## 2 Materials and Methods
### Method
#### 2.2.3. Verifying the Model  
According to the experiment matrix and measured responses value shown in Table 3, a statistical model is generated that covers different parameters. The model is optimized using stepwise regression, in which forward and backward regression are combined. This method can be used to eliminate insignificant terms automatically. The significance of the model and the affecting terms are tested by the analysis of variance (ANOVA). The ANOVA test results for the sidewall penetration \(S_{p}\) are presented in Table 4. The F value is \(25.77>F_{0.01}(6,43)=3.29\), and the values of Prob \(>\) F is less than 0.0001, which implies that the model is significant. The \"Lack of Fit\" of 0.7966 implies that the Lack of Fit is not significant relative to the pure error, which means the model is accurate for sidewall penetration \(S_{p}\). In addition, the value of Prob \(>\) F is less than 0.05, indicating that the model terms are significant, and if the value of Prob \(>\) F is less than 0.001, it means it is the most significant model term. In this case, \(S_{p}\), \(V\), and \(O_{W}\) are the most significant terms in the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Std** & **Run** & \multicolumn{6}{c}{**Code Variables**} & \multicolumn{6}{c}{**Response Parameters**} \\ \hline  & & \(W_{f}\) & \(V\) & \(W_{s}\) & \(O_{W}\) & \(\alpha\) & \(S_{p}\) & \(D_{p}\) & \(W_{h}\) & \(C\) \\  & & \(W_{f}\) & \(V\) & \(W_{s}\) & \(O_{W}\) & \(\alpha\) & (mm) & (mm) & (mm) & (mm) \\ \hline  
\hlinemodel. Moreover, \(W_{s}\) has a significant impact on \(S_{p}\). The ultimate statistical model for sidewall penetration \(S_{p}\) in terms of actual factors can be expressed in the form below:  
\[\ln(S_{p})=-3.5273+0.117V-0.00151359W_{s}+0.24786O_{W} \tag{2}\]  
Table 5 shows the ANOVA test result for the welding penetration \(D_{p}\). The \(p\)-value (Prob > F) of the model is less than 0.0001, indicating that the model is significant, and the Lack of Fit indicates that the welding penetration model is accurate. Furthermore, the \(p\)-value (Prob > F) of model terms shows that \(W_{f}\), \(V\), interaction items of oscillation width and welding position (\(O_{W}\times\alpha\)), and second-order of \(W_{s}(W_{s}^{2})\) are significant terms in modeling the welding penetration \(D_{p}\). After stepwise regression to the model, \(W_{s}\) should be added to the model as a hierarchical term to support the significant term \(W_{s}^{2}\), and the same as \(O_{W}\). The ultimate statistical models for welding penetration \(D_{p}\) in terms of actual factors is shown below:  
\[ \begin{array}{c}\ln(D_{p})=-4.82304+0.088931W_{f}+0.033066V+0.016112W_{s}+0. 31161O_{W}+0.010437\alpha\\ -0.00385421O_{W}\times\alpha-1.93449\times 10^{-5}W_{s}^{2}\end{array}  \tag{3}\]  
The results of ANOVA for the welding height \(W_{h}\) model is presented in Table 6. The \(p\)-value of model terms shows that \(W_{s}\) has the most significant effect on the welding height. The interaction items of oscillation width and welding position (\(O_{W}\times\alpha\)), and the second-order term \(O_{W}(O_{w}^{2})\) are significant terms in modeling welding height \(W_{h}\). The F value and \(p\)-value indicate that the model is significant, and the Lack of Fit item demonstrates that the model of the welding height \(W_{h}\) is adequate. The ultimate statistical model for the welding height \(W_{h}\) in terms of actual factors is as the following:  
\[ \begin{array}{c}\ln(W_{h})=1.17036+0.074716W_{f}-0.00245086W_{s}+0.49667O_{W }-0.010674\alpha+0.00588373O_{W}\times\alpha\\ -0.24781W_{s}^{2}\end{array}  \tag{4}\]