# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 4 Applications
### Launch Vehicle Design Problem
#### 4.2.3 Results Analysis and Discussion  
The proposed approach is repeated 10 times with different initial dataset \(\mathcal{U}_{M}\) and \(\mathcal{X}_{M}\) to evaluate the robustness of the proposed method to the initial dataset. For one representative repetition, Figure 16 presents the initial (upper left) and final (lower left) output stochastic process samples corresponding to the launch vehicle altitude as a function of percent of flight. Moreover, the centered altitude process samples are illustrated on the figures of the right. It can be seen that the added samples (in dashed red) are in the higher region of the samples in order to refine the 99% quantile of the altitude. Therefore, the refinement strategy selected appropriate new samples to improve the estimation of the 99% quantile.  
For one representative repetition, Figure 17 presents the pairplot associated with the initial and added samples for the input uncertain variables in dimension 7. The added samples by the refinement strategy are selected in specific regions of the input uncertain variable domain. For instance, trajectories with higher altitude result from the uncertainty on the specific impulse of the 2 nd stage that are above 1.5 s offering higher rocket engine efficiency. Similarly, the negative uncertainty on the residual mass of the structures of stages 1 and 2 allows trajectories with higher altitude. Therefore, the active-learning strategy  
Figure 16: Initial and final output stochastic process samples for the altitude (**a**,**c**) and the centered altitude (**b**,**d**).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number of KL Modes** & **2** & **5** & **10** & **20** & **40** \\ \hline \(Q2\) & \(-42.9737\) & 0.7874 & 0.9900 & 0.9999 & 1.0000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Predictivity factor \(Q2\) as a function of the number of KL modes.  
appropriately selects input uncertain variable realizations to be added to the surrogate model dataset to improve the 99% quantile estimation.  
For one representative repetition, the estimated 99% quantile before and after refinement are presented in Figure 18. It can be seen that the uncertainty on the quantile estimation is reduced after the enrichment of the dataset and that the estimated quantile converged to the validation quantiles.  
In Figure 19, boxplots of the 10 repetitions of the active-learning process are presented for the uncertainty on the quantile estimation (left) and the root mean square error (RMSE)  
Figure 17: Pairplot of the initial and added samples for the input uncertain variables.  
Figure 18: Quantile estimation and confidence area with initial dataset (**a**,**c**) and after 10 active-learning-based enriched samples (**b**,**d**).  
of the obtained quantile compared to the validation quantile (right). The uncertainty on the quantile estimation is reduced along the addition of new sample in the dataset. Similarly, the RMSE compared to the validation quantile is also reduced as the number of selected samples by active-learning increase. These boxplots illustrate the efficiency of the proposed approach and its robustness with respect to the initial dataset.