## 3 Results  
### Characteristics of the Microscopic Interactions  
The relative speed versus acceleration plot for an individual is shown in Figure 3. This is called the Lissajous diagram, and it can be observed that the correlation between instantaneous acceleration and relative speed improves when a time lag (\(\tau\)) is introduced to the acceleration. These plots are analogous to drivers' acceleration behaviors during car-following situations [48; 49]. For each individual, Lissajous diagrams were constructed, and relative speed versus acceleration plots were obtained. In this study, the time lag varied from 0.12 s to 0.68 s, and this difference could be due to individual characteristics. Even though such a time lag is incorporated in car-following models as \"the reaction time\", pedestrian simulation models do not specify such a time lag or reaction time. In pedestrian dynamics, this time lag may be comparable to the time delay specified as \"visual-motor delay\" in previous studies, e.g., [PERSON] et al. [50] and [PERSON] et al. [51]. This value varied depending on the individual characteristics and was approximately set as 0.40 s [50; 51]. However, [PERSON] et al. [43] reported a higher average delay time for the speed correlation between leading and following pedestrian dyads, and the average time delay value in that study ranged from 0.75 to 0.84 s. Furthermore, according to [PERSON] et al. [43], this time delay did not have any gender dependence. Aggregated time-delayed acceleration versus relative speed plots for different density levels and genders are compared in Figure 4. As can be observed from the figure, the acceleration tends to increase with increasing relative speed for both genders. This means  
Figure 3: Relative speed versus acceleration plots for an individual pedestrian without and with a time lag. that people accelerate or decelerate more when the speed difference between him/her and the leading pedestrian is larger. Further, it can be observed that the accelerations (and decelerations) of males are higher than females in all cases, which indicates that the acceleration (and deceleration) capacities of males are higher. ANCOVA (analysis of covariance) tests were performed to compare the regression lines for males and females for the four density levels separately. The test results indicated that, for all density levels, there is a significant difference between the slopes of the regression lines for males and females. The test statistics, i.e., (\(F\), \(p\)), for N = 14, N = 20, N = 24, and N = 30 scenarios were (12.14, 0.0005), (29.77, <0.0001), (42.66, <0.0001), and (16.97, 0.000038), respectively. It can further be noted that the slopes of the lines tend to decrease when the density level increases. This means that for a given relative speed, the acceleration and deceleration capacities decrease with increasing density. Relationships between acceleration and walking speed for different density levels and genders are compared in Figure 5. It is clear that the walking speeds decreased with  
Figure 4: Relative speed versus time-delayed instantaneous acceleration plots for different density levels: (**a**) N = 14; (**b**) N = 20; (**c**) N = 24; (**d**) N = 30.  
increasing density. Further, it can be observed that for all density levels and both genders the accelerations decrease with increasing instantaneous walking speed. In particular, when the speed is higher, people tend to decelerate more, and when the speed is lower, people tend to accelerate more. This observation is logical when general microscopic walking dynamics are considered. However, it should be noted that only walking speed may not determine the acceleration behaviors and the speed of the leading pedestrian, which is considered in relative speed, could also play a significant role. As verified with the \(t\)-test, the slopes of the regression lines for males and females were significantly different for all cases.