## 4 Results and Discussion
### Polynomial Regression Analysis and Response Surface Analysis
The response surface trajectory in the case of the leveling phase proves that feature \(F_{C}\) is more informational in predicting the condition changes in comparison to other features, as shown in Figure 9. The analysis shows a second-degree correlation in the polynomial dependence between the response error \(\Delta X_{C}\) and feature \(F_{C}\) in cases (a) and (b). In the test results for the initial frictional conditions (a) and the changed friction conditions (b), \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **RMSE** & \(R^{2}\) & **SSE** \\ \hline Case (a) & 0.05127 & 0.9920 & 7.8 \\ Case (b) & 0.04985 & 0.9924 & 7.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation metrics of regression models for the bending phase. Figure 8: Prediction surface plots for the bending phase: (**a**) initial friction conditions; (**b**) changed friction conditions. the positive gradient shows that a higher value of feature \(F_{C}\) is associated with a higher response error \(\Delta X_{C}\). This correlation is to be expected, as the higher load on the hydraulic press affects the intensity of the operating condition and thus causes a higher response error \(\Delta X_{C}\), ranging between +0.55 mm and +1.25 mm. The results indicate that the data points in case (b) correspond to three distinct what-if scenarios. For each scenario, three repetitions were conducted, with each repetition examining a specific condition of the hydraulic press. In the leveling phase, the forming force \(F_{C}\) has an overall impact on the hydraulic press response error \(\Delta X_{C}\) of the hydraulic press, so that a one-dimensional second-order polynomial function is selected (Equation (15)). \[\Delta X_{n}=\beta_{0}F_{n}^{2}+\beta_{1}F_{n}+\beta_{2} \tag{15}\]  
In the leveling phase, the evaluation metrics RMSE, \(R^{2}\), and SSE show a lower prediction performance in comparison to the bending phase, which is shown in Table 3. Furthermore, the regression models (a) and (b) built with experimental data yield a metric values \(R^{2}\) of 0.9539 and 0.9484. In addition, the SSE values of the metric are higher in the leveling phase than in the bending phase, which is due to the less stable operating condition of the hydraulic press in the leveling phase. However, the RMSE metric shows minimal deviation of the predicted values from the median value. Moreover, the prediction performance of the polynomial regression models is higher in case (b) in comparison to the prediction performance of the polynomial regression models in case (a). In the movement phase, polynomial regression and response surface analysis show the importance and relevance of two specific features in predicting condition changes in the hydraulic press. In addition, the input features \(v_{C}\) and \(F_{C}\) accurately reflect the tendency and sensitivity of the response \(\Delta X_{C}\) shown in Figure 10. In addition, feature \(F_{C}\) has a smaller impact on the response error \(\Delta X_{C}\) compared to feature \(v_{C}\) under conditions of low friction intensity and absence of a forming process during the movement phase. Alternatively, the positive gradient for feature \(v_{C}\) indicates that a higher value of feature \(v_{C}\) leads to an increasing value of the response error \(\Delta X_{C}\) and ranges from \(-\)4.94 mm to +5.85 mm. Analyzing the prediction surface plots for the experimental result shows that the relationship between the response \(\Delta X_{C}\) and the feature \(v_{C}\) follows a third-degree polynomial correlation. In addition, both the forming force \(F_{C}\) and the hydraulic cylinder velocity \(v_{C}\) have an effect on the response behavior of the hydraulic press during the movement phase. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **RMSE** & \(R^{2}\) & **SSE** \\ \hline Case (a) & 0.03858 & 0.9539 & 9.0 \\ Case (b) & 0.04089 & 0.9484 & 10.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation metrics of regression models for leveling phase.