## 2 Materials and Methods
### Selection of Machine Learning (ML) Methods for Grain Yield Predictions
To investigate the effectiveness of ML models, this study utilizes a set of twelve VIs based on the CD. These VIs were computed from a long-term series of collected images for each plot. The VIs were employed as independent variables, while the maize yields observed in each plot were utilized as dependent variables. This investigation made use of a diverse range of meticulously selected ML algorithms. The selection process took into consideration the distinctive characteristics of each algorithm, as well as insights derived from the relevant academic literature. The primary goal was to conduct a comprehensive analysis of multi-temporal satellite imagery data in order to provide accurate predictions of crop yields. Each chosen model underwent careful evaluation to ensure its suitability for the specific task at hand. **Model 1 (CatBoost regression)**[48]: This algorithm was chosen for its adeptness in handling categorical features prevalent in such data. By employing ordered boosting and symmetric trees, CatBoost effectively captured temporal variations and seasonality patterns, ensuring a high level of predictive accuracy. **Model 2 (decision tree regression)**[49; 50]: Selected to interpret the impact of different temporal features on crop yields, decision tree regression leverages its hierarchical structure to represent complex relationships within the data. Figure 4: The flow diagram of ML methods for grain yield prediction. **Model 3 (ElasticNet regression)**[51]: Addressing multicollinearity and performing feature selection simultaneously, ElasticNet regression has proven crucial for high-dimensional multi-temporal data. Its combination of L1 and L2 regularization techniques played a pivotal role in achieving these objectives. **Model 4 (gradient boosting regression)**[52; 53]: Demonstrating efficacy in handling complex relationships and temporal dependencies, gradient boosting regression build an ensemble of decision trees sequentially, enhancing predictive accuracy in the process. **Model 5 (Huber regression)**[54]: Chosen for its robustness in accommodating outliers prevalent in agricultural datasets, Huber regression minimizes the impact of extreme values through a parameter balancing the mean absolute error (MAE) and mean-squared error (MSE). **Model 6 (K-Nearest Neighbors regression)**[55]: Predicting grain yields based on similar multi-temporal patterns observed in the past, KNN regression relied on the assumption that analogous temporal patterns yield similar crop yields. However, the consideration of an appropriate K value and computational cost for larger datasets was warranted. **Model 7 (LASSO regression)**[56]: Serving as a feature selection technique, LASSO regression incorporates L1 regularization to encourage some feature coefficients to be precisely zero, facilitating the identification of vital temporal features. **Model 8 (linear regression)**[57]: Providing a baseline model to understand the overall trend between crop yields and multi-temporal features, linear regression assumed a linear relationship, offering valuable insights into the general temporal trend of grain yield. **Model 9 (M estimators regression)**[58]: Chosen for their robustness in handling outliers and noise in agricultural datasets, M estimators provided flexibility in choosing the loss function, making them suitable for data with non-Gaussian noise. **Model 10 (passive aggressive regression)**: As an online learning algorithm, passive aggressive regression adapted quickly to temporal changes in crop yield patterns, making it suitable for real-time predictions, particularly in rapidly evolving agricultural conditions. **Model 11 (random forest regression)**[59]: Handling high-dimensional multi-temporal data, RF provided robust predictions by building multiple decision trees on different data subsets and averaging their predictions.