## 4 Results and Discussions
### Engineering Problems  
In addition to numerical problems, one engineering problem, i.e., the prediction of the thrust (TH) on the whole rotor of a small aerial vehicle (UAV), is also employed to investigate and compare the model selection performance of MOA-CV and LOO-CV methods. The rotor blade and airfoil of a small (UAV) are shown in Figure 7. The whole rotor consists of three blades. In this problem, we focus on the thrust (TH) on the whole rotor. The airfoil (as shown in Figure 7b) is generated by using the class shape function transformation (CST) [26] which is a parametric modeling method. The airfoil has 16 parametric modeling variables. Six structural variables are presented in Figure 7a. Hence, the rotor blade of the UAV totally has 22 variables listed in Table 10. For the 16 parametric modeling variables, please see Reference [26]. Two sections are shown in Figure 7, i.e., blade tip and maximum-chord sections. \(l_{tip}\) and \(\varphi_{tip}\) are chord length and mounting angle of the blade tip section, respectively. \(l_{max}\) and \(\varphi_{max}\) are chord length and mounting angle of the maximum-chord section, respectively. The position of maximum-chord section is represented by \(d_{mt}\). \(\varepsilon\) is the forward sweep. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & \(\mathbf{5n}\) & \(\mathbf{8n}\) & \(\mathbf{10n}\) \\ \hline LOO-CV & 50\% & 50\% & 50\% \\ MOA-CV & 50\% & 25\% & 50\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Accurate rate of selecting best models by mean of LOO-CVerror and MOA-CVerror. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & \(\mathbf{5n}\) & \(\mathbf{8n}\) & \(\mathbf{10n}\) \\ \hline LOO-CV & 50\% & 75\% & 75\% \\ MOA-CV & 50\% & 75\% & \(\mathbf{100\%}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Accurate rate of removing worst models by mean of LOO-CVerror and MOA-CVerror. Figure 7: Rotor blade. (**a**) rotor blade and structural variables; (**b**) airfoil of rotor blade. We use the LHS approach built-in MATLAB to randomly generate 220 samples and get the corresponding TH. Then we choose 110, 176 and 220 (i.e., \(5n\), \(8n\) and \(10n\), where \(n\) is the number of variables) samples to construct PRS, RBF-MQ, RBF-TPS and KRG, respectively. The MOA-CV and LOO-CV methods are employed to select the most accurate and the worst accurate models. Then other 50 samples are generated for the calibrated criterion, i.e., \(R^{2}\). The results of \(R^{2}\), MOA-CVerror and LOO-CVerror are shown in Tables 11-13, respectively. The best model presented by each criterion are in bold; the worst model are shown in italic. From the result of \(R^{2}\) we can see that RBF-MQ has the best accuracy and PRS has the worst accuracy. Results of MOA-CVerror indicate RBF-MQ is the most accurate when the sample sizes are \(5n\) and \(10n\), which is consistent with those of \(R^{2}\), while results of LOO-CVerror show KRG is the most accurate, which is different from the results of \(R^{2}\). However, when the sample size is \(8n\), MOA-CV cannot select the best model, while LOO-CV can choose the best one. Both the MOA-CVerror and the LOO-CVerror indicate that the PRS is the worst model regardless of sample size. Overall, in this case, both the MOA-CV and the LOO-CV can choose the worst models, and in most cases, MOA-CV is likely to find the best model than LOO-CV.