## 3 Results
### Results of Ensemble Models  
In this subsection, we delve into the outcomes of our top-three ensemble models designed for predicting maize grain yield. These ensembles, meticulously curated by harnessing the strengths of various ML algorithms, underwent a comprehensive evaluation based on predefined performance metrics like the MSE, RMSE, MAE, and \(R^{2}\). Our analysis provides a detailed examination of each ensemble model's precision and effectiveness in capturing intricate connections within multi-temporal satellite imagery data, environmental variables, and maize yields. Additionally, we conducted a comparative assessment to identify the top-performing ensemble, characterized by its accuracy and reliability in predicting grain yields. Our presentation of these findings aims to offer valuable insights into the potential of ensemble modeling for maize grain yield prediction. We believe that such insights can prove instrumental for decision makers in agriculture and precision farming, guiding them toward more informed practices and strategies. Following thorough feature engineering and hyperparameter tuning, our results highlight the effectiveness of the Huber, M estimators, ridge regression, and linear regression models when paired with the features \"CIre_CD\" and \"NDRE_CD\" derived from the SHAP values technique for predicting maize grain yield. Building on these findings, we opted to utilize the VotingRegressor to create ensembles for each method, thereby enhancing the predictive capabilities of our model as follows:  
* PENemble 1: Using Huber, M estimators (RANSAC), linear regression, and ridge regression, the weighted ensemble prediction is given by \[P_{\text{Ensemble2}}=\frac{w_{1}P_{\text{Huber}}+w_{2}P_{\text{M-estimators}}+w_{3}P_{ \text{Linear\ Regression}}+w_{4}P_{\text{Ridge\ Regression}}}{w_{1}+w_{2}+w_{3}+w_{4}}.\] (6)
* PENemble 2: Using Huber, M estimators (RANSAC), and linear regression, the weighted ensemble prediction is given by \[P_{\text{Ensemble2}}=\frac{w_{1}P_{\text{Huber}}+w_{2}P_{\text{M-estimators}}+w_{3}P_{ \text{Linear\ Regression}}}{w_{1}+w_{2}+w_{3}}.\] (7)
* PENemble 3: Using [PERSON], M estimators (RANSAC), and ridge regression, the weighted ensemble prediction is given by \[P_{\text{Ensemble3}}=\frac{w_{1}P_{\text{Huber}}+w_{2}P_{\text{M-estimators}}+w_{3}P_{ \text{Ridge\ Regression}}}{w_{1}+w_{2}+w_{3}}.\] (8)
* PENemble 4: Using Huber and M estimators (RANSAC), the weighted ensemble prediction is given by \[P_{\text{Ensemble4}}=\frac{w_{1}P_{\text{Huber}}+w_{2}P_{\text{M-estimators}}}{w_{1}+w_{2}}.\] (9) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Parameter** & **Value** & \(\mathbf{R^{2}}\) \\ \hline \multirow{3}{*}{Huber} & alpha & 1 & \\  & epsilon & 1 & 0.926266 \\  & max\_iter & 100 & \\ \hline \multirow{2}{*}{M estimators} & min\_samples & 0.7 & 0.924029 \\  & stop probability & 0.96 & \\ \hline Linear regression & fit\_intercept & True & 0.918301 \\ \hline \multirow{3}{*}{Ridge regression} & alpha & 0.615848211 & \\  & fit\_intercept & True & 0.916892 \\ \cline{1-1}  & solver & Isqr & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary of hyperparameters. The results, reflecting the weights assigned to each model in the ensemble methods, are presented in Table 7.  
Method 1 (PEsemble1) employs four distinct models: Huber, M estimators, linear regression, and ridge regression, each assigned weights of 0.5, 0.3, 0.1, and 0.1, respectively. The Huber model is granted the highest weight, emphasizing its substantial impact on ensemble prediction. This aligns with the robust nature of [PERSON]'s regression, particularly beneficial in datasets susceptible to outliers. The obtained \(R^{2}\) score was 0.924202, signifying commendable predictive performance. In Method 2 (PEsemble2), three models--Huber, M estimators, and linear regression--were included, with weights of 0.45, 0.35, and 0.2, respectively. Huber remains dominant, though to a lesser extent than in Method 1.