Anticipatory DTW for Efficient Similarity Search in ... - Semantic Scholar
Anticipatory DTW for Efficient Similarity Search in Time Series Databases ◦ • Ira Assent Marc Wichterich • • Hardy Kremer Thomas Seidl •
RWTH Aachen University, Germany
◦
Ralph Krieger
Aalborg University, Denmark
VLDB 2009 Lyon, France
•
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Overview
1
Introduction
2
Dynamic Time Warping
3
Anticipatory pruning
4
Experiments
5
Conclusion
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
2 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Time series similarity search
time series from EEG data set 3000 2800
Time series Sequence of time related values Stock data, sensor data, EEG measurements, climate data, ...
value (ti) [[mV]
2600 2400 2200 2000 1800 1600 1400 1200 1000 800 0
128
256
384
512
point in time (i)
Similarity search Find time series with similar patterns over time
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
3 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Euclidean Distance Dynamic Time Warping (DTW)
Most widely used distance functions: Euclidean distance and Dynamic Time Warping Dynamic Time Warping allows scaling and stretching for better alignment, but computationally costly Euclidean Distance
Dynamic Time Warping
Figure: Euclidean distance (left) and Dynamic Time Warping (right)
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
4 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW definition k-band DTW DTW ([s1 , ..., sn ], [t1 , ..., tm ]) = DTW ([s1 , ..., sn−1 ], [t1 , ..., tm−1 ]) distband (sn , tm ) + min DTW ([s1 , ..., sn ], [t1 , ..., tm−1 ]) DTW ([s1 , ..., sn−1 ], [t1 , ..., tm ]) with ( distband (si , tj ) = DTW (∅, ∅) = 0,
Assent et al.
l m dist(si , tj ) |i − j·n m |≤k ∞ else
DTW (x, ∅) = ∞,
DTW (∅, y ) = ∞
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
5 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW Computation
tj
si
dist(si,tj) + min{ci-1,j-1,ci,j-1,ci-1,j} Figure: Cumulative warping matrix Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
6 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Existing DTW algorithms DTW is computationally expensive Many approaches use multistep filter-and-refine architecture If filter lower bounds DTW ⇒ lossless Different filters have been proposed and achieve substantial speed-ups
query index (filter) candidates
refinement
database (exact)
result
Figure: Multistep filter-and-refine architecture Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
7 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW properties for speed-up DTW is incremental. For any cumulative DTW matrix C = [ci,j ], the column minima are monotonically non-decreasing: mini=1,...,n {ci,x } ≤ mini=1,...,n {ci,y } for x < y . → Existing approach: early stopping /early abandon compute DTW cumulative matrix after each filled column (band), check threshold for pruning Not as tight as it could be ⇒ new anticipatory pruning
tj
si
dist(si,tj) + min{ci-1,j-1,ci,j-1,ci-1,j} Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Anticpatory pruning
DTW
query q
data
AP2 … APn AP1 filter result candidates anticipatory pruning
Figure: Anticipatory pruning in multistep filter-and-refine
Multistep with anticipatory pruning Benefit from work already done in filter step Low overhead, substantial speed-up
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
9 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
General idea Anticipatory pruning: As in early stopping, compute DTW incrementally Additionally: re-use filter information to anticipate full DTW (no additional computation necessary!) Requires: filter for remainder of time series
1..5
Assent et al.
1..6
es
...
TW
es
D
TW D
6..15
tim
at e tim
at e tim es TW D
...
at e
We characterize a class of filters to construct such anticipation: piecewise DTW property: reversible
7..15
1..7
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8..15
10 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Piecewise filter Piecewise DTW lower bound. A piecewise lower bounding filter for the DTW distance is a set f = {f0 , ..., fm } with the following property: j = 0 : fj (s, t) = 0 ∀j > 0 : fj (s, t) ≤
min
(i,j)∈bandj
DTW ([s1 , ..., si ], [t1 , ..., tj ])
1..5
Assent et al.
1..6
es
...
TW
es
D
TW D
6..15
tim
at e tim
at e tim es TW D
...
at e
Piecewise is the property of the filter that complements the incremental nature of DTW.
7..15
1..7
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8..15
11 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW is reversible DTW is reversible. For any two time series [s1 , ..., sn ] and [t1 , ..., tm ] their DTW distance is the same as for the reversed time series: DTW ([s1 , ..., sn ],[t1 , ..., tm ]) = DTW ([sn , ..., s1 ],[tm , ..., t1 ])
1..5
Assent et al.
1..6
es
...
TW
es
D
TW D
6..15
tim
at e tim
at e tim es TW D
...
at e
Reversible is the DTW property that allows alteration of the time series direction between filter and DTW.
7..15
1..7
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8..15
12 / 21
[5 , 3 , 4 , 5 , 3 , 4 , 8 , 3 , 9 ,
Introduction
Dynamic Time Warping
Anticipatory pruning
23 27 28
Experiments
Conclusion
19 18 21
Anticipatory pruning
16 17 21
Anticipatory Pruning qDistance.
14 15 17 12 13 16
9 13 n13and m, a cumulative distance matrix Given two time series s and t of length 6 8 10 C = [ci,j ], a piecewise lower bounding filter f for reversed time series, the j th 2 7 6 anticipatory pruning is 2 5 1 2 min 3 4 {c 5 } 6 + 7 f 8 (s 9 ← 10, t ← ). 0 := 6 10 13i,j 15 17 m−j 18 21 28 0 2 5i=1,...,n f : 18 18 15 14 11 10 10 10 10 7 0 AP: 18 20 20 20 21 23 25 27 28 28 28
stept) j: APj (s, min{}:
40
DTW
35
AP=min{} + f
distance
30 25
pruning threshold
20 15 10
f: anticipatory part on inverted time series min{}: early stopping part
5 0
step j: 0 Assent et al.
