Finding Similar Movements in
Positional Data Streams
Jens Haase and Ulf Brefeld
Knowledge Mining & Assessment
[email protected]
Prague, 27.9.2013
Monday, September 16, 13
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
2
B. Charlton v F. Beckenbauer
David Marsh
1966 World Cup Final, England - W. Germany
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Knowledge Mining & Assessment Group
3
Player Briefing
(coach, before game)
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4
Analyses
(newspapers, next day)
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Youth Soccer: Tactics and Paths
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
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Tactics and Trajectories
๏
Understanding player movements precondition for
analyzing tactics
๏
Requires efficient computation of similar movements
๏
This talk: Efficient retrieval of near-duplicate
trajectories given a query movement
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
7
and efficiently compute similarities between trajectories in D. That is, given
query trajectory q, we aim at finding the N most similar trajectories in D.
3.2
Representation
Representation
We aim to exploiting the symmetries of the pitch and use Angle/Arc-Length
(AAL) [11] transformations to guarantee translation, rotation, and scale invari
๏ Position of
= trajectories.
player coordinates
on the
pitchis to represent a move
ant representations
The main idea
of AAL
ment p = hx1 , . . . , xm i in terms of distances and angles
๏
๏
A game of soccer = positional data stream
p 7! p̄ = h(↵1 , kv 1 k), . . . , (↵m , kv m k)i,
Player trajectory = sequence of consecutive positions
(1
where v i = xi xi 1 . The difference v i is called the movement vector at tim
๏ Positions represented by angles wrt reference vector
i and the corresponding angle with respect to a reference vector v ref = (1, 0)>
is defined vasref (translation, rotation, scale invariant)

✓
◆
>
v i v ref
1
↵i = sign(v i , v ref ) cos
,
kv i k kv ref k
where the sign function computes the direction (clockwise ore counterclock
Vlachos et al. (KDD, 2004)
wise) of the movement with respect to the reference. In the remainder, we dis
card the
norms in EquationKnowledge
(1) and
represent
trajectories by their sequences
o
Ulf Brefeld
Mining
& Assessment Group
8
angles, p 7! p̃ = h↵1 , . . . , ↵m i.
Monday, September 16, 13
card the norms in Equation (1) and represent trajectories by their sequences of
Dynamic
Time
Warping
angles, p 7! p̃ = h↵1 , . . . , ↵m i.
this section, we propose a distance measure for trajectories. The propose
3.3 of
Dynamic
Time Warping
resentation
the previous
section fulfils the required invariance in term
In this
section,and
we propose
distance
measure some
for trajectories.
The proposed
translation,
rotation
scalinga [11].
However,
movements
may star
representation
the previous
required
in terms
w and end
fast, whileof others
start section
fast andfulfils
thenthe
slow
downinvariance
at the end.
Thus
of ๏translation, rotation and scaling [11]. However, some movements may start
should befor
independent
speed A remed
additionallyMovements
need to compensate
phase shiftsofofplayer
trajectories.
slow and end fast, while others start fast and then slow down at the end. Thus,
mes fromwethe
area of speech
recognitionforand
is called
dynamic
time
warpin
need
to
compensate
phase
shifts
of
trajectories.
A
remedy
๏additionally
Dynamic time warping compensates phase shifts
TW) [9].comes
Given
two
s=
hs1 , . . . and
, smisi called
and qdynamic
= hq1 ,time
. . . ,warping
qm i an
from
thesequences
area of speech
recognition
๏ Distance
(DTW)
[9]. Given
two sequences
=⇥
hsR
. . , sRm(e.g.,
i and qEuclidean
= hq1 , . . . ,distance)
qm i and
measure
element-wise
distance
function
dist s: R
1 , .!
an element-wise
distance
function distas
: Rfollows
⇥ R ! R (e.g., Euclidean distance),
define the
DTW
function
g
recursively
DTW
sequences
s and q defined
we๏define
thefor
DTW
function g recursively
as followsrecursively
g(;, ;) =g(;,
0 ;) = 0
g(s, ;) =g(s,
dist(;,
=1
;) = q)
dist(;,
q) = 1
8 8
9
9
hq2 ,hq. 2.,..,. q. ,mqm
i)i)
< g(s,
=
< g(s,
=
g(hs
,
.
.
.
,
s
i,
q)
q) =1 ,dist(s
q1 ) + min
2
m
g(hs
,
.
.
.
,
s
i,
q)
g(s, q) =g(s,
dist(s
q1 ) +1 , min
2
m
:
;
:
g(hs2 , . . . , sm i, hq2 , . . . , qm i) ;
g(hs2 , . . . , sm i, hq2 , . . . , qm i)
Dynamic Time Warping
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
9
card the norms in Equation (1) and represent trajectories by their sequences of
Dynamic
Time
Warping
angles, p 7! p̃ = h↵1 , . . . , ↵m i.
this section, we propose a distance measure for trajectories. The propose
3.3 of
Dynamic
Time Warping
resentation
the previous
section fulfils the required invariance in term
Rabiner & Juang (1993)
In this
section,and
we propose
a [11].
distance
measure some
for trajectories.
