# Hybrid Modeling for Stream Flow Estimation: Integrating Machine Learning and Federated Learning
## 1 Introduction  
Water, an essential component that supports life, poses both difficulties and opportunities as humanity grapples with the intricacies of its existence. In light of the enormous availability of freshwater resources on Earth, the susceptibility of these resources to wastageand disturbance poses a significant issue [1]. The competition among many living forms for resources has disrupted the balance of natural water flow, posing a complex challenge to human society. Hydrology is a field in which the intricate fabric of scientific inquiry unravels resolutions to predicaments about water [2]. The field of study, characterized by continuous scientific progress and the accumulation of knowledge, possesses the essential means to oversee and regulate this invaluable resource effectively. The hydrological cycle, characterized by its repetitive patterns, presents possibilities for estimating and optimizing water flow [3]. Nevertheless, the complicated interaction between water and environmental variables, which is frequently impacted by human activities and pollution, adds a layer of intricacy to forecasting irregular river flows. This necessitates a thorough investigation [4]. The hydrological equilibrium of localized basins in varied places is further disrupted by the rush of industrialization and urbanization, which is driven by population expansion [5]. Against the backdrop of these dynamic changes, two significant global issues come to the forefront: climate change and the increasing density of the population [6]. Collectively, these actions have significantly impacted global water supplies, revealing the imminent threat of an escalating water catastrophe [7]. Although the Earth possesses extensive reserves of water, the availability of accessible freshwater is limited and unevenly distributed [8]. The Turkish setting is a prime example of this contradiction, as it possesses significant water potential yet faces challenges in using this vital resource effectively [9]. Turkey's landscape indicates the impact of water shortage since the average yearly water availability per person continues to decline [10]. Addressing this predicament requires thoroughly utilizing available resources through a complete examination. This necessitates the exploration of an extensive collection of hydrological and meteorological data, encompassing groundwater and surface runoff, humidity, wind patterns, evaporation rates, and precipitation [11]. The ability to predict groundwater levels and estimate future water potential originates from mathematical models that fall within the domain of time series and regression [12]. Using wavelet models to analyze time series data reveals a wide range of valuable insights into repetitive patterns, enhancing estimation accuracy [13]. In this particular context, the primary focus lies in the examination of data about the flow of rivers. This crucial undertaking serves as the foundation for producing precise forecasts and making well-informed decisions [14]. The accurate calculation of river flow plays a crucial role in facilitating the effective planning and operation of essential water infrastructure, including dams and culverts [15]. Predicting river flows during abnormal occurrences serves as a proactive measure against imminent calamity, allowing for precautionary measures to be taken to protect both human lives and valuable assets [16]. The convergence of the empirical domain with theoretical frameworks has led to the prominent use of regression models in predicting hydrological data. Additionally, machine learning algorithms like random forest (RF) and support vector regression (SVR) have emerged as powerful tools in this field [17]. The wavelet transform (WT) approach is highly effective in enhancing estimation accuracy in studying river-accented flow, as demonstrated in several studies [18]. This research examines the Tunceli Munzur sub-basin of the Upper Euphrates Basin and the Besdegrimen rivers within the Kutahya Porsuk Stream Basin [19].