Computational Movement Analysis Lecture 4: Movement patterns Joachim Gudmundsson

1. Computational Movement AnalysisLecture 4: Movement patternsJoachim Gudmundsson

2. Movement patterns • Much of the early work on movement analysis focussed on finding movement patterns. • For example: • Flocks (group of entities moving close together) • Swarm • Convoys • Heards • Following (is an entity following another entity) • Leadership • Single file • Popular places (place visited by many) • …

3. Movement patterns: Challenge • Find definitions of movement patterns that are: • Useful and • 2. “Computable”

4. Movement patterns: Notation n – number of entities (trajectories)  - maximum complexity of a trajectory

5. Movement patterns: Groups

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

7. Movement patterns: Groups m – flock size Time = 0 1 2 3 4 5 6 7 8 m = 3

8. Movement patterns: Groups fixed subset variable subset flock meet examples for m = 3

9. Movement patterns: Convoy

10. Movement patterns: Convoy X – set of entities m – convoy size  – distance threshold Define a group: x,yX are directly connected if the -disks of x and y intersect. x1 and xk are -connected if there is a sequence x1,x2 ,…, xkof entities such that i, xi and xi+1 are directly connected. x  y x1 x5 x2 x4 x3

11. Movement patterns: Convoy A group of entities form a convoy if every pair of entities are -connected. [Jeung, Yiu, Zhou, Jensen & Shen, 2008][Jeung, Shen & Zhou, 2008]

12. Movement patterns: Leadership & Followers

13. front Movement patterns: Leadership & Followers • A leader? • - Should not follow anyone else! • Is followed by at least m • other entities. • For a certain duration.

14. Movement patterns: 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 log n) [Andersson et al. 2007]

15. Movement patterns: Popular places

16. Movement patterns: 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, G. & Wolle 2007]

17. Movement patterns: Popular places Continuous model: maximum number of visitors O(2n2) (2n2) Discrete model: maximum number of visitors O(n log n) (n log n)

18. Movement patterns: Single File

19. Movement patterns:Single File Single file: Intuitively easy to define Hard to define formally!

20. Movement patterns:Single File Single file: Intuitively easy to define Hard to define formally!

21. Movement patterns: Single File Single file: Intuitively easy to define Hard to define formally!

22. Towards a Formal Definition? We say that the 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.

23. Towards a Formal Definition? We say that the 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. Following behind? • Time t  Time t’  [t+min,t+max]

24. Define Following Behind Let x1 be an entity with parameterized trajectory f1 over the time interval [s1,t1] and let x2 be an entity with parameterized trajectory f2 over the time interval [s2,t2], where s2  [s1+min, s1+max] and t2 [t1+min, t1+max]. Entity x2 is following behind x1in [s1,t1] if there exists a continuous, bijective function : [s1,t1]  [s2,t2] such that (s1)=s2 and  t  [s1,t1]: (t)  [t-max,t-min]  d(f1((t)),f2(t)) . f2 f1

25. Define Following Behind Let x1 be an entity with parameterized trajectory f1 over the time interval [s1,t1] and let x2 be an entity with parameterized trajectory f2 over the time interval [s2,t2], where s2  [s1+min, s1+max] and t2 [t1+min, t1+max]. Entity x2 is following behind x1in [s1,t1] if there exists a continuous, bijective function : [s1,t1]  [s2,t2] such that (s1)=s2 and  t  [s1,t1]: (t)  [t-max,t-min]  d(f1((t)),f2(t)) . f2 f1

26. Free Space Diagram 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 P and Qhave Fréchet distance less than or equal to ε (p,q) (0,0)

27. Free Space Diagram and Following Behind The time delay means that the path will be restricted to a “diagonal” strip! FC(f1,f2) = {(s,t) | d(f1(s),f2(t))  t-s  [min,max]}. Running time: O(+k) where k is the complexity of the diagonal strip min max

28. Algorithm: Single file One can determine in O(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: Time: O(nk2)

29. No Order? If no order then compute the free space diagram between every pair of entities: Time: O(n2k2) x1 x2 x3 x4 x1 x2 x3 x4 ([3,8],[14,21]) [Buchin et al. 2008]

30. Trajectory grouping How to define and compute the structure of groups of moving entities, including merging and splitting? [Buchin, Buchin, van Kreveld, Speckmann and Staals 2013]

31. Grouping We do not want this to be considered: A merge of two groups into one, followed by a split of a group into two

32. Grouping We probably want this to be considered: A merge of two groups into one, followed by a split of a group into two

33. Grouping: definitions X – set of entities  - complexity of each input trajectory Define a group: x,yX are directly connected if the -disks of x and y intersect. x1 and xk are -connected if there is a sequence x1,x2 ,…, xkof entities such that i, xi and xi+1 are directly connected. x  y x1 x5 x2 x4 x3

34. Grouping: definitions A set S of entities is -connected if all entities in S are pairwise -connected.  The -disks of the entities in S form a connected component.

35. Grouping: definitions C(t) – the set of connected components at time t that forms a partition of X. C(t) consists of 5 components Time: t

36. Grouping: definitions What is a group? Three criteria for a group: • big enough (size m) • close enough (-connected) • long enough (duration δ) Only maximal groups are relevant (maximal in group size, starting time or ending time) m=4 

37. Grouping: definitions What is a group? Three criteria for a group: • big enough (size m) • close enough (-connected) • long enough (duration δ) Only maximal groups are relevant (maximal in group size, starting time or ending time) m=4 duration>δ

38. Grouping: definitions Note that an entity can be in several maximal groups at the same time!

39. Grouping:definitions Note that an entity can be in several maximal groups at the same time!

40. Grouping:definitions Note that an entity can be in several maximal groups at the same time!

41. Grouping: definitions Note that an entity can be in several maximal groups at the same time! t=0 t=1 t=3 t=2 t=[0,2] : red/blue t=[1,2] : red/blue/green t=[1,3] : red/green At time t=2 red is in 3 groups

42. Trajectory grouping structure

43. Trajectory grouping structure

44. Trajectory grouping structure blue t=10 red, blue red blue, green t=0 t=10 red, blue, green t=5 green t=0 red t=1 t=4 purple,green purple t=10 t=0 t=8

45. Grouping Questions: • Number of maximal groups? • How can we effectively compute all the maximal groups?

46. Grouping Idea: Consider the motion of the -disks of the entities over time.

47. Grouping • Idea: • Consider the motion of the -disks of the entities over time. • Union of n tubes. Denote this manifold by M. Each tube consists of  skewed cylinders with horizontal radius . We can see it as tracing the -disk of an entity over its trajectory.

48. Grouping We are interested in horizontal cross-sections, and the evolution of the connected components. This is captured by the so-called Reeb graphs.

49. Reeb graph Reeb graph, with maximal groups associated to edges captures the changes in connectivity of a process, using a graph • Edges are connected components • Vertices define changes in connected components (events)

50. Reeb graph Four types of vertices: start vertex (time t0) – in-degree 0, out-degree 1 end vertex (time t) – in-degree 1, out-degree 0 merge vertex - in-degree 2, out-degree 1 split vertex - in-degree 1, out-degree 2