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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, 2006
Benkert, G., Hubner & Wolle, 2006
Jensen, Lin & Ooi, 2007
Al-Naymat, Chawla & G., 2007
Vieria, Bakalov & Tsotras, 2009
Jeung, Yiu, Zhou, Jensen & Shen, 2008
Jeung, Shen & Zhou, 2008
...
Movement patterns: Groups
Time = 01 65432 7 8
m = 3
m – flock size
Movement patterns: Groups
flock
meet
fixed subset variable subset
examples for m = 3
Movement patterns: Groups
Movement patterns: Convoy
Movement patterns: Convoy
X – set of entities
m – 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
x2x3
x4
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: Convoy
Movement patterns: Leadership & Followers
front
A leader?
- Should not follow anyone else!
- Is followed by at least m other entities.
- For a certain duration.
Movement patterns: Leadership & Followers
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)
Movement patterns: Leadership & Followers
[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]
Movement patterns: Popular places
Movement patterns: Popular places
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!
Movement patterns: Single File
Single file: Intuitively easy to define
Hard to define formally!
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
x2x3
x4
x5
Grouping: definitions
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.
Grouping: definitions
C(t) – the set of connected components at time t that forms a partition of X.
C(t) consists of 5 components
Time: t
Grouping: definitions
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
Grouping: definitions
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
Grouping: definitions
Note that an entity can be in several maximal groups at the same time!
Grouping:definitions
Note that an entity can be in several maximal groups at the same time!
Grouping:definitions
Note that an entity can be in several maximal groups at the same time!
Grouping: definitions
Note that an entity can be in several maximal groups at the same time!
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
Trajectory grouping structure
purple
red, blue, greenblue, green
red greenpurple, green
red
blue
red, blue
t=0
t=1 t=4
t=5
t=8
t=10
t=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 1
end vertex (time t) – in-degree 1, out-degree 0
merge 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].
Complexity of the Reeb graph
Observation:
Reeb graph of M has (n2) vertices and edges.
Proof:
Construction for one time step [ti,ti+1].
Complexity of the Reeb graph
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. □
Complexity of 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.
Complexity of 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)
Computing the Reeb graph
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]
Computing the Reeb graph
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 and
ending at an end vertex.
purple
red, blue, greenblue, green
red greenpurple, green
red
blue
red, blue
t=0
t=1 t=4
t=5
t=8
t=10
t=0
t=0
t=10
t=10
Number of maximal groups
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, greenblue, green
red greenpurple, green
red
blue
red, blue
t=0
t=1 t=4
t=5
t=8
t=10
t=0
t=0
t=10
t=10
Number of maximal groups
Consider a merge vertex v.
Assume sets S and T merge at v.
S T
v
Number of maximal groups
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
Number of maximal groups
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.
Consider the union R’ of all such paths from entities in ST.
R’ is a directed acyclic subgraph of the Reeb graph.
S T
v
Number of maximal groups
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.
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’
Number of maximal groups
“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
Number of maximal groups
“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.
v
px
v
px pypy
Number of maximal groups
“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.
v v
Number of maximal groups
“Unravel” R’: if px and py split and then merge at a vertex u then duplicate the paths starting at u.
v v
Number of maximal groups
“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
Number of maximal groups
“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
Number of maximal groups
“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
Number of split vertices?
Every vertex has degree at most 3.
O(n) split vertices.
v
Number of maximal groups
“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
Number 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
Number of maximal groups
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
1
2
3
4
(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)
Computing the maximal groups
2. Maintain maximal groups
Each edge e=(u,v) is labelled with a set of maximal groups Ge.
Note: A group G becomes maximal at a vertex.
t0
tG
1
2
3
4
(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)
Computing the maximal groups
2. Annotate its edges and vertices
Each 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
Computing the maximal groups
2. Annotate its edges and vertices
Each 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)}b. Merge vertex: Propagate the maximal groups from “children” to e and add (Ce,tv).
v
u
Ge1Ge
2
Ge={Ge1Ge2
(Ce,tv)
}
Computing the maximal groups
2. Annotate its edges and vertices
Each 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)}b. Merge vertex: Propagate the maximal groups from “children” to e and add (Ce,tv).c. Split vertex:
v
Ge1
Ge
Ge2
Computing the maximal groups
2. Annotate its edges and vertices
Each 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)}b. Merge vertex: Propagate the maximal groups from “children” to e and add (Ce,tv).c. Split vertex:
- 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
v
Ge1
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,…}
Store maximal groups
e
Observation: Assume S, T in Ge. S starting at tS and T starting at tT.
• G ST in Ge with starting time tG max{tS,tT}.
tS
tT
tG Ge = {S,T,G,…}
Store maximal groups
e
Observation: Assume S, T in Ge. S starting at tS and T starting at tT.
• 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
1
2
3
4
(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)
Store maximal groups
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
1
2
3
4
(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}
Store maximal groups
Computing the maximal groups
2. Annotate its edges and vertices
Each edge e=(u,v) is labelled with a set of maximal groups Ge.
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:
- 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
(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
Computing the maximal groups
2. Annotate its edges and vertices
Each edge e=(u,v) is labelled with a set of maximal groups Ge.
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:
- 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
(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
Computing the maximal groups
2. Annotate its edges and vertices
Each edge e=(u,v) is labelled with a set of maximal groups Ge.
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:
- 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
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
Computing the maximal groups
2. Annotate its edges and vertices
Each edge e=(u,v) is labelled with a set of maximal groups Ge.
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:
- 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
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
Computing the maximal groups
Analysis
Start 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.