1. Marc van Kreveld (and Giri Narasimhan) Department of Information and Computing Sciences Utrecht University

4.  Are the people clustered in this room?  How do we define a cluster?  In spatial data mining we have objects/ entities with a location given by coordinates  Cluster definitions involve distance between locations  How do we define distance?

5.  Determine whether clustering occurs  Determine the degree of clustering  Determine the clusters  Determine the largest cluster  Determine the largest empty region  Determine the outliers

8.  Are the men clustered?  Are the women clustered?  Is there a co-location of men and women?  Determine regions favored exclusively by women. Men? Loners? Couples? Families?  Determine empty regions.

10.  Like before, we may be interested in  is there co-location?  the degree of co-location  the largest co-location  the co-locations themselves  the objects not involved in co-location  Regions with no (or little) co-location

12.  Locations have a time stamp  Interesting patterns involve space and time  Anomalies?

14.  Entities with a trajectory (time-stamped motion path)  Interesting patterns involve subgroups with similar heading, expected arrival, joint motion,...  n entities = trajectories; n = 10 – 100,000  t time steps; t = 10 – 100,000  input size is nt  m size subgroup (unknown); m = 10 – 100,000

15.  Tracked animals (buffalo, birds,...)  Tracked people (potential terrorists)  Tracked GSMs (e.g. for traffic purposes)  Trajectories of tornadoes  Sports scene analysis (players on a soccer field)

16.  What is the location visited by most entities? location = circular region of specified radius

17.  What is the location visited by most entities? location = circular region of specified radius 4 entities

18.  What is the location visited by most entities? location = circular region of specified radius 3 entities

19.  Compute buffer of each trajectory

20. 0 1 2 1 1 1 Compute the arrangement of the buffers and the cover count of each cell 1

21.  One trajectory has t time stamps; its buffer can be computed in O(t log t) time  All buffers can be computed in O(nt log t) time  The arrangement can be computed in O(nt log (nt) + k) time, where k = O( (nt) 2 ) is the complexity of the arrangement  Cell cover counts are determined in O(k) time

22.  Total: O(nt log (nt) + k) time  If the most visited location is visited by m entities, this is O(nt log (nt) + ntm)  Note: input size is nt ; n entities, each with location at t moments

23. Spatial data  n points (locations)  Distance is important  clustering pattern  Presence of attributes (e.g. man/woman):  co-location patterns Spatio-temporal data  n trajectories, each has t time steps  Distance is time- dependent  flock pattern  meet pattern  Heading and speed are important and are also time-dependent

24.  Also co-location pattern  Discovered simply by overlay E.g., occurrences of oaks on different soil types

25.  What if it is known that the entities only occur in regions of a certain type? bird nests radius of cluster Situation without subdivision

26.  What if it is known that the entities only occur in regions of a certain type? bird nests Situation with subdivision land-water radius of cluster

27. burglary house car

28.  Determine clusters in point sets that are sensitive to the geographic context (at least, for the relevant aspects)  Assume that a set of regions is given where points can only be, how should we define clusters? Joint research with Joachim Gudmundsson (NICTA, Sydney) and Giri Narasimhan (U of F, Miami), 2006

29.  Given a set P of points, a set F of regions, a radius r and a subset size m, a region-restricted cluster is a subset P’  P inside a circle C where  P’ has size at least m  C has radius at most 2r  C contains at most  r 2 area of regions of F ≤ 2r sum area ≤  r 2 r

30.  Given a set P of n points, a set F of polygons with n f edges in total, and values for r and m, report all region-restricted clusters of exactly m points  Exactly m points?  “Real” clustering (partition)?  Outliers?

31.  Exactly m points? Every cluster with >m points consists of clusters with m points with smaller circles  “Real” clustering (partition)?  Outliers? m = 5

32.  Exactly m points? Every cluster with >m points consists of clusters with m points with smaller circles  “Real” clustering (partition)?  Outliers? m = 5

33. 1. Determine all smallest circles with m points of P inside 2. Test if the radius is ≤ r (report) or > 2r (discard) 3. If the radius is in between, determine the area of regions of F inside

34. 1. Determine all minimal circles with m points of P inside 2. Determine all minimal circles with 3 points of P inside

35. ordinary = order-1 VD

36. 1. Determine all smallest circles with m points of P inside Use (m-2)-th order Voronoi diagram: cells where the same (m-2) points are closest Its vertices are centers of smallest circles around exactly m points

