## 4 Results

Table 2: Summary of performance metrics \(r^{2}\), RMSE, and MAPE achieved in the estimation of the HFF slope with the Linear, Lasso, Ridge, Bayesian Ridge and Decision Tree regression models. Figure 3: Estimation results for the Linear, Lasso, Ridge, Bayesian Ridge and Decision Tree regression models. Left panel: distribution of estimated HFF slopes versus true HFF slopes. Middle panel: mean and standard deviation of the estimated HFF slope values for the different true HFF slope values. Right panel: distribution of residuals. On the basis of these results, we conclude that the best performance in estimating the analytical slope of the HFF is achieved by the DNN model M3. This conclusion is based on the following observations: (i) The linear correlation coefficient is, on average (0.75), higher than that of the other models. (ii) The standard deviations (STD) of the estimates are smaller than those produced by the remaining models. (iii) The residuals associated with the M3 DNN model are narrower compared to those produced by the remaining regression algorithms (this observation is present in all DNN models and further supports the conclusion that DNN models are more accurate in estimating the analytical slope of the HFF). These factors collectively indicate that DNN model 3 provides the most accurate estimates of the analytical HFF slope among the models considered in this study. Figure 4: Estimation results for the Deep Neural Network models M1 to M5. Left panel: distribution of estimated HFF slopes versus the true HFF slopes. Middle panel: mean and standard deviation of the estimated HFF slope values for the different true HFF slope values. Right panel: distribution of residuals. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **M1** & **M2** & **M3** & **M4** & **M5** \\ \hline \(r^{2}\) & 0.69 & 0.72 & 0.75 & 0.74 & 0.74 \\ RMSE [Hz/s] & 691.28 & 654.20 & 611.90 & 624.98 & 633.75 \\ MAPE [\%] & 0.28 & 0.26 & 0.21 & 0.17 & 0.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of performance metrics \(r^{2}\), RMSE, and MAPE achieved in the estimation of the HFF slope with the Deep Neural Network regression models M1 to M5. The findings presented in this section show that the DNN models Table 1 are more proficient in estimating the HFF slope present in CCSN GW signals. While linear models emphasize the identification of linear relationships among variables in the data set, DNN models employ nonlinear transformations within their internal layers, making them more accurate and adept at identifying and quantifying features within the data set. This characteristic elucidates why DNN architectures outperform linear models, as illustrated in Tables 2 and 3. Additionally, figures in Table 2 show that Linear, Lasso, Ridge, Bayesian Ridge and Decision Tree models yield identical numerical results for each performance metric implemented in this study, indicating similar performance in the estimation of HFF with minimal variation across implementations. The Decision Tree regression algorithm, however, is the model that delivers the least accurate results, visually identifiable at the bottom of Figure 3 by the broader prediction bands for each HFF slope in the data set. To establish a more detailed representation of these results, Figure 5 shows the distribution of performance metrics obtained with the Deep Neural Network regression models M1 to M5. These results clearly show that model M3 stands out as the best architecture among the other DNN models for estimating the slope of the HFF (boxplots in green). It is important to mention that DNN model 5 (represented by purple boxplots) displays distributions of RMSE and MAPE with reduced values; however, this model also shows lower values of \(r^{2}\). Hence, it cannot be chosen as the model demonstrating the highest performance.