computational movement analysis lecture 4: movement patterns joachim gudmundsson

Computational Movement Analysis

Lecture 4: Movement patterns

Joachim Gudmundsson

Movement patterns

Much of the early work on movement analysis focussed on finding movement patterns.

For example:• Flocks (group of entities moving close together)• Swarm • Convoys • Heards• Following (is an entity following another entity)• Leadership• Single file• Popular places (place visited by many)• …

Movement patterns: Challenge

Find definitions of movement patterns that are:

1. Useful and

2. “Computable”

Movement patterns: Notation

n – number of entities (trajectories)

- maximum complexity of a trajectory

Movement patterns: Groups

t1

t2

t3t4

Kalnis, Mamoulis & Bakiras, 2005 G. & van Kreveld, 2006Benkert, G., Hubner & Wolle, 2006Jensen, Lin & Ooi, 2007Al-Naymat, Chawla & G., 2007Vieria, Bakalov & Tsotras, 2009Jeung, Yiu, Zhou, Jensen & Shen, 2008Jeung, Shen & Zhou, 2008...


Time = 01 65432 7 8m = 3

m – flock size


flock

meet

fixed subset variable subset

examples for m = 3


Movement patterns: Convoy


X – set of entitiesm – convoy size – distance threshold

Define a group:x,yX are directly connected if the -disks of x and y intersect.

x1 and xk are -connected if there is a sequence x1,x2 ,…, xk of entities such that i, xi and xi+1 are directly connected.

x

y

x1

x2 x3x4

x5

[Jeung, Yiu, Zhou, Jensen & Shen, 2008][Jeung, Shen & Zhou, 2008]

A group of entities form a convoyif every pair of entities are -connected.


Movement patterns: Leadership & Followers

front

A leader?

- Should not follow anyone else!- Is followed by at least m

other entities.- For a certain duration.


A leader?

- Should not follow anyone else!- Is followed by at least m

other entities.- For a certain duration.

Many different settings. Running time ~O(n2 log n)


[Andersson et al. 2007]

Movement patterns: Popular places

σ

A region is a popular place if at least m entities visit it.

σ is a popular place for m 5

[Benkert, Djordjevic, G. & Wolle 2007]



Continuous model: maximum number of visitors O(2n2)

(2n2)

Discrete model: maximum number of visitors O(n log n)

(n log n)

Movement patterns: Single File

Movement patterns: Single File

Single file: Intuitively easy to define

Hard to define formally!

Towards a Formal Definition?

We say that the entities x1, … , xm are moving in single file for a given time interval if during this time each entityxj+1 is following behind entity xj for j = 1, … ,m-1.

Towards a Formal Definition?

Time t’ [t+min,t+max]

Time t

We say that the entities x1, … , xm are moving in single file for a given time interval if during this time each entityxj+1 is following behind entity xj for j = 1, … ,m-1.

Following behind?

Define Following Behind

Let x1 be an entity with parameterized trajectory f1 over the time interval [s1,t1] and let x2 be an entity with parameterized trajectory f2 over the time interval [s2,t2], where s2 [s1+min, s1+max] and t2 [t1+min, t1+max].

Entity x2 is following behind x1 in [s1,t1] if there exists a continuous, bijective function : [s1,t1] [s2,t2] such that (s1)=s2 and t [s1,t1]: (t) [t-max,t-min] d(f1((t)),f2(t)) .

f1

f2

Define Following Behind

f1

f2

Let x1 be an entity with parameterized trajectory f1 over the time interval [s1,t1] and let x2 be an entity with parameterized trajectory f2 over the time interval [s2,t2], where s2 [s1+min, s1+max] and t2 [t1+min, t1+max].

Entity x2 is following behind x1 in [s1,t1] if there exists a continuous, bijective function : [s1,t1] [s2,t2] such that (s1)=s2 and t [s1,t1]: (t) [t-max,t-min] d(f1((t)),f2(t)) .

Free Space Diagram

If there exists a monotone path in the free-space diagram from (0, 0) to (p, q) which is monotone in both coordinates, then curves P and Q have Fréchet distance less than or equal to ε

(0,0)

(p,q)

Free Space Diagram and Following Behind

The time delay means that the path will be restricted to a “diagonal” strip!

