# Hybrid Modeling for Stream Flow Estimation: Integrating Machine Learning and Federated Learning
## 3 Material and Experimental Methods
### Federated Learning for Stream Flow Rate Estimation
This ingenious approach preserved the privacy of sensitive information related to flow rate, temperature, and precipitation, ensuring that such data remained securely stored at each station. Following the completion of local model training, the central server aggregated the model updates received from each station. An updated global model was derived by calculating the weighted average of the model parameters. This global model, enriched with insights from both stations, was then deployed back to each station for additional local training, continuing the iterative process. The success of the federated learning approach was contingent on evaluating the model's performance on each station's local validation set. This evaluation allowed us to monitor the model's convergence and assess its generalization capability. By embracing the federated learning approach, we harnessed the collective knowledge from both stream gauging stations while ensuring the confidentiality and decentralized nature of the sensitive data. Consequently, this method yielded a more robust and accurate stream flow rate estimation model compared with traditional centralized approaches that mandate the sharing of raw data. The advantages of employing FL-SFRE is manifold. Notably, this approach significantly enhances data privacy and security since it obviates the need for centralizing data collection and processing. Moreover, the collaboration fostered by this decentralized learning paradigm enables the utilization of data from diverse geographical locations, thereby capturing a broader spectrum of hydrological characteristics. This unique amalgamation of strengths from multiple stations resulted in an efficient model for accurately estimating stream flow rates, making it highly applicable in real-world water resource management and environmental monitoring scenarios. The federated learning approach holds promise for the sustainable management of water resources, where accurate and confidential estimation of stream flow rates is crucial for informed decision-making and environmental preservation. While this study exclusively employs a pair of predictors, the integration of multivariate methodologies remains profoundly advantageous. Such techniques hold the capacity to discern intricate interactions interwowoven within the two predictors. A prime illustration resides in the utilization of the SVR algorithm, which adeptly captures elusive non-linear relationships that might exist between the predictors. This augmentation significantly bolsters predictive accuracy. Moreover, the realm of multivariate methods extends its utility to the realm of managing incomplete datasets. The RF algorithm, for instance, serves as an adept tool for the imputation of absent values. This facet gains paramount significance in scenarios where datasets are plagued by substantial gaps in information. Through judicious imputation, the predictive accuracy witnesses a marked enhancement. In the broader context, the adoption of multivariate methods bestows a dual gift of precision and resilience upon predictive models. The consequential amelioration in predictions' fidelity and dependability underscores the continued merit of employing such techniques, even in cases where the model revolves around a mere two predictors. In the domain of predictive modeling, the very essence of multivariate methods encapsulates a dynamic interplay between various variables, granting a more comprehensive grasp of the underlying mechanisms. Within this interwoven fabric, intricate relationships often evade the discernment of univariate approaches, thereby amplifying the significance of their inclusion. Furthermore, the incorporation of multivariate methodologies unveils an avenue to combat the inherent incompleteness pervasive in real-world datasets. By resorting to intricate algorithms such as the random forest, the void left by missing data points can be artfully filled.