37. ordinary = order-1 VD

38. order-2 VD

39. order-3 VD

40.  The m-th order Voronoi diagram (or (m-2)) has O(nm) cells, edges, and vertices  It can be constructed in O(nm log n) time  we get O(nm) smallest circles with m points inside; for each we also know the radius

41. 2. Test if the radius is ≤ r (report) or > 2r (discard) Trivial in O(1) time per circle, so in O(nm) time overall

42. 3. Determine the area of regions of F inside Brute force: O(n f ) time per circle, so in O(nmn f ) time overall

43.  Complication: This need not give all region- restricted clusters!  Need to compute area of F inside a circle with moving center  Requires solving high-degree polynomials

44.  The anti-climax: we cannot give an exact algorithm!  If we takes squares instead of circles, we can deal with the problem....

45. 3. Determine the area of regions of F inside Brute force: O(n f ) time per square, so in O(nmn f ) time overall The total time for steps 1, 2, and 3 is O(nm log n) + O(nm) + O(nmn f ) = O(nm log n + nmn f ) time

46. 3. Determine the area of regions of F inside Using a suitable data structure (only possible for squares): O(log 2 n f ) time per square, so in O(nm log 2 n f ) time overall The total time becomes O(nm log n + n f log 2 n f + nm log 2 n f ) order- (m-2) VD construction preprocessing of data structure total query time in data structure

47.  The squares solution generalizes to regular polygons (e.g. 20-gons)  An approximation of the radius within (1+  )r gives a O(n/  2 + n f log 2 n f + n log n f /(m  2 )) time algorithm 16-gon

48.  Open problems:  Develop a region-restricted version of k-means clustering, single link clustering,...  Region-restricted co-location?  Replace region-restricted by gradual model 0 /unit 2 /unit5 /unit8 /unit typical:clusters:

49.  n trajectories, each with t time steps  n polygonal lines with t vertices  Already looked at most visited location

50. Patterns in trajectories Flock: near positions of (sub)trajectories for some subset of the entities during some time Convergence: same destination region for some subset of the entities Encounter: same destination region with same arrival time for some subset of the entities Similarity of trajectories Same direction of movement, leadership,...... flockconvergence

51. Patterns in trajectories Flocking, convergence, encounter patterns –Laube, van Kreveld, Imfeld (SDH 2004) –Gudmundsson, van Kreveld, Speckmann (ACM GIS 2004) –Benkert, Gudmundsson, Huebner, Wolle (ESA 2006) –... Similarity of trajectories –Vlachos, Kollios, Gunopulos (ICDE 2002) –Shim, Chang (WAIM 2003) –... Lifelines, motion mining, modeling motion –Mountain, Raper (GeoComputation 2001) –Kollios, Scaroff, Betke (DM&KD 2001) –Frank (GISDATA 8, 2001) –...

52. Patterns in trajectories Flock: near positions of (sub)trajectories for some subset of the entities during some time –clustering-type pattern –different definitions are used Given: radius r, subset size m, and duration T, a flock is a subset of size  m that is inside a (moving) circle of radius r for a duration  T

54. Patterns in trajectories Longest flock: given a radius r and subset size m, determine the longest time interval for which m entities were within each other’s proximity (circle radius r) Time = 016543278 longest flock in [ 1.8, 6.4 ] m = 3

55. Patterns in trajectories Meet: near some position of (sub)trajectories for some subset of the entities –clustering-type pattern Given: radius r, subset size m, and duration T, a meet is a subset of size  m that is inside a (stationary) circle of radius r for a duration  T this was “moving” for flock

57. Patterns in trajectories The same subset required for a flock or meet? Example: meet with m = 4; duration is 3 + time steps or 4 + time steps?

58. Patterns in trajectories flock meet fixed subsetvariable subset examples for m = 3

59. Patterns in trajectories Exact results ( input size is n  ) NP-hard O(n 3  log n) O(n 4  2 log n + n 2  3 ) fixed subsetvariable subset flock meet O(n 4  2 log n + n 2  3 )

60. Patterns in trajectories A radius-2 approximation of the longest flock can be computed in time O(n 2  log n)... meaning: if the longest flock of size m for radius r has duration T, then we surely find a flock of size m and duration  T for radius 2r longest flock for r at least as long a flock for 2r

61. Patterns in trajectories Approximate radius results ( input size is n  ) flock meet fixed subsetvariable subset O(n 2  log n) O((n 2  log n) /  2 ) O((n 2  log n) / (m  2 )) factor 2 factor 2+  factor 1+  NP-hard O(n 3  log n) O(n 4  2 log n + n 2  3 )

