# Visualization of Grouping Definitions

Intuitively, a Group is a set of entities that travel together for a sufficiently long period of time.
From many formal definitions of a group, we compare four grouping definitions: The Convoys[1], Swarms[2], (Original) Groups[3] and Refined Groups[4].
All four definitions require three parameters to define a group:

• minimum number entities in a group: m
• minimum time duration of a group: δ
• minimum distance between entities: ε (see [1][3][4] for details)
We apply algorithms to compute groups based on above definitions to data sets consist of real-life trajectories.
From the results, we visualize the trajectories and integrate them in the video from the same data set.
Our visualization program is based on the work of Maurice Marx .

# Grouping Information and Examples

### A Moving Entity

Head: current location of the entity, color indicates the grouping by one of the definition

Tail: previous locations of the entitiy, color indicates the grouping annotated by human

ID: unique for each entitiy

### Example 1

Human annotated groups: {1,2}

Groups by a grouping definition: none

### Example 2

Human annotated groups: {3,4,5}

Groups by a grouping definition: {4,5}

### Example 3

Human annotated groups: {8,9}
Groups by grouping definition: {7,8,9}, {8,9}, {9,10}

# Data Sets

### Vittorio Emanuelle II Gallery

Contains trajectories of 630 pedestrians tracked in the hallway.

Parameters: m=2,ε=0.963m,δ={17,38,58}

Source: here and here

### Crowds-By-Example

Pedestrian near a university building, contains 434 trajectories

Parameters: m=2,ε={1.22,1.52}m,δ={36,57,78}

Source: here and here

### Grand Central Terminal Set 1

There are 2592 trajectories in this 800 frames data set. We try different sampling rate of trajectories.

Parameters: m=2,ε=0.76m,δ={10,15,20}s

Sampling Rate: 100%, 50%, and 25%

Source: here

### Grand Central Terminal Set 2

There are 3313 trajectories in this 800 frames data set. We try different sampling rate of trajectories.

Parameters: m=2,ε=0.76m,δ={10,15,20}s

Sampling Rate: 100%, 50%, and 25%

Source: here

### ETH Walking Pedestrian Set 1

Pedestrians entering and exiting a university building, contains of 360 trajectories

Parameters: m=2,ε=1.24m,δ={72,89,105}

Source: here

### ETH Walking Pedestrian Set 2

Pedestrians in the sidewalk near the tram stop, contains 389 trajectories

Parameters: m=2,ε=0.94m,δ={20,58,96}

Source: here

#### References

• [1] Hoyoung Jeung, Man Lung Yiu, Xiaofang Zhou, Christian S.J ensen, and Heng Tao Shen. 2008. Discovery of convoys in trajectory databases. PVLDB 1, 1 (2008), 1068–1080.
• [2] Zhenhui Li, Bolin Ding, Jiawei Han, and Roland Kays. 2010. Swarm: Mining Relaxed Temporal Moving Object Clusters. PVLDB 3, 1 (2010), 723–734.
• [3] Kevin Buchin, Maike Buchin, Marc van Kreveld, Bettina Speckmann, and Frank Staals. 2015. Trajectory grouping structure. Journal of Computational Geometry 6, 1 (2015), 75–98.
• [4] Marc van Kreveld, Maarten Löffler, Frank Staals, and Lionov Wiratma. 2018. A Refined Definition for Groups of Moving Entities and Its Computation. Int. J. Comput. Geometry Appl. 28, 2 (2018), 181–196.