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Computational Geometry and Spatial Data Mining

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

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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 - How do we define distance?

Clustering - options

- Determine whether clustering occurs
- Determine the degree of clustering
- Determine the clusters
- Determine the largest cluster
- Determine the largest empty region
- Determine the outliers

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Co-location

- 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.

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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
- Regions with no (or little) co-location

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Spatio-temporal data

- Locations have a time stamp
- Interesting patterns involve space and time
- Anomalies?

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Trajectory data

- Entities with a trajectory (time-stamped motion

path) - Interesting patterns involve subgroupswith

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

Examples of trajectory data

- 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)

Example pattern in trajectories

- What is the location visited by most entities?

location circular region of specified radius

Example pattern in trajectories

- What is the location visited by most entities?

location circular region of specified radius

4 entities

Example pattern in trajectories

- What is the location visited by most entities?

location circular region of specified radius

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

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 bym

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

- 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

- Spatial data
- n points (locations)
- Distance is important
- clustering pattern
- Presence of attributes (e.g. man/woman)
- co-location patterns

Entities in subdivisions

- Also co-location pattern
- Discovered simply by overlayE.g., occurrences

of oakson 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

radius of 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 subdivisionland-water

radius of cluster

bird nests

Clustering entities in subdivisions

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, aregion-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 gtm 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 gtm points

consists of clusters with m points with smaller

circles - Real clustering (partition)?
- Outliers?

m 5

Region-restricted clustering

- Determine all smallest circles with m points of P

inside - Test if the radius is r (report) or gt 2r

(discard) - If the radius is in between, determine the area

of regions of F inside

Region-restricted clustering Step 1

- Determine all minimal circles with m points of P

inside - Determine all minimal circles with 3 points of P

inside

ordinary order-1 VD

Region-restricted clustering

- 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 radius

Region-restricted clustering

- 2. Test if the radius is r (report) or gt 2r

(discard) 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 isO(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)

total query time in data structure

order- (m-2) VD construction

preprocessing of data structure

Region-restricted clustering

- The squares solution generalizes toregular

polygons (e.g. 20-gons) - An approximation of the radius within (1?)r

gives a O(n/?2 nf log2 nf n log nf /(m ?2))

time algorithm

16-gon

Region-restricted clustering

- Open problems
- Develop a region-restricted version of k-means

clustering, single link clustering, ... - Region-restricted co-location?
- Replace region-restricted by gradual model

typical

clusters

0 /unit

2 /unit

5 /unit

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 (DMKD 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

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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 others proximity

(circle radius r)

Time 0

1

6

5

4

3

2

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

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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

variable subset

NP-hard

O(n3 ? log n)

flock

O(n4 ?2 log n n2 ?3)

meet

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 rhas

duration T, then we surely find a flock of size m

and duration ? T for radius 2r

Patterns in trajectories

Approximate radius results ( input size is n ? )

fixed subset

variable subset

flock

O(n2 ? log n)

O((n2 ? log n) / ?2)

factor 2

factor 2 ?

O(n3 ? log n)

NP-hard

meet

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

v7

v2

v4

For every node of the graph, make an entity with

a trajectory

v1

v3

v5

v1

v2

v3

v4

v5

v6

v7

v6

v1 is not adjacent to v4, v5, and v7

all nodes notadjacent to v1 go here

Fixed subset flock

v7

v2

v4

v1

v3

v1

v2

v3

v4

v5

v6

v7

v5

v6

Fixed subset flock

v7

v2

v4

v1

v3

v1

v2

v3

v4

v5

v6

v7

v5

v6

flock v4,v5,v7 of (full) duration 23 (372)

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 lt 2r

v4 in flock

v1

v2

v3

v4

v5

v6

v7

v4 not in flock

Flock and meet algorithms

- Go into 3D (space-time) for algorithms

time

4

3

duration

2

duration

1

0

flock

meet

Fixed subset flock, approximation

- An efficient radius-2 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

radius 2r, centered at 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 inO(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

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

Variable subset flock, exact

- Two entities define two cylinders through time by

tracing the two possible radius r disks

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 (ltm 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 ti1 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

A longest variable subset flock is a maximum

weight path in G

time

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

A longest variable subset flock is a maximum

weight path in G

time

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

Patterns in trajectories, summary

fixed subset

variable subset

apx

O(n2 ? log n)

O((n2 ? log n) / ?2)

factor 2

factor 2 ?

flock

NP-hard

O(n3 ? log n)

exact

O((n2 ? log n) / (m?2))

apx

O((n2 ? log n) / (m?2))

factor 1 ?

factor 1 ?

meet

O(n4 ?2 log n n2 ?3)

O(n4 ?2 log n n2 ?3)

exact

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

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

To conclude

- 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

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