# Hybrid Modeling for Stream Flow Estimation: Integrating Machine Learning and Federated Learning
## 4 Statistical Results  
For this research study, we utilized data from two stream gauging stations, namely Besdegirmen (E12A003) and Melekbahce (E21A033), to estimate stream flow rates using the federated learning approach. The Besdegirmen station is located in the Porsuk Stream sub-basin of the Sakarya Basin, while the Melekbahce station is situated in the Munzur water sub-basin of the upper Euphrates basin. At the Besdegirmen station, the dataset comprised daily average flow, daily average precipitation, and daily average air temperature data, spanning 1 October 1979 to 30 September 2022, with a total of 14,974 daily data points. For model training, 10,593 daily data points from 1 October 1979 to 30 September 2008 were used, and 5113 daily data points from 1 October 2008 to 30 September 2022 were employed for testing. Table 1 presents the statistical values derived from this dataset, providing valuable insights into the characteristics of the data. Similarly, at the Melekbahce station, the dataset included daily average flow, daily average precipitation, and daily average air temperature data, covering the period from 1 October 1986 to 30 September 2022, with a total of 8766 daily data points. For model training, 7305 daily data points from 1 October 1986 to 30 September 2006 were used, and 1461 daily data points from 1 October 2006 to 30 September 2022 were employed for testing. Table 2 presents the statistical values derived from this dataset, offering key insights into the data distribution and characteristics. By leveraging the federated learning approach, we aim to build predictive models for both stations while ensuring data privacy and collaboration between the stations. The statistical results obtained from these datasets will serve as the foundation for developing accurate and robust stream flow rate estimation models using advanced machine learning algorithms. Table 1 presents a comprehensive overview of the data collected from two stream gauging stations, namely Besdegirmen (E12A003) and Melekbahce (E21A033). The table contains essential parameters, including the average flow rate (Favg), average air temperature (Tavg), and total precipitation (Psum), recorded at each station on different dates. The \"Station\" column indicates the name or code of the respective station, while the \"Date\" column provides the specific date of the recorded data. The \"Average Flow Rate\" column represents the daily average water flow in the stream, measured in cubic millimeters per day (mm\({}^{3}\)/day). Additionally, the \"Average Air Temperature\" column displays the daily average temperature recorded at each station, measured in degrees Celsius (\({}^{\circ}\)C), while the \"Total Precipitation\" column shows the total amount of rainfall received on each date, measured in millimeters (mm). These valuable data will be employed to develop a predictive model for stream flow rate estimation, considering the interplay between flow rate and environmental factors such as air temperature and precipitation. The proposed federated learning approach will combine data from both stations, preserving data privacy while achieving accurate stream flow rate predictions. Table 2 presents a comprehensive overview of the federated learning process employed to estimate stream flow rates using data from two stream gauging stations, Besdegirmen (E12A003) and Melekbahce (E21A033). The table includes essential columns providing information on the iteration round, station name, the number of local data samples, local model training loss, and model updates (gradients) obtained after local model training for each station. During Round 1, both stations contributed their data for model training, resulting in specific model updates. Round 2 continued the process with updated models, showcasing further improvements. By collaboratively updating the global model while maintaining data privacy, the federated learning approach enhanced the accuracy and