## 5 Conclusions  
The current state of GW astronomy recognizes the HFF as a parameter present in all CCSN GW signals that can be related to the properties of the source associated with such emissions. A contemporary problem in this field is to determine the best method for estimating, at first-order approximation, the slope of the HFF. This study provides evidence to support the choice of a particular ML regression method, a DNN model, for this purpose. We conducted a study that compared the precision of different regression methods that better express the estimated slopes in the synthetic data set according to a supervised learning approach. Our results show that the best algorithm to characterize and determine the linear correlation between the training and testing data sets is provided by a DNN model. This model defines the best correlation between the variables considered in this study and has a fewer dispersion compared to the other ML model for the regression involved in this study. This study suggests that a DNN algorithm is reliable for estimating the slope of the HFF, which is more effective than the other methods discussed. Furthermore, the approach developed in this study, which uses analytical signals, could be extended to estimate the slope of the HFF in future studies. For example, in [43], the slope of the HFF is  
Figure 5: Distribution of performance metrics obtained with the Deep Neural Network regression models M1 to M5. The x-axis represents the model implemented based on the information in Table 1, while the y-axis shows the numerical results of each performance metric described in Section 3. The panel on the left displays the coefficient of determination (see Equation (15)), the middle panel shows the root mean square (see Equation (16)), and the panel on the right shows the MAPE (see Equation (17)). These values determine the most suitable model for estimating the slope of the HFF, with model 3 highlighted in a green box, demonstrating superior performance compared to other models developed for this study. estimated using an analytical data set to train a DNN model, which provided an explicit value of the parameter in real interferometric noise. An essential element of these computational studies aimed at developing robust and reliable computational instruments for estimating CCSN GW features lies in the future estimation of physical parameters from the GW source, using as input the numerical results, such as the slope of the HFF, obtained after successive evaluations of this parameter in hundreds of thousands of different LIGO noise realizations representing various scenarios for CCSN GW signal detection. The development, validation, and implementation of algorithms like the DNN models discussed in this paper will shape our ability in future research projects to efficiently derive astrophysical information from the GW source, providing insights into the physical structure of these objects, which have been thoroughly studied and characterized over the recent decade, with future prospects of detecting their signals in real interferometric noise. The research of [26] reveals that the estimation of the HFF slope is influenced by a variety of source parameters, such as the nuclear equation of state, mass, metallicity and rotation. This implies that the estimations made by certain regression techniques may be significantly impacted by these factors. This encourages the investigation of this topic by varying certain parameters and elucidating the effect that these parameters have on the estimation, which could provide a more accurate distribution of the estimations produced by the more reliable method. In contrast to other studies, our research provides numerical evidence to support the selection of a particular algorithm for regression over the most commonly used and applied methods. The description of these data can be used to guide future research in the field of GW astronomy. Conceptualization, A.C.-L., J.M.A. and C.M. ; methodology, A.C.-L., J.M.A., C.M. and R.F.-H.; software, A.C.-L. and J.M.A.