# Sensitivity Analysis for Microscopic Crowd Simulation
## 3 Results and Discussion
### Uncertain Input Parameters in the Bottleneck Scenario  
As briefly introduced in Section 2.2, we investigate a bottleneck scenario that is shown in Figure 3.  
We choose six uncertain parameters that are listed in Table 1. The first two, mean and standard deviation of the free-flow speed, refer to the truncated Gaussian distribution from which the free-flow speeds are drawn. Note that truncating the normal distribution means that the two parameters are not independent. Because the goal of our contribution is rather a proof of concept for a valuable methodology than an in-depth analysis of queuing behavior at a bottleneck, we decided to treat them as independent parameters. In the simulation, the free-flow speed of an agent is also used to derive a dependent parameter [11], the individual step size of each agent. As third parameter, we vary the number of agents between 160 and 200. Typically, bottleneck experiments in experimental pedestrian dynamics study the effect of door widths on the density in front of the opening. Thus, we also vary the bottleneck width. Lastly, we vary a parameter that corresponds to each agent's need for personal space, or distance to others: it is coded in the parameter \(h_{p}\) of a function that expresses the utility of a position for an agent. See Appendix A.1 for more details on \(h_{p}\).  
We want to point out that the parameters are of different character. Free-flow speed as well as personal space are properties of the agents, while the other two parameters, bottleneck width and number of agents, are parameters of the topography. It is important to keep this in mind when choosing the ranges for the parameters: Changes to topography parameters might change the dynamics of the system. Consequently, in our study, we had to make sure that within our parameter limits we maintain a bottleneck dynamic, that is, congestion in front of the constriction.  
Figure 4 shows snapshots of the simulation of the bottleneck scenario. On the left, the lower limits of all parameters are chosen, on the right the upper limits of all parameters are chosen. The topography changes with the bottleneck width.  
Because we study the feasibility of the methods in crowd simulation, we add a control parameter that is not actively used in the configuration of the simulation. We test whether the methods correctly detect the absence of an impact. Through the remaining \"impact\" of the control parameter, we also gain a measure for the inaccuracy that is caused by averaging several simulations. This procedure increases the number of evaluations and, therefore, the computation time, substantially. Hence, while a control parameter might be desirable, in principle, it will be out of question in computationally more demanding scenarios.  
\begin{table}
\begin{tabular}{c l l l l} \hline \hline
**Index** & **Parameter** & **Unit** & **Range** & **Type** \\ \hline
1 & Control parameter & & \(\mathcal{U}(1.0,5.0)\) & float \\
2 & Free-flow speed mean & m/s & \(\mathcal{U}(0.5,2.2)\) & float \\
3 & Free-flow speed std & m/s & \(\mathcal{U}(0.1,1.0)\) & float \\
4 & Number of agents & & \(\mathcal{U}(160,200)\) & int \\
5 & Bottleneck width & m & \(\mathcal{U}(1.6,3.0)\) & float \\
6 & Personal space strength \(h_{p}\) & & \(\mathcal{U}(30,70)\) & float \\ \hline \hline \end{tabular}
\end{table}
Table 1: Uncertain parameters and their distribution used for the sensitivity analysis.