1. 2IMG15 Algorithms for Geographic Data
Fall Lecture 7: Movement Patterns

2. Trajectory data
The data as it is acquired by GPS: sequence of triples (spatial plus time-stamp); quadruples for trajectories in 3D
(xn,yn,tn)
(x2,y2,t2)
(xi,yi,ti)
(x1,y1,t1)

3. Trajectory data
Typical assumption for sufficiently densely sampled data: constant velocity between consecutive samples ➨ velocity/speed is a piecewise constant function
(xn,yn,tn)
(x2,y2,t2)
(xi,yi,ti)
(x1,y1,t1)

4. Abstract / general purpose questions
Single trajectory
simplification, cleaning
segmentation into semantically meaningful parts
finding recurring patterns (repeated subtrajectories)
Two trajectories
similarity computation
subtrajectory similarity
Multiple trajectories
clustering, outliers
flocking/grouping pattern detection
finding a typical trajectory or computing a mean/median trajectory
visualization

5. Movement patterns
[Dodge, Weibel, Lautenschütz 2008]

6. Movement patterns Many, many possible movement patterns:
flocks (group of entities moving close together)
swarm
convoys
herds
following (is an entity following another entity)
leadership
single file
popular places (place visited by many) …
Challenge: useful definitions which are algorithmically tractable

7. Groups

8. Groups Kalnis, Mamoulis & Bakiras, 2005
Gudmundsson & van Kreveld, 2006
Benkert, Gudmundsson, Hubner & Wolle, 2006
Jensen, Lin & Ooi, 2007
Al-Naymat, Chawla & Gudmundsson, 2007
Vieria, Bakalov & Tsotras, 2009
Jeung, Yiu, Zhou, Jensen & Shen, 2008
Jeung, Shen & Zhou, 2008
... t2 t3 t1 t4

9. Groups
m group size
Time = 0 1 2 3 4 5 6 7 8
m = 3

10. Groups
fixed subset
variable subset
flock
meet
examples for m = 3

11. Convoy

12. Convoy X set of entities m convoy size ε distance threshold
Definitions x, y  X are directly connected if the ε-disks of x and y intersect x1 and xk are -connected if there is a sequence x1, x2 ,…, xk of entities such that for all i, xi and xi+1 are directly connected x ε y x1 x2 x3 x4 x5

13. Convoy
Definition
A group of entities forms a convoy if every pair of entities is -connected.
[Jeung, Yiu, Zhou, Jensen & Shen, 2008] [Jeung, Shen & Zhou, 2008]

14. Leadership & Followers

15. Leadership & Followers
A leader?
Should not follow anyone else!
Is followed by at least m other entities …
For a certain duration …
front

16. Leadership & Followers
A leader?
Should not follow anyone else!
Is followed by at least m other entities …
For a certain duration …
Many different settings …
Running time ~O(n2 t log n t) for n entities and t time steps
front
[Andersson et al. 2007]

17. Popular places

18. Popular places
A region is a popular place if at least m entities visit it σ
σ is a popular place for m  5
[Benkert, Djordjevic, Gudmundsson & Wolle 2007]

19. Single File

20. Single file
Single file intuitively easy to define … but hard to define formally!

21. Single file
Single file intuitively easy to define … but hard to define formally!

22. Single file
Single file intuitively easy to define … but hard to define formally!

23. Towards a formal definition?
Entities x1, … , xm are moving in single file for a given time interval if during this time each entity xj+1 is following behind entity xj for j = 1, … ,m-1

24. Towards a formal definition?
Entities x1, … , xn are moving in single file for a given time interval if during this time each entity xj+1 is following behind entity xj for j = 1, … ,n-1
Following behind?
Time t
Time t’ ∈ [t+min, t+max] ε

25. Following behind
x1 entity with parameterized trajectory f1 over time interval [s1, t1] x2 entity with parameterized trajectory f2 over time interval [s2, t2]
with s2 ∈ [s1+tmin, s1+tmax] and t2 ∈ [t1+tmin, t1+tmax]
x2 is following behind x1 in [s1,t1] if there exists a continuous, bijective function  : [s1, t1]  [s2, t2] such that (s1)=s2 and
 t ∈ [s1, t1]: (t) ∈ [t-tmax, t-tmin] & d(f1((t)), f2(t)) ≤ ε f2 f1

26. Following behind
x1 entity with parameterized trajectory f1 over time interval [s1, t1] x2 entity with parameterized trajectory f2 over time interval [s2, t2]
with s2 ∈ [s1+tmin, s1+tmax] and t2 ∈ [t1+tmin, t1+tmax]
x2 is following behind x1 in [s1,t1] if there exists a continuous, bijective function  : [s1, t1]  [s2, t2] such that (s1)=s2 and
 t ∈ [s1, t1]: (t) ∈ [t-tmax, t-tmin] & d(f1((t)), f2(t)) ≤ ε f2 f1

27. Following behind
x1 entity with parameterized trajectory f1 over time interval [s1, t1] x2 entity with parameterized trajectory f2 over time interval [s2, t2]
with s2 ∈ [s1+tmin, s1+tmax] and t2 ∈ [t1+tmin, t1+tmax]
x2 is following behind x1 in [s1,t1] if there exists a continuous, bijective function  : [s1, t1]  [s2, t2] such that (s1)=s2 and
 t ∈ [s1, t1]: (t) ∈ [t-tmax, t-tmin] & d(f1((t)), f2(t)) ≤ ε

28. Recap: Free space diagram
Definition If there exists a monotone path in the free-space diagram from (0, 0) to (p, q) which is monotone in both coordinates, then curves f and g have Fréchet distance less than or equal to ε
(0, 0)
(p, q)

29. Free space diagram and following behind
time delay ➨ the path will be restricted to a “diagonal” strip
Running time
O(t k)
where
k: number of cells
intersecting the diagonal strip
in a column or row
t: number of timestamps

30. Single file
k = #cells per row or column in strip, n = #timestamps, m = #entities
One can determine in O(n k2) time during which time intervals one trajectory is following behind the other.
If the order between the entities is specified then compute the free space diagram between every pair of consecutive entities ➨ O(m n k2)
If no order then compute the free space diagram between every pair of entities ➨ O(m2 n k2)
[Buchin et al. 2008]

31. Summary
Challenge: useful definitions which are algorithmically tractable
Examples:
Group
Convoy
Leadership/Following
Single-file/Following

32. Movement patterns Many, many possible movement patterns:
flocks (group of entities moving close together)
swarm
convoys
herds
following (is an entity following another entity)
leadership
single file
popular places (place visited by many) …
Challenge: useful definitions which are algorithmically tractable