The Impact of Position Errors on Crowd Simulation

• Recent stampedes killed or injured crowds at Shanghai’s New Year’s Eve, the Hajj, Turin, and Mumbai events.Source: §1 · p. 2
• GPS bracelets and rising smartphone ownership could enable location collection at large events, even without specialized devices.Source: §1 · p. 2
• Prior GPS-based early-warning research had not analyzed how position noise affects stampede prediction.GPS errors can arise from satellite synchronization, atmospheric disturbance, and reflections from buildings or other solid objects.Source: §1 · p. 2
• The paper incorporates three stampede assessment methods into a crowd simulator, models position errors, and evaluates their impact on simulated stampede probabilities.Source: §1 · p. 2

• GPS crowd-management systems use crowd-pressure heat maps or crowd-anomaly risk models for event monitoring and prediction.Source: §2 · p. 3
• Vision systems detect congestion and behavior in real time rather than proactively predicting stampedes, and face tracking, accuracy, and privacy challenges.Source: §2 · p. 3
• CNN-based video anomaly methods have been benchmarked on sparse UCSD crowds, not high-density stampede scenarios; GPS-video hybrids have been proposed.Source: §2 · pp. 3–4
• Agent-based frameworks support crowd simulation, including Menge and systems studied for emergency evacuation and Hajj crowd-management scenarios.Source: §2 · p. 4
• Prior stampede measures include crowd pressure, inter-person forces, social and physical forces, and density thresholds, but do not account for GPS noise [22].Source: §2 · p. 4
• Positioning research combines GPS with Wi-Fi, cameras, or sensors and uses filters such as Kalman; reported smartphone GPS errors range from 4.9 m typical accuracy to 5.0–8.5 m horizontal error [17, 9, 48, 56, 47, 13, 61, 30, 39, 64].Source: §2 · pp. 4–5

3 The ARES method

• ARES simulates pedestrian behavior from GPS locations to estimate future stampede probability and can alert on-site authorities when that probability exceeds a threshold.The procedure repeats as new location and velocity data arrive; the authors assume wearable GPS devices and knowledge of each agent’s next destination.Source: §3 · pp. 5–7

3.1 Simulation framework

• The authors use Menge’s PedVO model and a simplified Jamarat Bridge venue to simulate pedestrian behavior.PedVO computes collision-free velocities using preferred velocity and local neighborhood; the venue has three pillar obstacles where agents stop.Source: §3.1 · p. 8, Figure 2 · p. 9

3.2 Stampede assessment methods

• ARES extends Menge with three threshold-based stampede assessments: crowd pressure, physical force, and local density.A stampede is registered when a method’s specified threshold is exceeded.Source: §3.2 · p. 8

3.2.1 Crowd pressure method

• Crowd pressure is local density multiplied by local velocity variance; the stated thresholds are 0.02 s^-2 for turbulence and 0.04 s^-2 for stampede.The paper notes that stampedes may also occur without panic.Source: §3.2.1 · pp. 9–10

3.2.2 Physical force method

• The physical-force method flags a stampede when any monitored force exceeds its threshold; 4500 N is cited as a force that may cause an adult to fall.Agent mass is parameterized with a normal distribution, bounded from 50 kg to 100 kg.Source: §3.2.2 · pp. 10–11

3.2.3 Density method

• The density method estimates D=L/(πR^2) from neighbors within radius R and flags a stampede above seven agents per square metre.The calculation ignores obstacles; the authors conjecture that accounting for them might increase estimated stampede probabilities and false positives.Source: §3.2.3 · p. 12

3.3 Noise modeling

• Only horizontal errors are modeled because the venue map is two-dimensional; velocity errors are ignored as negligible compared with distance errors.Velocity errors could affect crowd-pressure and physical-force measurements, but not the velocity-independent density method.Source: §3.3 · pp. 12–13
• The model adds Rayleigh-distributed horizontal position errors to agents’ starting positions, varying E from 0 to 10 metres.E is the root-mean-squared distance error; E = 0 uses the actual starting positions.Source: §3.3 · p. 13
• Figures 4 and 5 show starting positions with E = 0 m and E = 5 m, respectively.The noisy positions are sampled from the Rayleigh distribution.Source: §3.3 · p. 13, Figure 4 · p. 14, Figure 5 · p. 14
• Some simulated agents shift outside the bridge, while their behavior remains unaltered to isolate the effect of positional spread.This setup is intended to assess how position spread affects the variation in simulated stampede probabilities.Source: §3.3 · p. 14

3.4 Simulated stampede probabilities

• The simulated stampede probability is estimated by averaging binary outcomes across n realizations, with each indicator equal to one for a stampede and zero otherwise.The authors use Monte Carlo simulation and the Law of Large Numbers to estimate the expected value and its confidence interval.Source: §3.4 · pp. 14–15
• The 95% confidence interval is reported for the expected probability; σ denotes the standard deviation of the realization indicators.The supplied passages do not state the interval’s formula.Source: §3.4 · p. 15

3.5 Discussion

• The three selected models are representative instances, so the authors state that these results cannot be generalized to other stampede assessment methods.They suggest other models may also be affected by measurement error, but do not establish that for the untested methods.Source: §3.5 · pp. 15–16
• Kalman filtering reduces noise on monotonic trajectories but struggles with abrupt movement changes, leaves average position error of 2.6 to 3.0 m, and can amplify small errors.The authors conclude that the remaining error may affect model performance.Source: §3.5 · p. 16