FC(f1,f2) = {(s,t) | d(f1(s),f2(t)) t-s [min,max]}.

min max

Running time: O(+k) where k is the complexity of the diagonal strip

Algorithm: Single file

One can determine in O(k2) time during which time intervals one trajectory is following behind the other.

If the order between the entities is specified then compute the free space diagram between every pair of consecutive entities:

Time: O(nk2)

No Order?

If no order then compute the free space diagram between every pair of entities:

Time: O(n2k2)

x1

x2

x3

x4

([3,8],[14,21])

x1

x2

x3

x4

[Buchin et al. 2008]

Trajectory grouping

How to define and compute the structure of groups of moving entities, including merging and splitting?

[Buchin, Buchin, van Kreveld, Speckmann and Staals 2013]

Grouping

We do not want this to be considered: A merge of two groups into one, followed by a split of a group into two

Grouping

We probably want this to be considered: A merge of two groups into one, followed by a split of a group into two

Grouping: definitions

X – set of entities - complexity of each input trajectory

Define a group:x,yX are directly connected if the -disks of x and y intersect.

x1 and xk are -connected if there is a sequence x1,x2 ,…, xk of entities such that i, xi and xi+1 are directly connected.

x

y

x1

x2 x3x4

x5


A set S of entities is -connected if all entities in S are pairwise -connected. The -disks of the entities in S form a connected component.


C(t) – the set of connected components at time t that forms a partition of X.

C(t) consists of 5 components

Time: t


What is a group?

Three criteria for a group:- big enough (size m)- close enough (-connected)- long enough (duration δ)

Only maximal groups are relevant (maximal in group size, starting time or ending time)

m=4


What is a group?

Three criteria for a group:- big enough (size m)- close enough (-connected)- long enough (duration δ)

Only maximal groups are relevant (maximal in group size, starting time or ending time)

m=4

duration>δ


Note that an entity can be in several maximal groups at the same time!

Grouping:definitions




t=0 t=1

t=2t=3

t=[0,2] : red/bluet=[1,2] : red/blue/greent=[1,3] : red/green

At time t=2 red is in 3 groups

Trajectory grouping structure

Trajectory grouping structure

purple

red, blue, greenblue, green

red greenpurple, green

red

blue

red, blue

t=0

t=1 t=4t=5

t=8

t=10t=0

t=0

t=10

t=10

Grouping

Questions: 1. Number of maximal groups?2. How can we effectively compute all the maximal

groups?

Grouping

Idea:Consider the motion of the -disks of the entities over time.

Grouping

Idea:Consider the motion of the -disks of the entities over time.

Þ Union of n tubes. Denote this manifold by M.

Each tube consists of skewed cylinders with horizontal radius .

We can see it as tracing the -disk of an entity over its trajectory.

Grouping

We are interested in horizontal cross-sections, and the evolution of the connected components.

This is captured by the so-called Reeb graphs.

Reeb graph

Reeb graph, with maximal groups associated to edges captures the changes in connectivity of a process, using a graph

– Edges are connected components– Vertices define changes in connected components (events)

Reeb graph

Four types of vertices: start vertex (time t0) – in-degree 0, out-degree 1end vertex (time t) – in-degree 1, out-degree 0merge vertex - in-degree 2, out-degree 1split vertex - in-degree 1, out-degree 2

Complexity of the Reeb graph

Observation:Reeb graph of M has (n2) vertices and edges.

Proof:Construction for one time step [ti,ti+1].


Observation:Reeb graph of M has (n2) vertices and edges.

Proof:Construction for one time step [ti,ti+1]. n/2 n/2 = (n2) components/ time step. □


Observation:The Reeb graph of M has O(n2) vertices.

Proof:Consider two entities during one time interval [ti,ti+1].

This interval can generate at most two vertices in the the Reeb graph.


Observation:The Reeb graph of M has O(n2) vertices.

Proof:Consider two entities during one time interval [ti,ti+1].

This interval can generate at most two vertices in the the Reeb graph.

O(n2) components/time step.

□

Computing the Reeb graph

Compute all events:For every pair of skewed cylinders compute the time of intersection.Time: O(n2)

Sort the events with respect to time:Time: O(n2 log n)


Compute & sort all events:Time: O(n2 log n)

Initialize graph:Time: O(n2)

Handle events by increasing time:Time: O(n2 log n)Using a dynamic ST-trees [Sleator & Tarjan’83]

[Parsa‘12]


Theorem:The Reeb graph of M can be computed in time O(n2 log n).

