## 7 Experimental Data and Model Validation
### Validation Method Durbin-Watson Test and Adjusted R\({}^{2}\)  
Figure 22 shows a quartile graph in which the drag residuals follow a line (at 45 degrees), indicating that its \"different\" values are distributed according to a normal law. The Durbin-Watson test predicts whether there is correlation in the order of appearance of the residuals for the drag force; given that the probability is 0.016% and less than 0.05%, there is evidence of correlation at a confidence level of 95%. We can reject the H\({}_{0}\) hypothesis (H\({}_{0}\) = there is no correlation between the residual drag values). The data show in Figure 23 show that the residuals for the lift force follow a normal distribution, and the histogram graph indicates a leftward bias. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test** & \
\begin{tabular}{c} **Residual Lift** \\ **(N)** \\ \end{tabular} &
\begin{tabular}{c} **Residual Drag** \\ **(N)** \\ \end{tabular} \\ \hline
1 & 0.35 & 0.00 \\  
16 \\
28 & 0.01 & 0.20 \\
29 & 0.05 & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Residual values of the experimental and simulated lift and drag forces. Linear regression analysis with adjusted R\({}^{2}\) is a statistical process that focuses on the relationships between sets of variables; in this case, the regression analysis allows for an estimation of the difference between the model's predicted values and the experimental values. The predicted and observed data are numerically sorted from small to large; the model data are placed on the x-axis and the observed data on the y-axis [45,46]. A line with the observed data on the y-axis is shown in Figure 23.  
Figure 23: Analysis of the lift forces’ residual values between the wind tunnel and model data. Figure 22: Analysis of the drag forces’ residual values between the wind tunnel and model data. a 45\({}^{\circ}\) slope is added to the plot where the dispersed data are shown. This line represents the ideal outcome where all the predicted values are 100% identical to the observed values. The magnitude of the correlation between the response and the predicted values can be quantified by calculating the adjusted R\({}^{2}\) (R-squared) of the model. This value denotes the proportion of the variance in the experimental data, which is predicted by the model data. An adjusted R\({}^{2}\) of 1 indicates that the model predicts all the experimental data perfectly, but in the real world, the models are never perfect, and the adjusted R\({}^{2}\) values may take any value from 0 to less than 1. This method shows in Figure 24 that the model predicted the lift and drag forces values within the 95% bounds for the linear regression method. The adjusted R\({}^{2}\) values are 0.9986 for the lift and 0.9945 for the drag forces, thus indicating that the predictions of the variance for the lift force are 99.86% and 99.45% for the drag force, as accounted for by the model predictions and as seen in Figures 22 and 23, respectively. To validate a statistical hypothesis on a group of data, if we set H\({}_{0}\) as the observed values and the simulated values from the same population, it is possible to assess at a confidence level of 95% whether this hypothesis is rejected or accepted. The weakness of this type of validation is that it is qualitative and not quantitative. This type of validation is mostly used to reject a model rather than to validate it at a high level of confidence [47]. Figure 24: Analysis of the lift and drag data with the adjusted R\({}^{2}\) validation method. The adjusted R\({}^{2}\) for the drag forces are strong and indicate that 87.24% of the experimental data can be explained by the model. The Durbin-Watson test makes it possible to check the correlation between the observed drag values and the simulated drag values. In this case, the test has a \(p\)-value much lower than its 0.05% threshold, which indicates a strong correlation. This hypothesis test does not reject H\({}_{0}\); so, the model is adequate for estimating drag forces.