient gradient boosting algorithm and regularization techniques, performs excellently in handling complex, multivariate nonlinear problems and effectively enhances the model's generalization ability. SVR, using kernel functions, handles nonlinear relationships, making it particularly suitable for small sample datasets and reducing the impact of noise. These three models, when used together, can capture data features from multiple angles, enhancing model stability and prediction accuracy. Finally, linear regression is used as the meta-model to integrate the results of these base models, optimizing weight distribution and further improving the accuracy and reliability of NDVI prediction. In our research, for hyperparameter optimization, this study adopted a combined strategy of grid search and random search. This approach allowed us to carefully adjust the parameters and determine the optimal parameter set based on the specific performance of each model. For the evaluation of the stacked ensemble model, this study used a stratified K-fold cross-validation method [90]. This strategy ensuredthe consistency of sample distribution between the training and validation sets during the cross-validation process, effectively reducing the risk of overfitting while enhancing the model's generalization ability.