# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 4 Applications
### Analytical Test Case
#### 4.1.2 Test Case Results  
The proposed approach is repeated 10 times with different initial datasets \(\mathcal{U}_{M}\) and \(\mathcal{X}_{M}\) to evaluate the robustness of the proposed method to the initial datasets. For one representative repetition, the initial samples of the design experiment and the added samples by the active learning strategy are represented in Figure 7. As the infill process starts with a limited number of samples, the refinement strategy adds samples in the dataset to improve the 95% quantile estimation. It can be seen that several (5 over 10) are added in the vicinity of the exact 95% quantile.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of KL Modes** & **1** & **2** & **4** & **6** \\ \hline Q2 & \(-\)489.28 & 0.9998 & 1.0000 & 1.0000 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Predictivity factor Q2 as a function of the number of KL modes.  
Figure 6: Karhunen–[PERSON] validation with respect to the number of modes: 1 (a), 2 (b), 4 (c), 6 (d).  
Moreover, in Figure 8, the evolution of the estimated 95% quantile with the surrogate model along with the iteration of the active learning process are presented. It can be seen that the first estimations of the quantile are far away from the exact \(q_{95}\) and as the refinement strategy updates the surrogate model, the estimate quantile converges to the exact quantile thanks to a limited number of exact mapping \(\mathcal{Q}\mathbf{x}\) evaluations. Therefore, the active learning process allows to estimate the 95% quantile of the output stochastic process \(\mathbf{X}(t,\mathbf{U})\) at an affordable computational cost.  
In addition, in Figure 9, the evolution of the quantile estimation and the confidence area are displayed before the refinement (left) and after the 10 active learning iterations (right). At the end of the refinement process, the estimated quantile converges to the exact quantile and the confidence area is tighten  
Figure 8: Evolution of the estimated \(q_{95}\) along the 10 active-learning iterations.  
Figure 7: Initial and added output stochastic process samples in the dataset.  
Figure 10 presents the boxplots over the 10 repetitions corresponding to the evolution of the estimated quantile confidence area (left) and the root mean square error (RMSE) on the right with respect to the exact quantile. Both the confidence area and the RMSE decrease as the refinement strategy adds new relevant samples into the input and output datasets. The RMSE between the estimated quantile and the exact quantile converges to 0. Moreover, it can be noted that the proposed approach is robust to the different initial datasets. Therefore, with only 14 evaluations of the exact model \(\mathcal{Q}_{\mathbf{X}}(\cdot)\) (4 initial samples and 10 added with the refinement strategy) it is possible to accurately estimate the 95% quantile while controlling the confidence in its estimation.  
In order to illustrate the interest of the surrogate model-based approach for the estimation of the 95% quantile, a Monte-Carlo simulation of the same number of samples (14 in the present case) may be performed to estimate the quantile. It is possible to determine the confidence area for the Monte-Carlo simulation as presented in Figure 11. For the same number of exact model evaluations \(\mathcal{Q}_{\mathbf{X}}(\cdot)\), the estimated 95% quantile is more accurate with the active learning approach compared to a Monte-Carlo estimation. It highlights the interest of surrogate-based estimation with a refinement strategy to estimate stochastic process quantile.