# Hybrid Modeling for Stream Flow Estimation: Integrating Machine Learning and Federated Learning
## 4 Statistical Results
### Machine Learning Applications Results  
In this research, we performed an extensive analysis using five different machine learning approaches to estimate flow rates. The study utilized data from daily average flow rate measurements at the Besdegirmen and Melekbahce stream gauging stations, along with the average daily air temperature and total precipitation data of their respective regions. Table 4 and Figure 9 present the specific results obtained for the Besdegirmen station. The findings demonstrated that the ETR and RF methods achieved the highest r\({}^{2}\) test values, performing impressively with scores of 88% and 85.8%, respectively. These high r\({}^{2}\)_test values indicate the models' excellent performance in accurately predicting the flow rate based on the input features. Our study delved into a comprehensive exploration of five distinct machine learning methodologies for the estimation of flow rates. Leveraging daily average flow rate measure \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **r\({}^{2}\)\_train** \%** & **r\({}^{2}\)\_test \%** & **d\_train** & **d\_test** & **rmse\_train** & **rmse\_test** \\ \hline LR & 39.5 & 35 & 354.2 & 359.2 & 499.9 & 515.5 \\ SVR & 90.9 & 73.5 & 121.4 & 179.9 & 193.3 & 327.6 \\ DT & 98.1 & 70.2 & 45.2 & 184.3 & 85.6 & 349.1 \\ RF & 97.9 & 85.8 & 54.4 & 144.2 & 92.5 & 240.7 \\ ETR & 98.2 & 88 & 40.3 & 135.3 & 82.8 & 220.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of Besdegirmen training and test results. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Data Type** & **Mean** & **Minimum** & **Maximum** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & **Distortion** & **Change** & **Mode** & **Median** \\ \hline Daily average flow (m\({}^{3}\)/s) & 5.94 & 1.01 & 95.10 & 6.29 & 4.00 & 39.59 & 5.94 & 3.85 \\ Daily total rainfall (mm = kg/m\({}^{2}\)) & 1.46 & 0.00 & 92.20 & 4.50 & 5.80 & 20.24 & 0.00 & 0.00 \\ Daily average temperature (‘C’) & 10.77 & \(-\)14.10 & 29.80 & 8.25 & \(-\)0.20 & 68.13 & 20.10 & 11.10 \\ Daily average flow (m\({}^{3}\)/s) & 89.38 & 17.70 & 739.00 & 82.21 & 2.11 & 6758.33 & 36.20 & 54.20 \\ Daily total rainfall (mm = kg/m\({}^{2}\)) & 2.24 & 0.00 & 74.60 & 6.33 & 4.25 & 40.04 & 0.00 & 0.00 \\ Daily average temperature (‘C’) & 12.85 & \(-\)18.90 & 33.50 & 10.77 & \(-\)0.14 & 115.99 & 2.20 & 13.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical values of Besdegirmen dataset. Figure 9: Besdegirmen training and test results. ments from the Besdegirmen and Melekbahce stream gauging stations, as well as average daily air temperature and total precipitation data from their respective regions, our analysis was multi-faceted. The results, meticulously detailed in Table 4 and graphically depicted in Figure 9 for the Besdegirmen station, served as a mosaic, illustrating the performance of each method. The outcome of our analysis unveiled the striking prowess of the ETR and RF techniques. With r2_test values of 88% and 85.8%, respectively, ETR and RF displayed a marked capacity to predict flow rates with a commendable degree of precision. This substantiates the efficacy of these methodologies in capturing the intricate nuances within the dataset and attaining accurate estimations. The results of this study underscore the effectiveness of machine learning techniques in accurately estimating flow rates and emphasize the superiority of the ETR and RF methods in this particular scenario. These findings hold immense significance for water resource management and hydrological analyses, providing valuable insights into the potential of machine learning to address complex real-world challenges. By employing such advanced approaches, we can enhance our understanding of water flow patterns and support informed decision-making for sustainable water resource management. Table 4 presents a comparison of the training and test results for different machine learning models applied to the Besdegirmen dataset.