4.1 Crowd pressure method

• At R = 1 m, the crowd pressure method predicts a stampede with p = 1 even at 20 agents, while increasing R makes the method less sensitive.With fixed R and E, p tends to rise with N, though this is not universal; PedVO velocity changes and the resulting velocity variance are cited as a mechanism.Source: §4.1 · pp. 18–19, Figure 7 · p. 18
• Larger position errors generally make crowd-pressure probability estimates less accurate, and their effects on p can be positive or negative; the reported confidence intervals are narrow.Spreading positions can reduce physical contacts, while random hot zones can raise p.Source: §4.1 · pp. 19–20, Figure 8 · p. 20

4.2 Physical force method

• The physical force method is sensitive to agent count, with simulated stampede probabilities around 0.45 at N = 40; position errors particularly affect results at N = 20.Its confidence intervals are narrow, indicating consistency across repeated simulations.Source: §4.2 · p. 21, Figure 9 · p. 21

4.3 Density method

• In the density method, larger R means lower sensitivity; position errors may affect p, which ranges from 8% to 32.7% at N = 320; confidence intervals are narrow, indicating high precision and reproducibility.Source: §4.3 · p. 23

4.4 Comparison

• Position errors can affect simulated stampede probabilities, especially for the crowd pressure and density methods at intermediate agent counts; probabilities generally rise as agent count increases.At very small or large counts, probabilities are almost certainly 0 or 1, respectively; thresholds and radius R also affect results.Source: §4.4 · p. 23

5 Conclusions and future work

• The authors propose ARES, a real-time stampede-prediction method implemented as a prototype in Menge, with position errors added to three stampede-assessment methods.They compare simulated stampede probabilities to analyze the impact of position errors.Source: §5 · p. 24
• Position errors can significantly change simulated stampede probabilities, even when a standard Kalman filter is applied.The authors stress the issue especially when GPS signal interference is pronounced.Source: §5 · p. 24
• The authors identify position-error mitigation and position-error distributions from hybrid sources such as video cameras and GPS as future research directions.They describe their results as a starting point for controlling position errors in stampede assessments.Source: §5 · p. 24

A Noise reduction using Kalman filter

• The Kalman-filter evaluation compares filtered estimates with noisy and true positions using the same venue map and pedestrian model as the simulations.The filter's state includes each agent's coordinates and velocity at time s.Source: A Noise reduction using Kalman filter · p. 25
• The paper examines both a set of agents and two individual agents, with findings summarized in appendix section A.3.Source: A Noise reduction using Kalman filter · p. 25
• The evaluation procedure adds position noise, filters the noisy positions, and compares measured and estimated errors against true positions.Algorithm 2 takes N, S, and E as inputs and outputs MAE and KDE.Source: A Noise reduction using Kalman filter · p. 25

A.1 Estimation of positions for a set of agents

• The experiment uses N = 10240 agents over S = 150 seconds, with 1500 simulation steps per agent to provide filter observations and spread agents along the bridge.Source: §A.1 · p. 26
• For E = 1–10 m, Rayleigh noise is added to positions while velocities remain accurate; the filter estimates each agent's final position.Measured and estimated final-position errors are computed against the actual positions.Source: §A.1 · p. 26
• The Kalman filter yields average error ranging from 2.6 to 3.0 m, and estimated error increases monotonically with E.The MAE averages Euclidean distance errors across agents' final coordinates; Table 1 presents measured and estimated MAEs for E = 1, 2, . . . , 10 m.Source: Table 1 · p. 26, §A.1 · p. 27
• When E < 4 m, measured error is smaller than filtered error; at E ≥ 4 m, filtering may be beneficial, but some position errors exceed 20 m despite an average near 3 m.The KDEs have heavy right tails, and errors around 3 m may still significantly vary agents' relative positions and reduce prediction-model performance.Source: §A.1 · pp. 27–28, Figure 11 · p. 27

A.2 Estimation of positions for two individual agents

• The authors conjecture that difficulty with non-monotonic trajectories helps explain why the filter cannot reduce MAE below ≈3 m.The discussion examines two agents from a simulation of 10,240 agents with E = 1 m.Source: §A.2 · p. 28
• For the other agent, the monotone trajectory’s Kalman-filter estimate and true trajectory begin overlapping near (180, -9.5).The agent slowly approaches the first pillar while moving from right to left.Source: §A.2 · p. 28
• Abrupt stops and direction changes go undetected, leaving the Kalman-filter estimate noticeably distant from the true trajectory.In the illustrated case, the filter predicts continued straight-line motion as the agent stops near pillars and changes direction.Source: §A.2 · p. 28

A.3 Summary

• E ≥4 m: the standard Kalman filter may improve GPS positional accuracy when position errors are high.Figure 12 gives example filter performance at E = 1 m.Source: §A.3 · p. 28, Figure 12 · p. 29
• ≈3 m: position error retained by the filter means stampede probabilities remain affected.The authors suggest this residual may be explained by difficulty handling changing movement patterns.Source: §A.3 · p. 28
• Extensions to the standard Kalman filter and integration with techniques such as mean shift are proposed, with their applicability left for future research.The cited proposals include extending the standard filter and integrating mean shift [31].Source: §A.3 · pp. 29–30

B The PedVO model

• PedVO captures physiological and psychological factors in addition to the fundamental diagram.The paper adopts PedVO as its pedestrian model in the experiments.Source: B The PedVO model · p. 30
• The fundamental diagram serves as the velocity modifier, representing the relationship between speed and density.Source: B The PedVO model · p. 30
• The experiments use PedVO’s default settings from the fundamental-diagram example provided by Menge.The default parameters are presented in Table 2.Source: B The PedVO model · p. 30, Table 2 · p. 30