## 2 Materials and Methods
### Selection of Best Suited Models  
The selection of the most appropriate models depends on the specific characteristics of the multi-temporal imagery and the nature of the grain yield data. Models such as CatBoost regression and gradient boosting regression are well suited to capturing temporal patterns and seasonality effects. ElasticNet and LASSO regression offer valuable features for feature selection in high-dimensional data. Huber regression and M estimators are effective at handling outliers within the crop yield data. Given the diverse range of scenarios and variable combinations explored in this study, it is crucial for researchers to rigorously evaluate each algorithm's performance using defined evaluation metrics. This meticulousevaluation process enables researchers to identify the most effective models for accurately predicting grain yield within their unique agricultural context, thus enhancing the reliability and accuracy of the predictions.  
To evaluate the performance of these various algorithms [62; 63; 64; 65], the authors used several metrics, including mean-squared error (MSE), root-mean-squared error (RMSE), mean absolute error (MAE), and R-squared (\(R^{2}\)) as  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}, \tag{1}\]  
\[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(P_{i}-M_{i})^{2}}, \tag{2}\]  
\[MAE=\frac{\sum_{i=1}^{N}|P_{i}-M_{i}|}{N}, \tag{3}\]  
and  
\[R^{2}=1-\frac{\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{N}(y_{i}-\hat {y})^{2}}, \tag{4}\]  
where \(N\) is the number of all samples, \(M\) and \(P\) are the true values and predicted values of the yields, \(y_{i}\) represents the actual values, \(\hat{y}_{i}\) represents the predicted values, \(|\cdot|\) represents the absolute value, and \(\hat{y}\) represents the mean of the actual values, respectively. These metrics provide insights into the accuracy and predictive power of each algorithm in estimating the growth stage and productivity of maize.  
The study encompasses an extensive set of forty features utilized as inputs for the ML models to facilitate the accurate prediction of crop growth and yield. This comprehensive set of features comprises eight soil-related factors, eight parameters derived from environmental data, and twenty-four features derived from twelve VIs. The soil-related factors encompass N, P, K, pH level, soil temperature, soil humidity, EC, and maize height. These factors provide insights into the soil composition, fertility, and moisture content, as well as the growth status of the maize plants. On the other hand, the parameters derived from environmental data capture important variables such as water feed, UV radiation, evapotranspiration, daily rainfall, rain rate, humidity, wind speed, and temperature. These environmental factors play a crucial role in influencing crop growth and yield, and their inclusion in the analysis allows for a comprehensive understanding of the interactions between soil conditions and the surrounding environment.  
Furthermore, the study incorporates twenty-four features derived from twelve VIs, specifically the Clgr, Clre, EVI2, GNDVI, MCARI2, MTVI2, NDRE, NDVI, NDWI, OSAVI, RDVI, and RGBVI, accompanied by the CD and CV. Through the incorporation of a diverse set of forty features, ML models can capture the complex relationships between soil properties, environmental conditions, and vegetation characteristics, thereby exerting a significant influence on crop health and a holistic analysis of crop growth and productivity.