## 2 Materials and Methods
### Predictions  
Based on the correlation between SAR variables and their different speckle filters and yield, the speckle filter that provided the best performance to do the predictions was chosen. Additionally, when correlating the SAR variables, those with correlations higher than 80% among themselves were discarded to avoid collinearity issues in the prediction models. From there, predictions using only the selected filter and the SAR variables with the lowest correlations among themselves were carried out, which could result in complementary information.  
Random Forest (RF) was the machine learning algorithm used in this study to estimate soybean yield, due to its robustness and widespread use in yield predictions. For this, the set of 15,000 yield data points for each harvest was randomly divided into 70% for training and 30% for testing. The parameters used for the regressions were the default settings of the algorithm: nodesize = 5, mtry = p/3, where p is the number of predictors, with the only modification being the number of trees (ntree) reduced from 500 to 150 for reducing operational costs. The RF models were evaluated using the coefficient of determination (R\({}^{2}\)) and the root mean squared error (RMSE) for the testing set as error metrics.  
For the predictions, models were evaluated exclusively with the optical vegetation index (using only EVI) and models including EVI and SAR variables (Figure 4), to understand whether SAR variables improve the performance of soybean yield prediction. Initially, a scenario was evaluated including all SAR variables and EVI from the three development stages together, compared to using only EVI at all three stages (Figure 4a). Subsequently, models were created using all SAR variables for each of the three stages separately, to determine if including SAR variables at separate stages optimizes performance and which stage is better for predicting soybean yield (Figure 4b). Using the growing stage that showed the best performance, models were evaluated with each SAR variable and EVI separately to evaluate the performance of each individual SAR variable and further optimize predictions (Figure 4c). This process led to the identification of the stage and SAR variable that provided the best performance for soybean yield prediction.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Statistics**} & \multicolumn{2}{c}{**Harvest 1**} & \multicolumn{2}{c}{**Harvest 2**} & \multicolumn{2}{c}{**Harvest 3**} \\ \cline{2-7}  & **Full Data** & **15 k Points** & **Full Data** & **15 k Points** & **Full Data** & **15 k Points** \\ \hline Mean & 3.50 & 3.51 & 2.97 & 2.97 & 5.08 & 5.08 \\ Median & 3.44 & 3.44 & 2.99 & 2.99 & 5.05 & 5.05 \\ Standard & 0.77 & 0.78 & 0.86 & 0.86 & 0.72 & 0.72 \\ deviation & 0.72 & 22.23 & 28.98 & 29.14 & 14.20 & 14.14 \\ CV\% & 22.20 & 22.23 & 28.98 & 29.14 & 14.20 & 14.14 \\ Minimum value & 1.14 & 1.18 & 1.01 & 1.04 & 2.84 & 3.00 \\ Maximum value & 5.95 & 5.86 & 5.47 & 5.46 & 7.93 & 7.93 \\ Q1 & 2.93 & 2.93 & 2.38 & 2.38 & 4.58 & 4.58 \\ Q3 & 4.04 & 4.05 & 3.53 & 3.54 & 5.54 & 5.55 \\ Interquartile range & 1.11 & 1.11 & 1.14 & 1.16 & 0.95 & 0.96 \\ n & 71,832 & 15,000 & 59,141 & 15,000 & 59,568 & 15,000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Descriptive statistics of the full datasets in each area (after cleaning) and the 15,000 randomly selected yield points. Data in tons per hectare.