# Scenario Optimisation and Sensitivity Analysis for Safe Automated Driving Using Gaussian Processes
## 6 Results
### Probabilistic Sensitivity Analysis of Scenario Data Sets
#### 6.3.3 Sensitivity Analysis of the Traffic Jam Scenario  
The results of the SA for the traffic jam scenario are visualised in Figure 7, where the main effect plot reflects what is already known from studying the visualisation of the three scenario parameters as discussed and illustrated in Reference [8]. The influence of \(S_{1}\) on the scenario outcome is contrary to the influence of \(S_{2}\) and \(AA\), with all of them having linear effects over the entire input domain. In concrete terms, this means, the logical scenario results in a collision for higher values of \(S_{1}\) and lower values of \(S_{2}\) and \(AA\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Simulation Parameters** & **Lower Limit** & **Upper Limit** \\ \hline \(S_{1}\) [km/h] & 40 & 70 \\ \(S_{2}\) [km/h] & 5 & 20 \\ \(AA\) [deg] & 10 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input parameters for the traffic jam approach scenario and their ranges. Figure 6: Traffic jam approach scenario with the blue target vehicle and the yellow ego vehicle equipped with a radar sensor. The box plot of the first-order sensitivity indices in the middle of Figure 7 shows that parameter \(S_{1}\) has the highest effect on the scenario outcome, followed by \(AA\) and \(S_{2}\). It is to be noted that the uncertainty in the box plots includes the uncertainty about the input-output relationship as well as the uncertainty of the integration estimates for the emulator. For uncorrelated input parameters and a deterministic response, the first-order sensitivity index would be equal to the total effect sensitivity index, \(S_{i}=S_{T_{i}}\). A comparison of the first-order sensitivity indices with the respective total effect sensitivity indices therefore presents a measure of the proportion of variability due to interaction between input parameters. From Figure 7 and the values listed in Table 4, it is discernible that the total effect of all three parameters is significantly higher than the first-order sensitivity indices. This indicates interaction effects between the input parameters causing a portion of the variability in the scenario outcome. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Variance (\%)** & \(S_{i}\) & \(S_{T_{i}}\) \\ \hline Speed of ego vehicle (\(S_{1}\)) & 28.88 & 0.3122 & 0.6160 \\ Speed of traffic jam (\(S_{2}\)) & 16.64 & 0.1289 & 0.4088 \\ Sensor Aperture Angle (\(AA\)) & 31.18 & 0.2427 & 0.5438 \\ (\(S_{1}\), \(S_{2}\)) & 3.72 & & \\ (\(S_{1}\), \(AA\)) & 6.21 & & \\ (\(S_{2}\), \(AA\)) & 3.87 & & \\ Total Variance (\%) & 90.49 & & \\ \hline Estimated mean output & 0.537633 \(\pm\) 8.1986 \(\times\) 10\({}^{-5}\) & & \\ Estimated variance output & 0.136 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The emulator-based SA of the traffic jam scenario (c.f. Section 6.3.2). Figure 7: The estimated main effects, 1 st and 2 nd order sensitivity indices of the parameters in the traffic jam scenario as discussed in Table 3.  
Table 4 reports the probabilistic SA measures for the scenario outcome (collision or no-collision) with respect to the changes of the input parameters discussed in Table 3. From the second column of Table 4, we can say that 31.2% of total variance (also known as the main effect and explained in Section 4) can be explained by \(AA\) (sensor aperture angle) based on 100 data points selected using the LHC sampling method, while main effects of 'Speed of ego vehicle' (\(S_{1}\)) and 'Speed of traffic jam' (\(S_{2}\)) are describing 28.9% and 16.6% of total variance, respectively. We can then conclude that the outcome of the scenario is more sensitive to variation of \(AA\) than \(S_{2}\). Note that the main effect proportions do not sum to 100% of the variance. The remaining variance after the main effects are deducted from the total variance is thus explained by the first-order interactions (13.8%) and the third-order interaction (9.51%), as explained in Appendix A.