# Hybrid Modeling for Stream Flow Estimation: Integrating Machine Learning and Federated Learning
## 3 Material and Experimental Methods
### Support Vector Regression  
SVR stands as a sophisticated and specialized machine learning model, renowned for its ability to generate regression within specific data ranges. As discerned in Figure 4, data points that fall outside this predetermined range are deemed irrelevant and are omitted from the SVR analysis. A salient feature of SVR lies in its adeptness at constructing individual vectors, emanating from the system's origin, to encapsulate the data points residing outside the established range. At its core, SVR operates on the tenet of structural risk minimization, seeking not only to minimize empirical risk but also to curtail the upper bound of generalization error. This principle ensures the model's enhanced ability to generalize and make accurate predictions on unseen data. As a testament to its exceptional prowess, SVR has proven its mettle across a myriad of diverse applications, reinforcing its significance in the realm of advanced machine learning models. Within the ambit of our study, we further augmented  
Figure 4: Threshold and error definition in the SVR method for an example scatter plot. the SVR model's capabilities by blending it with the formidable radial basis function (RBF) kernel. This blending process is instrumental in calculating the proximity and similarity between two data points, a critical aspect that underpins SVR's predictive capabilities in our research endeavors [58; 59; 60]. The judicious combination of SVR and the RBF kernel fortifies the model's capacity to capture intricate patterns and relationships within the data, culminating in a potent tool for accurate prediction and forecasting in our study. The practical applications of SVR using the RBF kernel demonstrate its effectiveness across several circumstances. The potential applications of this technology include estimating real estate prices based on property qualities, analyzing film ratings based on reviews, evaluating product demand based on pricing dynamics, and forecasting a company's sales trajectory. In situations when the data landscape exhibits non-linearities or is affected by noise, SVR with the RBF kernel is a suitable option that should be taken into account and put into practice. When confronted with complex non-linear data or noisy data, it is appropriate to consider using SVR with the RBF kernel. This choice deserves careful consideration and practical implementation. In situations where the data environment is characterized by complex non-linear relationships or vulnerability to disruptive noise, the application of SVR with the RBF kernel proves to be a suitable option, deserving thorough evaluation and purposeful deployment. Figure 4 illustrates the threshold and error definition within the SVR method through the depiction of an exemplary scatter plot. In this context, SVR endeavors to generate a regression model that accurately predicts the relationship between two variables. The scatter plot showcases data points representing the relationship between the independent variable (X-axis) and the dependent variable (Y-axis). SVR's primary objective is to fit a regression line that best represents this relationship. However, it is essential to define specific data ranges that contribute to the generation of the regression model while neglecting data points outside these ranges. The threshold in Figure 4 delineates this defined data range, segregating the data points that fall within its boundaries from those that lie outside. Data points within the threshold are considered relevant and influential for building the regression model. Conversely, data points outside the threshold are treated as outliers and are not factored into the model's construction. The error bars in the figure represent the magnitude of error associated with each data point concerning the threshold. SVR seeks to minimize these errors to achieve a more accurate regression model. The error bars indicate how much each data point deviates from the regression line, with larger error bars signifying higher discrepancies.