The proposed
translation,
rotation
scaling
However,
movements
may star
representation
the previous
required
in terms
w and end
fast, whileof others
start section
fast andfulfils
thenthe
slow
downinvariance
at the end.
Thus
of ๏translation, rotation and scaling [11]. However, some movements may start
should befor
independent
speed A remed
additionallyMovements
need to compensate
phase shiftsofofplayer
trajectories.
slow and end fast, while others start fast and then slow down at the end. Thus,
mes fromwethe
area of speech
recognitionforand
is called
dynamic
time
warpin
need
to
compensate
phase
shifts
of
trajectories.
A
remedy
๏additionally
Dynamic time warping compensates phase shifts
TW) [9].comes
Given
two
s=
hs1 , . . . and
, smisi called
and qdynamic
= hq1 ,time
. . . ,warping
qm i an
from
thesequences
area of speech
recognition
๏ Distance
(DTW)
[9]. Given
two sequences
=⇥
hsR
. . , sRm(e.g.,
i and qEuclidean
= hq1 , . . . ,distance)
qm i and
measure
element-wise
distance
function
dist s: R
1 , .!
an element-wise
distance
function distas
: Rfollows
⇥ R ! R (e.g., Euclidean distance),
define the
DTW
function
g
recursively
DTW
sequences
s and q defined
we๏define
thefor
DTW
function g recursively
as followsrecursively
g(;, ;) =g(;,
0 ;) = 0
O(|s||q|)
g(s, ;) =g(s,
dist(;,
=1
;) = q)
dist(;,
q) = 1
8 8
9
9
hq2 ,hq. 2.,..,. q. ,mqm
i)i)
< g(s,
=
< g(s,
=
g(hs
,
.
.
.
,
s
i,
q)
q) =1 ,dist(s
q1 ) + min
2
m
g(hs
,
.
.
.
,
s
i,
q)
g(s, q) =g(s,
dist(s
q1 ) +1 , min
2
m
:
;
:
g(hs2 , . . . , sm i, hq2 , . . . , qm i) ;
g(hs2 , . . . , sm i, hq2 , . . . , qm i)
Dynamic Time Warping
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
10
Approximate DTW
The time complexity of DTW is O(|s||q|) which is clearly intractable for c
puting similarities of thousands of trajectories. However, recall that we ai
finding the N best matches for a given query. This allows for pruning s
๏ Approximate
DTW
computationsDTW
using lower
bounds
f , i.e., f (s, q)  g(s, q), with an
by lower
bounds
propriate function f that can be more efficiently computed than g [10]. We
๏ Focus on characteristic values
two different lower bound functions, fkim [6] and fkeogh [5], that are defi
as
follows: fkim focuses on the first, last, greatest, and smallest values of
๏ Kim et al.
(ICDE, 2001)
sequences [6]
๏
๏
first, last, greatest, smallest value
fkim (s, q) = max {|s1
q1 |, |sm
Keogh (VLDB, 2002)
qm |, | max(s)
max(q)|, | min(s)
min(q)|}
and can be computed in O(m). However, the greatest (or smallest) entr
minimum/maximum
values
the ๏transformed
paths is always
closeof
or subsequences
identical to ⇡ (or ⇡) and can
be ignored. Consequentially, the time complexity reduces to O(1) [10].
๏ Complexity in O(|s|)
second lower bound fkeogh [5] uses minimum `i and an maximum values u
subsequences of the query q given by
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
`i = min(qi
r , . . . , qi+r )
and
ui = max(qi
11
r , qi+r ),
ing them with Algorithm 1. The idea
same bucket, so that all objects of a b
near-neighbours. An interesting equiv
based
hashes
(DBH)
[1]
that can be a
two randomly drawn
members
s
,
s
2
D
we
define
the
function
h
as
follows:
Athitsos et al. (2008),
1 2 Gionis et al., (1999)
metric) distance measures.
2 the function h a
two randomly
drawn
2 dist(s,
D we
define
dist(s,
s1 )2members
+ dist(s1s,1s,To
)22 define
s
)
2s
2
a hash .family for our pu
hs1 ,s2 (s) =
2
dist(s
,
s
)
1
2
๏ randomly
two
drawn
members
s
,
s
2
D
we
define
the
function
h as2 follows:
1
2
h
:
D
!
R
that
maps
trajectory
Distance-based hash function
2
dist(s, s1 ) + dist(s1 , s2 ) a dist(s,
s2 )s2 2
hsuse
(s)2identity
=
.
1 ,s2the
In the remainder, we will
dist(s
,
s
)
=
f
(s
,
s
)
for
sim2
2
1 2
kims1)
2 dist(s
dist(s, s1 ) + dist(s1 , s21) 2 dist(s,
2 , s2 )
Locality Sensitive Hashing
.
plicity. Tohscompute
a discrete hash value for s we verify whether h(s)
lies in a
1 ,s2 (s) =
2 dist(s1 , s2 )
certain interval
[t1 ,remainder,
t2 ],
In the
we will use the identity dist(s1 , s2 ) = fkim (s1 , s
In the
we will
use the identity
dist(s
(swe
plicity.
