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```Computational Geometry and
Spatial Data Mining
Marc van Kreveld
Department of Information and
Computing Sciences
Utrecht University
Clustering?
• 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
Clustering - options
•
•
•
•
Determine whether clustering occurs
Determine the degree of clustering
Determine the clusters
Determine the largest cluster
• Determine the outliers
Co-location
• Are the men clustered?
• Are the women clustered?
• Is there a co-location of men and women?
Co-location
• 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
Spatio-temporal data
• Locations have a time stamp
• Interesting patterns involve space and
time
Trajectory data
• Entities with a trajectory (time-stamped
motion path)
• Interesting patterns involve subgroups
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
Examples of trajectory data
•
•
•
•
•
Tracked animals (buffalo, birds, ...)
Tracked people (potential terrorists)
Tracked GSMs (e.g. for traffic purposes)
Sports scene analysis (players on a
soccer field)
Example pattern in trajectories
• What is the location visited by most
entities?
location =
circular region of
Example pattern in trajectories
• What is the location visited by most
entities?
location =
circular region of
4 entities
Example pattern in trajectories
• What is the location visited by most
entities?
location =
circular region of
3 entities
Example pattern in trajectories
• Compute buffer of each trajectory
Example pattern in trajectories
• Compute buffer of each trajectory
• Compute the arrangement
of the buffers and the cover
count of each cell
1
1
1
2
0
1
1
Example pattern in trajectories
• 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
Example pattern in trajectories
• 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
Patterns in entity data
Spatial data
Spatio-temporal data
• n points (locations)
• Distance is important
• n trajectories, each
has t time steps
• Distance is timedependent
– clustering pattern
• Presence of attributes
(e.g. man/woman):
– co-location patterns
– flock pattern
– meet pattern
are important and are
also time-dependent
Entities in subdivisions
• Also co-location pattern
• Discovered simply by overlay
E.g., occurrences of oaks
on different soil types
Clustering entities in subdivisions
• What if it is known
that the entities only
occur in regions of a
certain type?
Situation without
subdivision
cluster
bird nests
Clustering entities in subdivisions
• What if it is known
that the entities only
occur in regions of a
certain type?
Situation with
subdivision
land-water
cluster
bird nests
Clustering entities in subdivisions
house
car
burglary
Region-restricted clustering
Joint research with Joachim Gudmundsson (NICTA,
Sydney) and Giri Narasimhan (U of F, Miami), 2006
• 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?
Region-restricted clustering
• 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 r2 area of regions of F
r
≤ 2r
sum area ≤ r2
Region-restricted clustering
• Given a set P of n points, a set F of
polygons with nf 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?
Region-restricted clustering
• Exactly m points?
Every cluster with >m points
consists of clusters with m
points with smaller circles
• “Real” clustering
(partition)?
• Outliers?
m=5
Region-restricted clustering
• Exactly m points?
Every cluster with >m points
consists of clusters with m
points with smaller circles
• “Real” clustering
(partition)?
• Outliers?
m=5
Region-restricted clustering
1. Determine all smallest circles with m
points of P inside
2. Test if the radius is ≤ r (report) or > 2r
3. If the radius is in between, determine the
area of regions of F inside
Region-restricted clustering
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
ordinary =
order-1 VD
order-2 VD
order-3 VD
Region-restricted clustering
• 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
Region-restricted clustering
2. Test if the radius is ≤ r (report) or > 2r
Trivial in O(1) time per circle, so in O(nm)
time overall
Region-restricted clustering
3. Determine the area of regions of F inside
Brute force: O(nf) time per circle, so in
O(nmnf) time overall
Region-restricted clustering
• 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
Region-restricted clusters
• The anti-climax: we cannot give an exact
algorithm!
• If we takes squares instead of circles, we
can deal with the problem ....
Region-restricted clustering
3. Determine the area of regions of F inside
Brute force: O(nf) time per square, so in
O(nmnf) time overall
The total time for steps 1, 2, and 3 is
O(nm log n) + O(nm) + O(nmnf) =
O(nm log n + nmnf) time
Region-restricted clustering
3. Determine the area of regions of F inside
Using a suitable data structure (only
possible for squares): O(log2 nf) time per
square, so in O(nm log2 nf) time overall
The total time becomes
O(nm log n + nf log2 nf + nm log2 nf)
order- (m-2)
VD construction
preprocessing
of data structure
total query time
in data structure
Region-restricted clustering
• The squares solution generalizes to
regular polygons (e.g. 20-gons)
16-gon
• An approximation of the radius within (1+)r
gives a O(n/2 + nf log2 nf + n log nf /(m 2))
time algorithm
Region-restricted clustering
• Open problems:
– Develop a region-restricted version of k-means
– Region-restricted co-location?
