# Modeling Spatio-Temporal Evolution of Urban Crowd Flows

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Methodology

#### 3.1. Urban Lattice

#### 3.1.1. Spatial Partition

#### 3.1.2. Crowd Level

- Speed$$\begin{array}{c}\hfill {v}_{x,y}^{t}=\frac{{\sum}_{({x}^{{t}_{i}^{\prime}},{y}^{{t}_{i}^{\prime}})\in \overline{(x,y)}}{v}^{{t}_{i}^{\prime}}}{{\sum}_{({x}^{{t}_{i}^{\prime}},{y}^{{t}_{i}^{\prime}})\in \overline{(x,y)}}1}.\end{array}$$
- Volume (or density)$$\begin{array}{c}\hfill i{n}_{x,y}^{t}=|\{{\mathbb{P}}^{t}:({x}^{{t}_{1}^{\prime}},{y}^{{t}_{1}^{\prime}})\notin \overline{(x,y)},({x}^{{t}_{L}^{\prime}},{y}^{{t}_{L}^{\prime}})\in \overline{(x,y)}\}|,\end{array}$$$$\begin{array}{c}\hfill ou{t}_{x,y}^{t}=|\{{\mathbb{P}}^{t}:({x}^{{t}_{1}^{\prime}},{y}^{{t}_{1}^{\prime}})\in \overline{(x,y)},({x}^{{t}_{L}^{\prime}},{y}^{{t}_{L}^{\prime}})\notin \overline{(x,y)}\}|,\end{array}$$$$\begin{array}{c}\hfill pas{s}_{x,y}^{t}=|\{{\mathbb{P}}^{t}:({x}^{{t}_{1}^{\prime}},{y}^{{t}_{1}^{\prime}})\notin \overline{(x,y)},\exists ({x}^{{t}_{i}^{\prime}},{y}^{{t}_{i}^{\prime}})\in \overline{(x,y)},({x}^{{t}_{L}^{\prime}},{y}^{{t}_{L}^{\prime}})\notin \overline{(x,y)}\}|,\end{array}$$$$\begin{array}{c}\hfill sta{y}_{x,y}^{t}=|\{{\mathbb{P}}^{t}:({x}^{{t}_{1}^{\prime}},{y}^{{t}_{1}^{\prime}})\in \overline{(x,y)},({x}^{{t}_{L}^{\prime}},{y}^{{t}_{L}^{\prime}})\in \overline{(x,y)}\}|,\end{array}$$
- Flux$$\begin{array}{c}\hfill {f}_{x,y}^{t}=i{n}_{x,y}^{t}+ou{t}_{x,y}^{t}+pas{s}_{x,y}^{t}+sta{y}_{x,y}^{t}.\end{array}$$
- Crowd rate$$\begin{array}{c}\hfill {s}_{x,y}^{t}=\frac{i{n}_{x,y}^{t}+sta{y}_{x,y}^{t}}{i{n}_{x,y}^{t}+ou{t}_{x,y}^{t}+pas{s}_{x,y}^{t}+sta{y}_{x,y}^{t}}.\end{array}$$

- Free flow: ${I}_{x,y}^{t}=0$ for ${v}_{x,y}^{t}>\u03f5$;
- Slowed flow: ${I}_{x,y}^{t}=1$ for ${v}_{x,y}^{t}\le \u03f5$ and ${s}_{x,y}^{t}<\lambda $;
- Crowded flow: ${I}_{x,y}^{t}=2$ for ${v}_{x,y}^{t}\le \u03f5$ and ${s}_{x,y}^{t}\ge \lambda $;

#### 3.2. Urban Crowd Hotspot

#### 3.2.1. Connectivity

#### 3.2.2. Connected Component

#### 3.2.3. Crowd Region

#### 3.3. Spatio-Temporal Evolution

#### 3.3.1. Mask Region

#### 3.3.2. Crowd Morphology

Algorithm 1: Morphological analysis. |

#### 3.3.3. Nested Crowd Evolution

Algorithm 2: Nested morphological analysis. |

## 4. Case Study

#### 4.1. Scenario I: Simulation

#### 4.1.1. Synthetic Data

#### 4.1.2. Pattern Assignments

#### 4.2. Scenario II: Real Observation

#### 4.2.1. Case Study Area

#### 4.2.2. Citywide Crowd Hotspots

#### 4.2.3. Morphological Evolutionary Patterns

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Overview of the proposed methodology for analyzing the morphological evolutionary patterns of urban crowd flows.

**Figure 2.**The temporal series of synthetic urban crowd distributions and the exemplary associated mask regions between the given consecutive time frames.

**Figure 3.**Assignment paths along the decision tree for the simulated crowd regions. Note that A, B, C, D, E, E’, F on the vertical axis are condition labels as defined in Algorithm 1.

**Figure 5.**Citywide crowd hotspots for the slowed flows (

**left**) and the crowded flows (

**right**). The color indicates the crowd rate of each cell. The subplot on the right demonstrates the derived crowded regions for slow flows, while the subplot on the left demonstrates the derived regions that were severely crowded.

**Figure 6.**Statistics of the morphological evolutionary patterns for the slowed flows (

**left**) and between the slowed and the crowded flows (

**right**).

**Figure 7.**Temporal transitions between the morphological evolutionary patterns for the slowed flows (

**top**) and the crowded flows (

**bottom**) at distinct time scales—that is, middle of the night, morning rush hours, afternoon rush hours, and evening rush hours. Note that the node size denotes the relative frequency of each pattern, the link width denotes the transmission probability between two nodes, and the link color denotes the originating node.

**Figure 8.**Temporal transitions between the morphological evolutionary patterns of the slowed flows (

**left**) and the crowded flows (

**right**) at five typical locations.

Morphology (t → t+1) | ||
---|---|---|

Centroid (x, y) | Area (Number of Cells) | |

Newly Occurring | None → Exist | Zero → Non-Zero |

Disappearing | Exist → None | Non-Zero → Zero |

Splitting and Merging | Multiple → Multiple | — |

Splitting | Single → Multiple | — |

Merging | Multiple → Single | — |

Stable | No Change | No Change |

Stable and Moving | Cell A → Cell B | No Change |

Shrinking | No Change | Large → Small |

Shrinking and Moving | Cell A → Cell B | Large → Small |

Growing | No Change | Small → Large |

Growing and Moving | Cell A → Cell B | Small → Large |

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## Share and Cite

**MDPI and ACS Style**

Qin, K.; Xu, Y.; Kang, C.; Sobolevsky, S.; Kwan, M.-P. Modeling Spatio-Temporal Evolution of Urban Crowd Flows. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 570.
https://doi.org/10.3390/ijgi8120570

**AMA Style**

Qin K, Xu Y, Kang C, Sobolevsky S, Kwan M-P. Modeling Spatio-Temporal Evolution of Urban Crowd Flows. *ISPRS International Journal of Geo-Information*. 2019; 8(12):570.
https://doi.org/10.3390/ijgi8120570

**Chicago/Turabian Style**

Qin, Kun, Yuanquan Xu, Chaogui Kang, Stanislav Sobolevsky, and Mei-Po Kwan. 2019. "Modeling Spatio-Temporal Evolution of Urban Crowd Flows" *ISPRS International Journal of Geo-Information* 8, no. 12: 570.
https://doi.org/10.3390/ijgi8120570