## 4 Results  
In this section, we present the results of our analyses, which focus on estimating the HFF slope using a data set of analytical CCSN GW signals embedded in noise. To assess the precision and effectiveness of our estimation process, we evaluate the performance of the different regression models introduced in previous sections. The results of the implementation of the HFF slope estimation for the Linear, Lasso, Ridge, Bayesian Ridge, and Decision Tree regression models are presented in Figure 3. The left panel shows the distribution of estimated HFF slopes (on the y-axis) versus the true HFF slopes (on the x-axis). These plots are used to correlate the true analytical slope values of HFF (represented as \(s\)) with their respective estimated slopes (represented as \(\hat{s}\)). In every figure, the coefficient of determination value is reported in the top left corner, providing a quantitative measure of the model's performance in capturing this correlation. These results show that these regression models produce a highly diverse range of estimated HFF slopes. This indicates that these models provide high variability in the estimations of the HFF slope. Note that around 2000 Hz/s, the Linear, Lasso and Ridge algorithms reveal a distribution of estimations that deviates significantly from the true value, indicating higher variability and thus lower accuracy. The Decision Tree algorithm at the bottom shows a wider distribution of estimated values, indicating a less precise model for analyzing the slope of the HFF. The central panel of Figure 3 displays the error in estimating the slope of the HFF based on the estimated mean and standard deviation (std), that is, \(\hat{s}\pm std\). This is represented by the black vertical bars, visually representing the extent to which the estimated slopes deviate from the estimated mean HFF. Bars representing estimates with greater uncertainty are identified by having a greater distance from the dashed black line compared to narrower bars. Note that all models provide average estimations around the true HFF with some bias and more variability at the extremes of the true values (i.e., at the lower and higher values of the true HFF). Finally, the right panel in Figure 3 shows the distribution of the residuals, \((s-\hat{s})\). These histograms represent the HFF slope errors provided by the models. These plots demonstrate a Gaussian shape, except for the Decision Tree algorithm. This algorithm exhibits a different and less favorable distribution compared to the other models. The organization depicted in Figure 3 is mirrored in Figure 4, showcasing the outcomes of our investigation across the five distinct architectures (refer to Table 1) of DNN algorithms used for regression to predict the HFF slope. In contrast to the results with the Linear, Lasso, Ridge, Bayesian Ridge and Decision Tree, these results show a lower dispersion in the distribution of the predicted slopes (left panel) and lower standard deviation around the estimated mean (central panel) in the five DNN models. In particular, the DNN model M3 with 64 neurons in the input layer demonstrates superior performance compared to the other DNN architectures, as it presents more robust estimations characterized by a narrow distribution around the mean. Furthermore, the standard deviation bars in the central panel reveal a lower variance compared to the other architectures and regression models. Table 2 shows the results of the performance metrics (\(r^{2}\), RNSE, MAPE) for the Linear, Lasso, Ridge, Bayesian Ridge and Decision Tree models, while Table 3 shows such metrics for the DNN models. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Model** & **Linear** & **Ridge** & **LASSO** & **Bayesian Ridge** & **Decision Tree** \\ \hline \(r^{2}\) & 0.70 & 0.70 & 0.70 & 0.70 & 0.66 \\ RMSE [Hz/s] & 648.09 & 691.44 & 676.18 & 678.04 & 719.09 \\ MAPE [\%] & 0.29 & 0.29 & 0.29 & 0.17 & 0.24 \\ \hline \end{tabular}
\end{table}