To compute
a⇢discrete
hash
whether h(
1 , svalue
2 ) = ffor
1 , sverify
2 ) for simkims
s1 remainder,
and s2 randomly
Kim2et[tal.
1 : hs1use
(s)
,(ICDE,
t2 ] 2001)
[t1 ,t2 ]
,s
1
2
plicity.
Tofrom
compute
a interval
discrete
hash
for s as
wedistance
verify function
whether h(s) lies in a
hs1 ,s2 (s)
certain
[t1=
, tvalue
drawn
database
2 ],
0:
otherwise
certain interval [t1 , t2 ],
⇢
1 : hsso
2 [t
[t1and
,t2 ] t2 are chosen
⇢
1 , t2 ]
1 ,sthat
2 (s)the
Optimally,
thedetermined
interval boundaries
t
probability
๏ Bucket
1
by 1h:s1h,ss12,s(s)
= 1 , t2 ]
[t1 ,t2 ]
2 (s) 2 [t
:
otherwise
that a randomly drawn
s 22(s)
X lies
and
with 50% chance
hs1 ,s
= with 50% chance 0within
0:
otherwise
outside
of
the
interval.
The
set
T
defines
the
set of admissible intervals,
๏ Set of admissible intervals
Optimally,
the interval boundaries t1 and t2 are chosen so
that the
n
o
Optimally, the interval boundaries t[t11 ,tand
t2 are chosen so that
probability
[t1 ,t2the
]
2]
X=
lies
T (s1 , s2 )that
= a[trandomly
, t2 ] : P rdrawn
(hs1 ,ss2 2
(s))
0)with
= P r50%
(hschance
(s))within
= 1) and
. with 5
,s
1
D
D
1
2
that a randomly drawn s 2 X lies with 50% chance within and with 50% chance
thesetinterval.
The
the set
of admissible interva
outside of the outside
interval.of
The
T defines
theset
setTofdefines
admissible
intervals,
Given h and T we can now ndefine the DBH hash family that can be directly
n
o [t1 ,t2 ]
[t
,t
]
1
2
[t1,,tt2 ]] : P r (hs ,s (s))
[t1=
,t2 ]0) = P r (hs ,s (s))
integrated in standard
LSH
algorithms:
,
s
)
=
[t
1
2
1
2
D
D 12
1
2
T (s1 , s2 ) = T[t(s
,
t
]
:
P
r
(h
(s))
=
0)
=
P
r
(h
. 1 2
Ulf Brefeld
Knowledge
Mining
&
Assessment
Group
1 2
D s1 ,s2
D s1 ,s2 (s)) = 1)
n
o
Monday, September 16, 13
[t1 ,t2 ]
Empirical Evaluation
๏
DEBS Grand Challenge
http://www.orgs.ttu.edu/debs2013/index.php?goto=cfchallengedetails
๏
8 vs. 8 soccer game recorded by Fraunhofer IIS
๏
In total 33 sensors
๏
Ulf Brefeld
Monday, September 16, 13
๏
1 sensor per shoe (200Hz)
๏
1 sensor in the ball (2000Hz)
15,000 positions per second (3 dimensional)
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Coordinates on the Pitch
๏
Coordinate system, origin (0,0) is at kick-off
๏
Discarding additional data, players are represented
by triplet:
(sensor/player id, timestamp, player coordinates)
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Representation
๏
๏
Further preprocessing:
๏
Discarding positions outside of the pitch
๏
Removing half-time effect of changing sides
๏
Averaging player positions over 100ms
Trajectory windows of size 10
t+1
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Monday, September 16, 13
t+2
Knowledge Mining & Assessment Group
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Evaluation
๏
Given: a query trajectory
๏
Task: Find near-duplicates
๏
๏
Focus on 15k consecutive positions of one player
๏
Ulf Brefeld
Monday, September 16, 13
(i.e., N=1000 most similar trajectories)
(for baseline comparisons)
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Run-time
๏
Exact computation infeasible
๏
Dynamic time warping very effective
๏
DBH adds only little
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Monday, September 16, 13
Knowledge Mining & Assessment Group
17
nof. trajetories
Pruned Trajectories
Kim
Keough
DBH
total
1000
0.00%
0%
11.42%
11.42%
5000
0.28%
34.00%
16.33%
50.61%
10000
9.79%
41.51%
17.80%
60.10%
15000
17.5%
46.25%
11.82%
75.57%
๏
Effectiveness of DBH depends only on data
๏
Kim and Keogh effective for constant N
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
18
DBH Accuracy
๏
On average DBH performs very accurate
๏
However, worst cases clearly inappropriate
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Knowledge Mining & Assessment Group
19
Example
Query:
Ulf Brefeld
Monday, September 16, 13
Retrieved:
Knowledge Mining & Assessment Group
20
Conclusion
๏
Efficient computation of near duplicate movements
in positional data streams
๏
Dynamic time warping (DTW)
๏
Distance-based hashing (DBH)
๏
(Super-)linear complexity
๏
Accurate results
Ulf Brefeld
Monday, September 16, 13
Knowledge Mining & Assessment Group
21