Computational Geometry and Spatial Data Mining

Marc van Kreveld (and Giri Narasimhan)Department of Information and Computing SciencesUtrecht University

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 occursDetermine the degree of clusteringDetermine the clustersDetermine the largest clusterDetermine 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 timeAnomalies?

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 tornadoesSports scene analysis (players on a

soccer field)

What is the location visited by most entities?

location = circular region of specified radius



4 entities



3 entities

Compute buffer of each trajectory

Compute buffer of each trajectory

0

1

2

1

11

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

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 by

m 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 patternDiscovered simply by overlay

E.g., occurrences of oakson different soil types

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

bird nestsradius of cluster

Situation without subdivision

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

bird nests

Situation with subdivisionland-water

radius of cluster

burglary

housecar

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

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

≤ 2r sum area ≤ r2

r

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

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

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

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

ordinary =order-1 VD

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

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 insideBrute 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 insideUsing 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 becomesO(nm log n + nf log2 nf + nm log2 nf)

order- (m-2)VD construction

preprocessingof data structure

total query timein 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 log2 nf + 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

0 /unit 2 /unit 5 /unit 8 /unit

typical: clusters:

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 65432 7 8

longest flock in [ 1.8 , 6.4 ]

m = 3

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

flock

meet

fixed subset variable subset

examples for m = 3

Patterns in trajectories

Exact results ( input size is n )

NP-hard O(n3 log n)

O(n4 2 log n + n2 3)


flock

meet O(n4 2 log n + n2 3)

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

longest flock for r at least as long a flock for 2r

Patterns in trajectoriesApproximate radius results ( input size is n )

flock

meet


O(n2 log n) O((n2

log n) / 2)

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

log n) / (m2))

factor 2 factor 2+

factor 1+ factor 1+

NP-hard O(n3 log n)

O(n4 2 log n + n2 3) O(n4 2 log n + n2 3)

v3

Fixed subset flock• It is NP-complete to decide if a graph has a subgraph

with m nodes that is a clique

v1 v2 v3 v4 v5 v6 v7

For every node of the graph,make an entity with a trajectory

all nodes notadjacent to v1 go here

v1

v2 v4

v5v6

v7

v1 is not adjacent tov4, v5, and v7

r

v3

Fixed subset flock

v1 v2 v3 v4 v5 v6 v7

v1

v2 v4

v5v6

v7

v4 not in flock

v4 in flock

v3

Fixed subset flock

v1 v2 v3 v4 v5 v6 v7

v1

v2 v4

v5v6

v7

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 {v4,v5,v7} of (full) duration 23 (3·7+2) and size 3

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 v3 v4 v5 v6 v7

v4 not in flock

v4 in flock

Flock and meet algorithms• Go into 3D (space-time) for algorithms

time

0

1

2

4

3

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

radius 2r, centered at vi

vi

flock with 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

r

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• 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 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

flock

meet


O(n2 log n) O((n2

log n) / 2)

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

log n) / (m2))

factor 2 factor 2+

factor 1+ factor 1+

NP-hard O(n3 log n)

apx

exact

apx

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

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

Computational Geometry and Spatial Data Mining

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