## 3 Regression Analysis
### Identification of the Validity of the Regression Equations  
Confirmatory experiments have been conducted to identify the validity of the developed regression equations by comparing the experimental results with the predicted values. The results are listed in Table 5, where percent deviation is calculated by,  
\[\text{Percent~{}Deviation}=\frac{\text{EV}-\text{PV}}{\text{EV}}\times 100\% \tag{13}\]  
where EV is the experimental value; PV is the predicted value. From the results, it can be found that the predicted values by regression equations fit the experimental results within a reasonable error. Therefore, the regression equations can be used to predict the response of tensile properties to \(f_{\rm L}\)/\(f_{\rm H}\), and \(\psi\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Num.** & \(f_{\rm L}\)**/\(kHz\)** & \(f_{\rm H}\)**/\(kHz\)** & \(\psi\)** & **EV** & **PV** & **Percent Deviation** \\ \hline \hline
**Tensile strength** & & & & & & \\ \hline
1 & 0.2 & 10 & 0.98 & 324 & 327.7 & 1.2 \\
2 & 0.2 & 30 & 0.98 & 343 & 331.4 & 3.3 \\
3 & 0.7 & 10 & 4.2 & 330 & 332 & 0.6 \\
4 & 0.7 & 30 & 4.2 & 341 & 331 & 2.9 \\ \hline \hline
**Yield strength** & & & & & & \\ \hline
1 & 0.2 & 10 & 0.98 & 213 & 215.6 & 1.3 \\
2 & 0.2 & 30 & 0.98 & 223 & 232.7 & 4.2 \\
3 & 0.7 & 10 & 4.2 & 219 & 226.3 & 3.2 \\
4 & 0.7 & 30 & 4.2 & 240 & 242.1 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Validation of regression models by HPV-GTAW for AA 5456 aluminum alloy. The response surfaces and contour plots of percent elongation and ratio of reduction in area are presented in Figures 7-10, respectively. It is observed that \(f_{\rm L}\) and \(f_{\rm H}\) regarding to both the percent elongation and the ratio of reduction in area own the same changing tendency as tensile strength. As a consequence, 1 kHz and 20 kHz are the optimum parameters of \(f_{\rm L}\) and \(f_{\rm H}\), respectively. However, as shown in Figures 8 and 10, the response of percent elongation and the ratio of reduction in area versus  
Figure 4: Contour plot of surface on tensile strength (\(TS\)): (**a**) contour plot of surface that \(f_{\rm L}\) is 1 kHz; (**b**) interaction of \(f_{\rm H}\) and \(\psi\). Figure 5: Response surface on yield strength (\(TS\)). Figure 6: Contour plot of surface on yield strength (\(YS\)): (**a**) contour plot of surface that \(f_{\rm L}\) is 1 kHz; (**b**) interaction of \(f_{\rm H}\) and \(\psi\). \(\psi\) show completely different changing tendency. The response values are at the maximum when \(\psi\) is 0.6 under the condition that \(f_{\rm L}\) is 1 kHz and \(f_{\rm H}\) is 20 kHz. As a result, the optimum parameter of \(\psi\) is 0.6. In general, the optimum parameters on both percent elongation and the ratio reduction in area are \(f_{\rm L}=1\) kHz,\(f_{\rm H}=20\) kHz and \(\psi=0.6\) (i.e., \(I_{\rm b}\) is 75 A, \(I_{\rm p}\) is 180 A and \(\delta\) is 0.2). Figure 8: Contour plot of surface on percent elongation (_PE_): (**a**) contour plot of surface that \(f_{\rm L}\) is 1 kHz; (**b**) interaction of \(f_{\rm H}\) and \(\psi\). Figure 7: Response surface on percent elongation (_PE_). Figure 9: Response surface on ratio of reduction in area (_RA_). The optimum parameter of \(\psi\) to tensile strength and yield strength is 7.33, while that to percent elongation and ratio of reduction in area is 0.6. With every factor considered comprehensively, it can be concluded that plastic properties, including percent elongation and ratio of reduction in area, play a more important role in tensile properties than tensile strength because all the values of tensile strength in these experiments are more than 300 MPa, which is high enough in most applications. Plastic properties restrict the further improvement of mechanical properties of AA 5456 aluminum alloy [4]. Therefore, take 0.6 as the optimal parameter of \(\psi\) when plastic properties are at maximum.