Figure 5.10: Fitness values for di erent parameter values A and B, when pedestrian trajectories are simulated with the social-force model and the force speci - cation in Eq. (5.5). The darker the shade is, the better the tness, for the corresponding combination of A and B values. Next, let us look at the resulting parameter values for the best performing model speci cation, which is the Elliptical II speci cation with a variable parameter N and a variable . The resulting parameter values of our three datasets are put after each other in matrix P , where each row corresponds to a pedestrian and each column corresponds to a parameter. The covariance C = cov(P ) is computed and in order to analyze how di erent parameters correlate with each other, we compute the correlation coe cient R(i; j) which measures how well parameter i correlates with parameter j. It is de ned via the covariance C as p R(i; j) = C(i; j) (5.14) : C(i; i)C(j; j) The correlation coe cient spans from perfect correlation R(i; j) = 1 (rows i and j di er only by a scaling constant) via zero correlation R(i; j) = 0 (C(i; j) = 0) to perfect de-correlation R(i; j) = 1, which means that columns i and j are orthogonal. The resulting correlation coe cients R(i; j) are in Tab. 5.4 with exceptional values highlighted in boldface. We see for example that parameter A correlates strongly 91
8 Conclusions and Outlook It has been demonstrated how di erent parts of pedestrian research, such as modeling, video tracking, optimization and applications to real-world scenarios, can be brought together. Part of the previous research has been re-evaluated by means of empirical data and tools that were not available a few years ago. By measuring only things relevant to crowd dynamics and by making use of the rich amount of known properties of pedestrians and crowds, a method was introduced which can simultaneously evaluate thousands of pedestrians from videos. For some aspects such as the distance-dependence of the repulsive forces between pedestrians, empirical data supported an exponential decay. The angular-dependent interactions turned out to be asymmetric and stronger in front of pedestrians than behind their backs. However, it is surprising that this happens to the degree that pedestrians almost do not react at all to what happenens behind their backs. By evaluating various speci cations of the social-force model, it turned out that velocity-dependent forces greatly increase the performance of the model, in terms of a better t to empirical trajectory data. The reason for this is that the velocity re ects the pedestrians anticipatory behavior. Previously, it was assumed that pedestrians would interact with all surrounding pedes-trians. We have seen that this is not the case. Rather pedestrians typically interact with their eight closest neighbors only. Also, when it comes to evacuations, it has been shown that when evacuees are socially in uenced by their eight closest people, evacu-ation curves similar to empirical data can be obtained, without having to assume any arbitrary distributions of pre-movement time. For high-density data, in contrast, new results are cantly di erent from what has been assumed before. Rather than a smooth pedestrian ow in accordance with the previously measured fundamental diagrams, it turns out that the pedestrian ow enters two consecutive dynamic regimes at high densities. The stop-and-go waves and turbulent regimes have been reported before at an anecdo-tal level of detail. Given a unique dataset, including extremely high crowd densities, it 144