– Replace region-restricted by gradual model
typical:
0 /unit
2 /unit 5 /unit
clusters:
8 /unit
Patterns in trajectories
• n trajectories, each with t time steps
 n polygonal lines with t vertices
• Already looked at most visited location
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, ......
flock
convergence
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)
...
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
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 = 0 1 2 3 4 5 6 7 8
m=3
longest flock
in [ 1.8 , 6.4 ]
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
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?
Patterns in trajectories
fixed subset
variable subset
flock
meet
examples for m = 3
Patterns in trajectories
fixed subset
flock
meet
NP-hard
O(n4 2 log n + n2 3)
variable subset
O(n3  log n)
O(n4 2 log n + n2 3)
Exact results ( input size is n  )
Patterns in trajectories
• A radius-2 approximation of the longest flock can be
computed in time O(n2  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
Patterns in trajectories
Approximate radius results ( input size is n  )
fixed subset
flock
meet
O(n2  log n)
variable subset
O((n2  log n) / 2)
factor 2
factor 2+ 
NP-hard
O(n3  log n)
O((n2  log n) / (m2)) O((n2  log n) / (m2))
factor 1+ 
factor 1+ 
O(n4 2 log n + n2 3)
O(n4 2 log n + n2 3)
Fixed subset flock
• It is NP-complete to decide if a graph has a subgraph
with m nodes that is a clique
v2
For every node of the graph,
make an entity with a trajectory
v1
v2
v3
v4
v5
v1
v6
v7
v4
v7
v3
v6
v5
r
all nodes not
v4, v5, and v7
Fixed subset flock
v2
v4 in flock
v1
v2
v3
v4
v5
v1
v6
v7
v4
v3
v6
v4 not in flock
v7
v5
Fixed subset flock
v2
v1
v1
v2
v3
v4
v5
v6
v7
v4
v7
v3
v6
v5
flock {v4,v5,v7} of
(full) duration 23
(3·7+2) and size 3
The trajectories have a fixed flock of size m and full
duration if and only if the graph has a clique of size m
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
v1
v2
v4 in flock
v3
v4
v5
v4 not in flock
v6
v7
Flock and meet algorithms
• Go into 3D (space-time) for algorithms
time
4
3
2
1
0
flock
meet
Fixed subset flock, approximation
algorithm of longest fixed flock exists
• Idea: if some vi is in the longest flock,
then all other entities are within
distance 2r from vi
flock
with vi
vi
2r
Fixed subset flock, approximation
• For each vj, we can determine the
O() time intervals where vj is in
the column of vi
• Maintain the intersections for all
entities in an augmented tree in
O(n  log n) time
• Do this for all columns (role of vi)
and report longest overall pattern
Total: O(n2  log n) time
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
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• Two entities define two cylinders
through time by tracing the two
Variable subset flock, exact
• A critical moment is where another
entity is on the boundary of the disk;
it may go outside or inside
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 ti and ti+1 there
are O(n3) critical moments  in total
there are O(n3 ) critical moments
2r
Variable subset flock, exact
• Let the O(n3 ) 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
Variable subset flock, exact
• The graph G can be built in O(n3  log n) time
• A maximum weight path can be found in
O(n3  log n) time
time
A longest variable
subset flock is a
maximum weight
path in G
Patterns in trajectories, summary
• Flock and meet patterns require algorithms in 3dimensional 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
Patterns in trajectories, summary
fixed subset
apx
flock
exact
O(n2  log n)
factor 2
NP-hard
variable subset
O((n2  log n) / 2)
factor 2+ 
O(n3  log n)
apx O((n2  log n) / (m2)) O((n2  log n) / (m2))
factor 1+ 
factor 1+ 
meet
exact
O(n4 2 log n + n2 3)
O(n4 2 log n + n2 3)
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
Patterns in trajectories
• Flock and meet patterns require algorithms in 3dimensional space (space-time)
• Exact algorithms are inefficient  only suitable
for smaller data sets
• Approximation can reduce running time with an
order of magnitude
To conclude
• With an exact definition of a spatial or spatiotemporal 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
To discuss
• 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
```
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