# Sensitivity Analysis for Microscopic Crowd Simulation
## 3 Results and Discussion
### First Eigenvector and Activity Scores  
When looking for active subspaces in crowd simulations, it is important to keep in mind that the method cannot be used for purely rule-based models where the concept of gradients is meaningless. This especially applies for cellular automaton models. For most other models, as for the Optimal Steps Model used here, gradients for the quantity of interest may not be directly available, but they can be approximated. We use central difference quotients for this. According to [26], we normalize each input parameter to the interval \([-1,1]\). For each parameter, the step-size \(h\) for the central difference is scaled by the parameter's range. We choose the smallest possible step size for \(h\): since the number of agents is a natural number, \(h\) has to scale to at least one agent. Based on the range of the parameter \"number of agents\", the step size is set to \(h=0.025\). All of the sensitivity results are presented for the normalized model. Figure 8a presents the eigenvalues of the matrix \(\hat{C}\), obtained from the gradient samples according to (9). A spectral gap is present between the first and second eigenvalue. That means we identify a one-dimensional active subspace defined by the first eigenvector. Since the gap is clearly above the chosen step size (including the first eigenvalue below the gap), we can assume that the gap is real and not based on a phantom eigenvalue. The first eigenvector is shown in Figure 8b. From the scatterplots in Figure 5, we know that the personal space strength correlates negatively with the maximum Voronoi density. Accordingly, we rotate the first eigenvector. Now, the first eigenvector entries show the direction in which each parameter contributes to the quantity of interest. The standard deviation of the free-flow speed, bottleneck width, and personal space strength have a negative impact on the outcome. That means, if these parameters are increased, a reduced (maximum Voronoi) density will be observed. The first eigenvector is the most dominant parameter direction in the input parameter space. Figure 7: Normalized Sobol’ total indices for bottleneck scenario. Calculated with Monte Carlo approach while using [PERSON]’s method [53]. For each sampling factor \(M\), three runs were performed. The parameter index is mapped as follows: 1—control parameter, 2—free-flow speed mean, 3—free-flow speed standard deviation, 4—number of agents, 5—bottleneck width, 6—personal space strength. Independent of the size of the active subspace, the first eigenvector entries also constitute a sensitivity metric. Therefore, we normalized the absolute first eigenvector according to [26] by the length of the vector. The resulting measures are shown in Figure 9.  
From the active subspace we identified, we can now derive the activity scores while using (12). Figure 10 presents the activity scores for different oversampling factors \(\alpha\). The oversampling factor determines the number of gradient samples used for the approximation of \(C\): \(M=ak\text{ln}(m)\). Here, \(m\) is the number of parameters and \(k-1\) is the expected size of the subspace. According to ([51], p. 35), \(M\) evaluations of the gradient approximate the first \(k\) eigenvalues of \(C\) sufficiently accurate. The oversampling factor \(\alpha\) is typically chosen between 2 and 10 ([51], p.35). As expected, the variation of the activity scores for a fixed oversampling factor, represented by the area around the mean, diminishes  
Figure 8: Eigenvalues and first eigenvector components of the matrix \(C\). A one-dimensional subspace was identified based on the spectral gap between the first and second eigenvalue. The parameter index is mapped as follows: 1—control parameter, 2—free-flow speed mean, 3—free-flow speed standard deviation, 4—number of agents, 5—bottleneck width, 6—personal space strength. The index for the eigenvalue plot refers to the directions identified by the decomposition. Figure 9: Normalized first eigenvector metric for the bottleneck scenario.