## 4 Results and Discussion
### Discussion on the Performances of the Algorithms  
Table 3 presents a comparison of the predictive performances of different models evaluated in terms of Maximum Absolute Error (MAE) and Root-Mean-Squared Errors (RMSE). The best result with LCRM was an RMSE of around 0.432, while the best result with a conventional global model, NN, was an RMSE of about 1.229. This result demonstrates that the proposed LCRMs scheme with switching is superior to a single global nonlinear model.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**290 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**360 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**430 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**500 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**Average**} \\ \cline{2-11}  & **MAE** & **RMSE** & **MAE** & **RMSE** & **MAE** & **RMSE** & **MAE** & **RMSE** & **MAE** & **RMSE** \\ \hline Arrhenius equation & 12.872 & 13.296 & 0.769 & 0.903 & 2.052 & 2.324 & 2.367 & 2.769 & 4.515 & 6.904 \\ NN & 1.448 & 1.594 & 1.621 & 1.653 & 0.73 & 0.831 & 0.238 & 0.286 & 1.009 & 1.229 \\ LCRMs (ours) & 0.525 & 0.818 & 0.133 & 0.169 & 0.145 & 0.183 & 0.105 & 0.127 & 0.227 & 0.432 \\ \hline \hline \end{tabular}
\end{table}
Table 3: MAE and RMSE for three methods.  
Figure 6: True stress–strain curves by Neural Network at (**a**) 290 \({}^{\circ}\)C, (**b**) 360 \({}^{\circ}\)C, (**c**) 430 \({}^{\circ}\)C, and (**d**) 500 \({}^{\circ}\)C.  
In this research, the accuracy of the different models was further evaluated by calculating the correlation coefficient denoted as R, as well as the Average Absolute Relative Error (AARE). Figure 8 shows the correlation between the experimental value and the predicted value by different models along with the R and AARE values for each model. The R and AARE values of the LCRMs are observed to be superior compared to those of the Arrhenius equation and NN. This indicates that the LCRMs have the most accurate performance.  
Studies were performed to obtain the relative percentage errors of the three different models. Figure 9 shows the relationships between relative percentage error and relative frequency for the (a) Arrhenius equation, (b) NN, and (c) LCRMs. In each plot, the relative percentage errors were indicated by the mean error value \(\upmu\) and the standard deviation w. The absolute value of the mean of the relative percentage error by LCRMs was 0.029 while the absolute values of the mean obtained by the Arrhenius equation and NN were 2.381 and 3.165, respectively. The standard deviation of the relative percentage error obtained by the LCRMs was 0.998 while the standard deviations of the error derived by the Arrhenius equation and NN were 14.192 and 2.308, respectively. The results showed that the errors by the Arrhenius equation and NN were dispersed. The study of the relative percentage error indicates that the performances of the LCRMs were the most reliable compared to the Arrhenius equation and the NN. Meanwhile, it should be noted that the absolute value of the mean of the relative percentage error obtained by the traditional Arrhenius equation is smaller than that obtained by the NN, while the standard deviation of the relative percentage error obtained by the Arrhenius equation is larger than the standard deviation obtained by the NN. The current research implies that the presented LCRMs are superior to the Arrhenius equation and NN in terms of prediction error and reliability. Further, the modeling using the LCRMs showed the best efficiency.  
Figure 7: True stress–strain curves by LCRMs at (a) 290 \({}^{\circ}\)C, (b) 360 \({}^{\circ}\)C, (c) 430 \({}^{\circ}\)C, and (d) 500 \({}^{\circ}\)C.  
Figure 8: Correlation between experimental value and predicted value by (**a**) Arrhenius equation, (**b**) NN, and (**c**) LCRMs.  
Figure 9: Relative percentage error by (**a**) Arrhenius equation, (**b**) NN, and (**c**) LCRMs.