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Modeling pedestrian fundamental diagram based on Directional Statistics

Kota Nagasaki, Keiichiro Fujiya, Toru Seo

TL;DR

This work tackles the challenge of modeling pedestrian fundamental diagrams across diverse flow types by introducing Directional Statistics into the FD framework. It defines angular-variance measures, including a novel $p$th angular variance, to quantify directional dispersion and multi-peak distributions, and embeds these into a smooth, capacity-penalized FD that also accounts for walls. Validation on real trajectory data shows the model captures key phenomena such as capacity reductions from conflict in crossing flows and capacity gains from lane formation in bi-directional flows, while remaining applicable across multiple flow types. The approach offers a unified, direction-aware view of pedestrian dynamics with potential for extension to directional-dependent predictions and other geometries.

Abstract

Understanding pedestrian dynamics is crucial for appropriately designing pedestrian spaces. The pedestrian fundamental diagram (FD), which describes the relationship between pedestrian flow and density within a given space, characterizes these dynamics. Pedestrian FDs are significantly influenced by the flow type, such as uni-directional, bi-directional, and crossing flows. However, to the authors' knowledge, generalized pedestrian FDs that are applicable to various flow types have not been proposed. This may be due to the difficulty of using statistical methods to characterize the flow types. The flow types significantly depend on the angles of pedestrian movement; however, these angles cannot be processed by standard statistics due to their periodicity. In this study, we propose a comprehensive model for pedestrian FDs that can describe the pedestrian dynamics for various flow types by applying Directional Statistics. First, we develop a novel statistic describing the pedestrian flow type solely from pedestrian trajectory data using Directional Statistics. Then, we formulate a comprehensive pedestrian FD model that can be applied to various flow types by incorporating the proposed statistics into a traditional pedestrian FD model. The proposed model was validated using actual pedestrian trajectory data. The results confirmed that the model effectively represents the essential nature of pedestrian dynamics, such as the capacity reduction due to conflict of crossing flows and the capacity improvement due to the lane formation in bi-directional flows.

Modeling pedestrian fundamental diagram based on Directional Statistics

TL;DR

This work tackles the challenge of modeling pedestrian fundamental diagrams across diverse flow types by introducing Directional Statistics into the FD framework. It defines angular-variance measures, including a novel th angular variance, to quantify directional dispersion and multi-peak distributions, and embeds these into a smooth, capacity-penalized FD that also accounts for walls. Validation on real trajectory data shows the model captures key phenomena such as capacity reductions from conflict in crossing flows and capacity gains from lane formation in bi-directional flows, while remaining applicable across multiple flow types. The approach offers a unified, direction-aware view of pedestrian dynamics with potential for extension to directional-dependent predictions and other geometries.

Abstract

Understanding pedestrian dynamics is crucial for appropriately designing pedestrian spaces. The pedestrian fundamental diagram (FD), which describes the relationship between pedestrian flow and density within a given space, characterizes these dynamics. Pedestrian FDs are significantly influenced by the flow type, such as uni-directional, bi-directional, and crossing flows. However, to the authors' knowledge, generalized pedestrian FDs that are applicable to various flow types have not been proposed. This may be due to the difficulty of using statistical methods to characterize the flow types. The flow types significantly depend on the angles of pedestrian movement; however, these angles cannot be processed by standard statistics due to their periodicity. In this study, we propose a comprehensive model for pedestrian FDs that can describe the pedestrian dynamics for various flow types by applying Directional Statistics. First, we develop a novel statistic describing the pedestrian flow type solely from pedestrian trajectory data using Directional Statistics. Then, we formulate a comprehensive pedestrian FD model that can be applied to various flow types by incorporating the proposed statistics into a traditional pedestrian FD model. The proposed model was validated using actual pedestrian trajectory data. The results confirmed that the model effectively represents the essential nature of pedestrian dynamics, such as the capacity reduction due to conflict of crossing flows and the capacity improvement due to the lane formation in bi-directional flows.
Paper Structure (23 sections, 6 theorems, 25 equations, 11 figures, 8 tables)

This paper contains 23 sections, 6 theorems, 25 equations, 11 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Model
  4. Angular variance
  5. $p$th angular variance
  6. Formulation of FD
  7. Base function
  8. Incorporating $p$th angular variance into FD function
  9. Incorporating the effects of the walls into FD function
  10. Proposed FD model
  11. Case study
  12. Data
  13. Parameter estimation
  14. Ablation study
  15. Discussion
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\nu_p(\bm{\theta})$ denote the $p$th angular variance for a dataset $\bm{\theta}=\{\theta_j\,|\,j=1,2,\dots,N\}$ and $\tilde{\bm{\theta}}^m$ denote a dataset with $2\pi/m~(m\in\mathbb{N}, m\geq2)$ period, the following formulas hold

Figures (11)

  • Figure 1: Four examples of flow types and their histograms. The data is a 10-second sample for each flow type from the data used in the case study.
  • Figure 2: The example of angular mean and arithmetic mean for angle.
  • Figure 3: Angular data and corresponding angular variance $\nu_1$ and second angular variance $\nu_2$. The dots around the unit circle represent the angular data $\theta_j$ and the arrows represent the unit vector $(\cos \theta_j, \sin \theta_j)^{\top}$ corresponding to the data $\theta_j$.
  • Figure 4: Histograms for each flow type and corresponding values for each $p$th angular variance.
  • Figure 5: Examples of wall ratio $r$.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5