## 3 Design of Experiment
### Design of Experiment Steps
#### 3.1.5 Execute Design of Experiment and Analysis Data
The coefficient of determination (\(R^{2}\)) and residual are the indices used to determine whether the response surface modeled through the central composite design is appropriate. The coefficient of determination is used as an indicator when evaluating the fitness of a regression model for appropriateness, or determining the relationship between a target design parameter and an independent design parameter. The formula of the coefficient of determination is as follows [23]. \[R^{2}=1-\frac{\sum\left(Y_{j}-\hat{Y}_{j}\right)^{2}}{\sum\left(Y_{j}-\overline {Y}\right)^{2}} \tag{24}\]  
where \(Y_{j}\) is the exact response, \(\hat{Y}_{j}\) is the predicted response, and \(\overline{Y}\) is the mean value of the exact response. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **MTOW** & **P/W** & **Span** & **L/D Cruise** \\ \hline Intercept & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ Range & 0.9853 & 0.5335 & 0.9975 & 0.9830 \\ Max speed & 0.9706 & 0.7371 & 0.9943 & 0.9715 \\
**Cruise speed** & 0.1127 & 0.8855 & \textless{}0.0001 & \textless{}0.0001 \\
**Cruise altitude** & \textless{}0.0001 & 0.8855 & \textless{}0.0001 & 0.0038 \\
**Passengers** & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\
**Baggage** & \textless{}0.0001 & \textless{}0.0001 & 0.0037 & \textless{}0.0001 \\ Rate of climb & 0.9853 & 0.8104 & 0.9982 & 0.9866 \\
**Stall speed** & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ Service ceiling & 0.9853 & 0.8855 & 0.9992 & 0.9953 \\ Turn speed & 0.9853 & 0.5335 & 0.9992 & 0.9953 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Independent design parameter \(p\)-value. However, in general, the coefficient of determination tends to increase as the number of design independent parameters increases, so it may not be an appropriate standard. Therefore, it is necessary to introduce a new indicator, the adjusted coefficient of determination (\(AR^{2}\)). The adjusted coefficient can be supplemented by considering the size of the sample and the number of independent design parameters. The equation of the adjusted coefficient of determination is as follows [23]. \[AR^{2}=1-\frac{\Big{(}1-R^{2}\Big{)}(n-1)}{(n-k-1)}=\frac{(n-1)R^{2}-k}{n-k-1} \tag{25}\]  
where \(n\) is the sample size and \(k\) is the number of independent design parameters. The adjusted coefficient of determination has a value between 0 and 1, and it can be judged that the closer to 1, the better the relationship between the target design parameter and the independent design parameter. The coefficient of determination and the adjusted coefficient of determination are shown in Table 16 for a Vahana type vehicle. The graph of the residual is shown in Figure 15. In the residual graph, if the results are clustered or if there is a specific pattern, it can be considered a poor-fit. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Objective Design Parameter** & \(R^{2}\) & \(AR^{2}\) \\ \hline MTOW & 0.99 & 0.99 \\ P/W & 0.99 & 0.99 \\ Span & 0.99 & 0.99 \\ L/D cruise & 0.99 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: \(R^{2}\), \(AR^{2}\) of response surface method. Figure 14: Response surface method graph. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Quantiles**} & \multicolumn{2}{c}{**Predicted Value [-]**} & \multicolumn{2}{c}{**Objective Value [-]**} \\ \hline
100.0\% & Maximum & 14.0 & \\
75.0\% & Quartile & 12.6 & \\
50.0\% & Median & 12.1 & \\
41.9\% & & 12.0 & \\
25.0\% & Quartile & 11.6 & \\
0.0\% & Minimum & 10.0 & \\ \hline \hline \end{tabular}
\end{table} 
Table 20: L/D cruise percentile. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Independent Design Parameter** & **Desirability** & **Optimized Value** \\ \hline Cruise speed & \(-\)0.095 & 109.05 [kt] \\ Cruise altitude & 0.079 & 7697.5 [ft] \\ Passengers & 1 & 3 [PAX] \\ Baggage & 1 & 120 [lb] \\ Stall speed & 1 & 40 [kt] \\ \hline \hline \end{tabular}
\end{table} 
Table 21: Desirability function.