## 6 Regulator Structure Optimization Analysis
### Analysis of Optimization Results  
The coefficients of the response surface model are shown in Table 4 and indicate a 95% confidence interval for each parameter. This interval can be used to determine the significance of each parameter. If the upper and lower bounds of the interval do not contain 0, then the term is significant. If it contains 0, the term is insignificant, and the corresponding parameter should be excluded. Whether the exclusion is determined will be further judged according to the analysis of variance (ANOVA). After obtaining the relationship between input variables, response values, and variables, the significance test of the regression using ANOVA is conducted to determine the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **Regression** & **Degree of** & **Standard** & **95\% CI Lower** & **95\% CI Upper** \\  & **Coefficient** & **Freedom** & **Error** & **Bound** & **Bound** \\ \hline intercept & 204.94 & 1 & 8.72 & 184.32 & 225.56 \\ distance & \(x_{1}\) & \(-\)5.02 & 1 & 6.90 & \(-\)8.32 & \(-\)1.71 \\ \(x_{2}\) & \(-\)3.20 & 1 & 6.90 & \(-\)4.51 & \(-\)1.90 \\ \(x_{3}\) & 19.90 & 1 & 6.90 & 3.60 & 36.21 \\ \(x_{1}\) \(x_{2}\) & 2.18 & 1 & 9.75 & 0.13 & 4.24 \\ \(x_{1}\) \(x_{3}\) & 7.48 & 1 & 9.75 & 1.43 & 13.54 \\ \(x_{2}\) \(x_{3}\) & 13.87 & 1 & 9.75 & 0.81 & 26.93 \\ \(x_{1}^{3}\) & \(-\)10.68 & 1 & 9.50 & \(-\)13.16 & \(-\)8.21 \\ \(x_{2}^{5}\) & \(-\)17.62 & 1 & 9.50 & \(-\)30.09 & \(-\)5.14 \\ \(x_{3}^{2}\) & \(-\)13.03 & 1 & 9.50 & \(-\)23 & \(-\)3.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Response surface model coefficients. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & **Number of** & **Sleeve Wall** & **Valve Hole** & **Outlet Valve Seat** & **Equivalent Stress** \\  & **Calculations (Times)** & **Thickness (mm)** & **Diameter (mm)** & **Wall Thickness (mm)** & **(MPa)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Test schemes and corresponding response values of the gas pressure regulator. significance of each item (Table 5). The value of F indicates the significance of the whole fitted equation, and the larger the F, the more significant the equation and the better it fits. However, to determine the significance of the variables, the value of \(p\) must be analyzed. The \(p\) of the regression model measures the magnitude of the difference between the control and experimental groups. If \(p\) is less than 0.0001, it indicates that the choice of the quadratic polynomial for the response surface model is very reasonable and that the relationship between the regression equation and the equivalent stresses is significant. It also indicates that the presence of variables has a positive effect on the significance of the model. The regulator model is analyzed to verify the applicability of the model using a residual-based diagnostic plot obtained from regression analysis methods. A normal probability plot is considered the ideal normal probability if the reference points are distributed along a straight line. Figure 8 shows the residual normal probability plot, which shows that the reference points are uniformly distributed on both sides of the straight line. Although individual reference points can be observed with the offset, the impact on the model is not significant or acceptable. Hence, the data for calculating the results of the equivalent stress in this paper are accurate and meet the requirements. Figure 8: Positive probability of residual error.