# Hybrid Modeling for Stream Flow Estimation: Integrating Machine Learning and Federated Learning
## 4 Statistical Results
### Machine Learning Applications Results
The table showcases the performance metrics for each model, including the coefficient of determination (r2) for both training and test datasets, as well as the explained variance (d) and RMSE for training and test datasets. Figure 9 depicts the training and test results for the Besdegirmen dataset using different machine learning models. The figure illustrates the performance of each model in predicting the flow rate based on the input features. The x-axis of the graph represents the different machine learning models: LR, SVR, DT, RF and ETR. The y-axis represents the coefficient of determination (r2) values, which indicate the goodness of fit of the models. Another field of study is the analysis result performance comparison with machine learning for Melekbahce, which is given in Table 5 and Figure 10. According to the analysis results, the RF, ETR and DVR algorithms were the most successful algorithms in the estimation. The forecast success rate was 70.9 for RF and 68.6 for ETR and SVR. Table 5 presents a comparison of the training and test results for various machine learning models applied to the Melekbahce dataset. The table showcases performance metrics for each model, including the coefficient of determination (r2) for both training and test datasets, as well as the explained variance (d) and RMSE for training and test datasets. WT was applied to improve the results of LR, SVR, DT, RF, and ETR machine learning methods used in the analysis. Comparative results for the Besdegirmen stream gauging stations of the Badger Creek in the hybrid model are given in Table 6 and Figure 11. The r2_test values of the three best performing methods in analysis using DT were 97.4, 97.7, and 95.3 for LR, ETR, and RF, respectively. From Table 6, it becomes feasible to compare the model performance across a range of measures. Upon closer examination of the SVR model, it becomes apparent that it has a significantly high R-squared value (99.9%) on the training dataset. Nevertheless, the performance of this model significantly decreases when evaluated on the test dataset, with an accuracy of 85.8%. This observation indicates a tendency of the model to overfit the training data. The observed disparity between the R-squared values of the training and test sets suggests that the SVR model overfits the training data, leading to a diminished ability \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **r2\_train \%** & **r2\_test \%** & **d\_train** & **d\_test** & **rmse\_train** & **rmse\_test** \\ \hline LR & 48.2 & 49.1 & 4024.6 & 4130.5 & 5736.5 & 5897.4 \\ SVR & 71.3 & 68.6 & 2331.8 & 2567.7 & 4269.4 & 4927.1 \\ DT & 99.8 & 54.2 & 27.6 & 3351.2 & 284.5 & 5593.9 \\ RF & 96.1 & 70.9 & 907.6 & 2690.4 & 1571.2 & 4456.7 \\ ETR & 99.8 & 68.6 & 27.9 & 2719.5 & 284.6 & 4630.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of Melekbahce training and test results. Table 7 presents a comparison of the training and test results for the Melekbahce dataset using the WT in conjunction with different machine learning models. The table includes various performance metrics for each model, which are important indicators of the model's predictive capabilities and generalization performance on unseen data. Based on the results, it can be observed that the ETR model demonstrates the best performance, achieving high r\({}^{2}\)_train and r\({}^{2}\)_test values of 98.5% and 84.8%, respectively. Additionally, ETR yields the lowest RMSE values for both training and test datasets, indicating superior predictive accuracy. The DT and RF models also exhibit strong predictive capabilities, while LR and SVR show relatively lower r\({}^{2}\) values and higher RMSE values. After analyzing the above studies, the results in Table 5 highlight the effectiveness of the ETR model when combined with the WT for predicting flow rates in the Melekbahce dataset. These findings can assist researchers in selecting the most appropriate machine learning approach to estimate flow rates accurately in similar environmental datasets.