Number of maximal groups

Each entity follows a directed path in the Reeb graph starting at a start vertex andending at an end vertex.

purple



red

blue

red, blue

t=0

t=1 t=4t=5

t=8

t=10

t=0

t=0

t=10

t=10


Theorem: There are O(n3) maximal groups.

Proof:There are O(n2) vertices in the Reeb graph. Prove that O(n) maximal groups can start at a start or merge vertex.

purple



red

blue

red, blue

t=0

t=1 t=4t=5

t=8

t=10

t=0

t=0

t=10

t=10


Consider a merge vertex v. Assume sets S and T merge at v.

S Tv


Consider a merge vertex v. Assume sets S and T merge at v.Let px denote the path of entity x ST starting at v.

S T

px

v



Consider the union R’ of all such paths from entities in ST.R’ is a directed acyclic subgraph of the Reeb graph.

S Tv



Consider the union R’ of all such paths from entities in ST.R’ is a directed acyclic subgraph of the Reeb graph.“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u. S T

v

px

py

R’


“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.

v

px py

v

px py



v

px

v

px pypy



v v


“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.A maximal group can only end at an end or split vertex.

v


“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.A maximal group can only end at an end or split vertex. Number of leaves = |S|+|T| n

v


“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.A maximal group can only end at an end or split vertex. Number of leaves = |S|+|T| nNumber of split vertices?Every vertex has degree at most 3. O(n) split vertices.

v


“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.A maximal group can only end at an end or split vertex. Number of leaves = |S|+|T| nNumber of split vertices?Every vertex has degree at most 3. O(n) split vertices.There can be at most one maximal group starting at v and ending at a split vertex w. O(n) maximal groups starting at v.

v


Theorem: There are O(n3) maximal groups.

Proof:There are O(n2) vertices in the Reeb graph. There are O(n) maximal groups can start at a start or merge vertex. □

v

Computing the maximal groups

Input: Given a set of trajectories and three parameters m, and .

1. Compute the Reeb graph using distance . Time: O(n2 log n)2. Process the vertices in time-order, maintaining maximal

groups

t0

tG

1234

(1,t0)

(2,t0)(3,t0)

(4,t0)

t1 t2

(1,t0)(2,t0)

(12,t1)

(3,t0)(4,t0)

(34,t1) t3 t4

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)

(1234,t2)

1,3

2,4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)


2. Maintain maximal groupsEach edge e=(u,v) is labelled with a set of maximal groups Ge.Note: A group G becomes maximal at a vertex.

t0

tG

1234

(1,t0)

(2,t0)(3,t0)

(4,t0)

t1 t2

(1,t0)(2,t0)

(12,t1)

(3,t0)(4,t0)

(34,t1) t3 t4

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)

(1234,t2)

1,3

2,4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)


2. Annotate its edges and verticesEach edge e=(u,v) is labelled with a set of maximal groups Ge.Note that a group G becomes maximal at a vertex v.

a. Start vertex: Set Ge = {(Ce,tv=t0)}

v

Ge={(Ce,tv=t0)}

u



a. Start vertex: Set Ge = {(Ce,tv=t0)}b. Merge vertex: Propagate the maximal groups from “children” to e and add (Ce,tv).

v

u

Ge1 Ge

2

Ge={Ge1Ge2(Ce,tv)}



a. Start vertex: Set Ge = {(Ce,tv=t0)}b. Merge vertex: Propagate the maximal groups from “children” to e and add (Ce,tv).c. Split vertex:

vGe1

Ge

Ge2




- A group may end at v (if group splits)- A group may start at v on either e1 or e2 - A group may continue on e1 or e2

vGe1

Ge

Ge2

Question: How can we compute these groups efficiently?

Store maximal groups

e

Observation: Assume S, T in Ge. S starting at tS and T starting at tT.

tS

tT

Ge = {S,T,…}


e


• G ST in Ge with starting time tG max{tS,tT}.

tS

tT

tG Ge = {S,T,G,…}


e


• G ST in Ge with starting time tG max{tS,tT}.

• If ST then ST or T S.

tS

tT

tG Ge = {S,T,G,…}

S and T are disjoint, ST or T S

Represent Ge by a tree Te.

