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Computational Geometry and Spatial Data Mining

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Marc van Kreveld (and Giri Narasimhan)

Department of Information and Computing Sciences

Utrecht University

Computational Geometry and Spatial Data Mining

- 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?

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

- 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.

- 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

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

- 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

- 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)

- What is the location visited by most entities?

location = circular region of specified radius

- What is the location visited by most entities?

location = circular region of specified radius

4 entities

- What is the location visited by most entities?

location = circular region of specified radius

3 entities

- Compute buffer of each trajectory

1

- Compute buffer of each trajectory

- Compute the arrangement of the buffers and the cover count of each cell

1

1

1

2

0

1

- 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

- 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

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

- Also co-location pattern
- Discovered simply by overlayE.g., occurrences of oakson different soil types

- What if it is known that the entities only occur in regions of a certain type?

Situation without subdivision

radius of cluster

bird nests

- 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

house

car

burglary

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?

- 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

- 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?

- Exactly m points?Every cluster with >m points consists of clusters with m points with smaller circles
- “Real” clustering (partition)?
- Outliers?

m = 5

- Exactly m points?Every cluster with >m points consists of clusters with m points with smaller circles
- “Real” clustering (partition)?
- Outliers?

m = 5

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

- Determine all minimal circles with m points of P inside
- Determine all minimal circles with 3 points of P inside

ordinary =

order-1 VD

- 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

- 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

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

3. Determine the area of regions of F inside

Brute force: O(nf) time per circle, so in O(nmnf) time overall

- 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

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

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

3. Determine the area of regions of F inside

Using a suitable data structure (only possible for squares): O(log2nf) time per square, so in O(nm log2 nf) time overall

The total time becomes

O(nm log n + nflog2 nf +nm log2 nf)

total query time

in data structure

order- (m-2)

VD construction

preprocessing

of data structure

- The squares solution generalizes toregular polygons (e.g. 20-gons)
- An approximation of the radius within (1+)r gives a O(n/2 + nf log2nf + n log nf /(m 2)) time algorithm

16-gon

- 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

- n trajectories, each with t time steps n polygonal lines with t vertices
- Already looked at most visited location

- 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

- 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)
- ...

- 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

- 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 ]

- 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

- The same subset required for a flock or meet?

Example: meet with m = 4; duration is 3+ time steps or 4+ time steps?

fixed subset

variable subset

flock

meet

examples for m = 3

fixed subset

variable subset

O(n3 log n)

NP-hard

flock

meet

O(n42 log n + n23)

O(n42 log n + n23)

Exact results ( input size is n )

longest flock for r

at least as long a flock for 2r

- 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

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) / (m2))

O((n2 log n) / (m2))

factor 1+

factor 1+

O(n42 log n + n23)

O(n42 log n + n23)

r

- 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

v4 in flock

v4 not in flock

v7

v2

v4

v1

v3

v1

v2

v3

v4

v5

v6

v7

v5

v6

v7

v2

v4

v1

v3

v1

v2

v3

v4

v5

v6

v7

v5

v6

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

- Longest fixed flock is NP-hard
- Max clique has no approximation cannot approximate duration, nor flock size
- The reduction applies for all radii < 2r

v4 in flock

v1

v2

v3

v4

v5

v6

v7

v4 not in flock

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

time

4

3

duration

2

duration

1

0

flock

meet

- 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

- 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

r

defining entities

- 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

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- Two entities define two cylinders through time by tracing the two possible radius r disks

- A critical moment is where another entity is on the boundary of the disk; it may go outside or inside

- 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

- Let theO(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

- The graph G can be built inO(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

- 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

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

apx

O((n2 log n) / (m2))

O((n2 log n) / (m2))

factor 1+

factor 1+

meet

O(n42 log n + n23)

O(n42 log n + n23)

exact

- 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

- 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

- 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

- 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