62. v3v3 Fixed subset flock It is NP-complete to decide if a graph has a subgraph with m nodes that is a clique v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 For every node of the graph, make an entity with a trajectory all nodes not adjacent to v 1 go here v1v1 v2v2 v4v4 v5v5 v6v6 v7v7 v 1 is not adjacent to v 4, v 5, and v 7 r

63. v3v3 Fixed subset flock v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v1v1 v2v2 v4v4 v5v5 v6v6 v7v7 v 4 not in flock v 4 in flock

64. v3v3 Fixed subset flock v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v1v1 v2v2 v4v4 v5v5 v6v6 v7v7 The trajectories have a fixed flock of size m and full duration if and only if the graph has a clique of size m flock {v 4,v 5,v 7 } of (full) duration 23 (3·7+2) and size 3

65. Fixed subset flock Longest fixed flock is NP-hard Max clique has no approximation  cannot approximate duration, nor flock size The reduction applies for all radii < 2r v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v 4 not in flock v 4 in flock

66. Flock and meet algorithms Go into 3D (space-time) for algorithms time 0 1 2 4 3 flockmeet

67. Fixed subset flock, approximation An efficient radius-2 approximation algorithm of longest fixed flock exists Idea: if some v i is in the longest flock, then all other entities are within distance 2r from v i radius 2r, centered at v i vivi flock with v i 2r2r

68. Fixed subset flock, approximation For each v j, we can determine the O(  ) time intervals where v j is in the column of v i Maintain the intersections for all entities in an augmented tree in O(n  log n) time Do this for all columns (role of v i ) and report longest overall pattern Total: O(n 2  log n) time

69. Variable subset flock, exact The subset that forms the flock may change entities, but must stay of size  m Any flock subset at any instant has a disk D of radius r with at least 2 entities on the boundary  defining entities r defining entities

70. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

71. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

72. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

73. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

74. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

75. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

76. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

77. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

78. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

79. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

80. Variable subset flock, exact Two entities define two cylinders through time by tracing the two possible radius r disks

81. Variable subset flock, exact A critical moment is where another entity is on the boundary of the disk; it may go outside or inside

82. Variable subset flock, exact At a critical moment: –a variable subset flock may start (m entities) –a variable subset flock may stop (<m entities) –Three pairs of defining entities have disks that coincide There are also critical moments when two entities are at distance exactly 2r Between two time steps t i and t i+1 there are O(n 3 ) critical moments  in total there are O(n 3  ) critical moments 2r2r

83. Variable subset flock, exact Let the O(n 3  ) critical moments be the nodes in a directed acyclic graph G Edges of G are between two consecutive critical moments of the same two defining entities –directed from earlier to later –weight is time between critical moments –only if at least m entities are inside the disk time A longest variable subset flock is a maximum weight path in G

84. Variable subset flock, exact The graph G can be built in O(n 3  log n) time A maximum weight path can be found in O(n 3  log n) time time A longest variable subset flock is a maximum weight path in G

85. Patterns in trajectories, summary Flock and meet patterns require algorithms in 3- dimensional space (space-time) Exact algorithms are inefficient  only suitable for smaller data sets Approximation can reduce running time with one or two orders of magnitude

86. Patterns in trajectories, summary flock meet fixed subsetvariable subset O(n 2  log n) O((n 2  log n) /  2 ) O((n 2  log n) / (m  2 )) factor 2 factor 2+  factor 1+  NP-hard O(n 3  log n) apx exact apx exact O(n 4  2 log n + n 2  3 )

87. Future research on longest trajectories Faster exact and approximation algorithms Better approximation factors Remove restriction of fixed shape of flocking region (compact or elongated both possible during same flock) Longest duration convergence longest convergence

88.  Flock and meet patterns require algorithms in 3- dimensional space (space-time)  Exact algorithms are inefficient  only suitable for smaller data sets  Approximation can reduce running time with an order of magnitude

89.  With an exact definition of a spatial or spatio- temporal pattern, geometric algorithms can be used to compute all patterns  Many known structures from computational geometry are useful (Voronoi diagrams, arrangements,...)  Since the (exact) algorithms may be inefficient, approximation may be a solution

90.  What patterns must be detected in practice (both spatial and spatio-temporal)?  What is the most appropriate definition (formalization) of these?  Spatial association rules, auto-correlation, irregularities, classification,... and other computable things in spatial/spatio-temporal data mining