Each node v in Te represents a group Gv in Ge.The children of v are the largest subgroups of of Gv.

t0

tG

1234

(1,t0)

(2,t0)(3,t0)

(4,t0)

t1 t2

(1,t0)(2,t0)

(12,t1)

(3,t0)(4,t0)

(34,t1) t3 t4

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)

(1234,t2)

1,3

2,4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)


Represent Ge by a tree Te.

Each node v in Te represents a group Gv in Ge.The children of v are the largest subgroups of of Gv.

t0

1234

(1,t0)

(2,t0)(3,t0)

(4,t0)

t1 t2

(1,t0)(2,t0)

(12,t1)

(3,t0)(4,t0)

(34,t1) t3 t4

1,3

2,4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)

(1234,t2)

{1,2,3,4}

{1,2} {3,4}

{1} {2} {3} {4}



2. Annotate its edges and verticesEach edge e=(u,v) is labelled with a set of maximal groups Ge.



(1234,t2)

(12,t1) (34,t1)

(1,t0) (2,t0) (3,t0) (4,t0)t2 t3 t4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)(1234,t2)

1,3

2,4





(1234,t2)

(12,t1) (34,t1)

(1,t0) (2,t0) (3,t0) (4,t0)

(1,t0)

(2,t0)

(3,t0)

(4,t0)t2 t3 t4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)(1234,t2)

1,3

2,4





t2 t3 t4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)(1234,t2)

(1234,t2)

(12,t1) (34,t1)

(1,t0) (2,t0) (3,t0) (4,t0)

(1,t0)

(2,t0)

(3,t0)

(4,t0)

1,3

2,4





t2 t3 t4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)(1234,t2)

(1234,t2)

(12,t1) (34,t1)

(1,t0) (2,t0) (3,t0) (4,t0)

(1,t0)

(2,t0)

(3,t0)

(4,t0)

(13,t3)

(24,t3)

1,3

2,4


AnalysisStart vertex: O(1)Merge vertex: O(1)Split vertex: O(|Te|) and |Te|=O(n)Total time: #vertices in the Reeb graph O(n) = O(n3)

+ the total output size

t2 t3 t4

(1,t0)(3,t0)

(13,t2)

(2,t0)(4,t0)

(24,t2)

(1,t0)(2,t0) (3,t0)(4,t0)

(12,t1)(34,t1)(1234,t2)

(1234,t2)

(12,t1) (34,t1)

(1,t0) (2,t0) (3,t0) (4,t0)

(1,t0)

(2,t0)

(3,t0)

(4,t0)

(13,t3)

(24,t3)

1,3

2,4

Changing group size and duration

minimum group size 3

For illustration: x-coordinate is time

smaller minimum group size (2)

larger minimum group size (5)

larger minimum duration

smaller minimum duration

Robustness

If a group of 6 entities has 1 entity leaving very briefly, should we really see this as

- two maximal groups of size 6, and- one maximal group of size 5 ?

6 entities

6 entities5 entities

Robust grouping structure

Entities are -connected at time t if they are directly connected at some time t‘ in [t-/2, t+/2]

Robust grouping structure

Modify Reeb graph

› at most O(n3) changes in the Reeb graph› compute robust maximal groups in O(n3 log n) time (plus output

size)

Passing encounter

Collapse encounter

Summary

Grouping structures [Buchin et al.’13]- Model for the grouping structure of moving entities- Algorithms for computing the grouping structure

Many different movement patterns• Moving groups• Convoys • Following/Leadership• Single file• Popular places (place visited by many)

Open Problem

• Interesting movement patterns?

• Query versions?

• Automatically set parameters?

References

• K. Buchin, M. Buchin, M. van Kreveld, B. Speckmann and F. Staals. Trajectory Grouping Structure. WADS, 2013.

• H. Jeung, M. L. Yiu, X. Zhou, C. S. Jensen and H. T. Shen. Discovery of Convoys in Trajectory Databases. PVLDB, 2008.

• K. Buchin, M. Buchin and J. Gudmundsson. Constrained Free Space Diagrams: a Tool for Trajectory Analysis, International Journal of Geographical Information Science, 2010.

• M. Andersson, J. Gudmundsson, P. Laube and T. Wolle. Reporting leaders and followers among trajectories of moving point objects. GeoInformatica, 2008.

computational movement analysis lecture 4: movement patterns joachim gudmundsson

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