1
2
3
4
5
6
7
8
9 10
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
13 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Example
q
step j: min{}: f: AP:
[5 , 3 , 4 , 5 , 3 , 4 , 8 , 3 , 9 , 10]
t [3 , 8 , 2 , 0 , 2 , 6 , 6 , 4 , 0 , 2]
0 0 18 18
33 35 23 27 28 19 18 21 16 17 21 14 15 17 12 13 16 9 13 13 2
6
8 10
7
6
2 5 1 2 3 4 5 6 2 5 6 10 13 15 18 15 14 11 10 10 20 20 20 21 23 25
7 17 10 27
8 9 10 18 21 28 10 7 0 28 28 28
40 35 Figure: 30 Assent et al.
DTW
Anticipatory pruning example AP=min{} + f
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
14 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Anticipatory pruning is lossless Theorem: Anticipatory Pruning lower bounds DTW. Anticipatory pruning between two time series s, t with respect to a bandwidth k and a lower bounding filter f lower bounds the DTW: APj (s, t) ≤ DTW (s, t) ∀ j ∈ {1, . . . , n} Proof sketch Anticipatory pruning: series of partial DTW paths plus lower bound estimate of the remainder from the previous filter step (1) For each possible step r , 1 ≤ r ≤ n, column minima lower bound the true path (2) DTW is reversible (3) Combine (1),(2): lower bound of DTW Lower bounding means: lossless pruning, i.e. speed-up + no loss of accuracy Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
15 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Existing piecewise lower bounds
Linearization LBKeogh : Euclidean distance to envelopes (upper, lower bound of segments) [Keogh, VLDB 2002; Zhu/Shasha, SIGMOD 2003] Corner boundaries LBHybrid : piecewise corner-like shapes in the warping matrix through which every warping path has to pass [Zhou/Wong, ICDE 2007] Path Approximation FTW (Fast search method for Dynamic Time Warping): go from coarser DTW (less costly) to finer as needed: [Sakurai/Yoshikawa/Faloutsos, PODS 2005] → are all piecewise lower bounds as required for anticipatory pruning (details in paper)
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
16 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Experimental setup Synthetic and real world data sets SignLanguage: 1, 400 multivariate (11 attributes) of sign language finger tracking data[1] , length 64 to 512 TRECVid: 650 to 2, 000 benchmark data [2] NEWSVid: 2, 000 to 8, 000 TV news recorded at 30 fps (20 attributes), length 64 to 2048 Random walk RW1/RW2: zero normalized time series of length 512 and of cardinality 10,time000 (1 to 50 attributes) time series generated via RW2 series generated via RW1 40
6000
30
4000
20
value (tti)
value (tti)
2000 10 0
0 -2000
-10 -4000
-20 -30
-6000 0
[1]
(a)
128
256
384
RW1 t(i+1) ti + point = in time (i)N(0, 1) j j
www.cse.unsw.edu.au/ waleed/tml/data/ campaigns and TRECVid, MIR 2006
Assent et al.
0
(b) [2]
512
128
256
384
512
RW2 t(i+1) = tpoint in time ij + N(t ij (i)− t(i−1)j , 1) j
Smeaton/Over/Kraaij, Evaluation
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
17 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Experiments
relative Improvement [percent]
100
80
60 AP_FTW ES_FTW
40
AP_LBKeogh ES_LBKeogh
20
AP_LBHybrid ES_LBHybrid 0 1
5
10
20
30
40
50
number of attributes
Figure: Relative improvement (#calc.) for varying number of attributes on RW2
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
18 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Experiments 2
relative Improvement [percent]
65 AP_LBKeogh ES_LBKeogh
55
AP_FTW ES_FTW 45
AP_LBHybrid ES_LBHybrid
35
25 10
30
50
70
90
110
130
150
k
Figure: Efficiency improvement (#calc.) for varying DTW bandwidths on NEWSVid
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
19 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
1
LBKeogh ES_LBKeogh AP_LBKeogh
0.5
0.25
0.125 128
64 32 reduced length average query time [s] log. scale
256
16
8
average query time [s] log. scale
average query time [s] log. scale
Experiments 3 100
10
1
FTW ES_FTW AP_FTW
0.1 256
1
128
64 32 reduced length
16
8
LBHybrid ES_LBHybrid AP_LBHybrid
0.5
0.25
0.125 256
128
64 32 reduced length
16
8
Figure: (log. scale) Absolute improvement (average query time), reduction, RW2 Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
20 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Conclusion Anticipatory pruning Speed up DTW (Dynamic Time Warping) Widely used distance function for time series similarity search
Our novel anticipatory pruning makes best use of A family of multistep filter-and-refine approaches Compute an estimated overall DTW distance from already available filter information: series of lower bounds of the DTW DTW
query q
data
AP2 … APn AP1 filter result candidates anticipatory pruning
Experiments demonstrate substantially reduced runtime AP can be flexibly combined with existing and future DTW lower bounds AP is orthogonal to speed-up via indexing etc.
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
21 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Conclusion Anticipatory pruning Speed up DTW (Dynamic Time Warping) Widely used distance function for time series similarity search
Our novel anticipatory pruning makes best use of A family of multistep filter-and-refine approaches Compute an estimated overall DTW distance from already available filter information: series of lower bounds of the DTW DTW
query q
data
AP2 … APn AP1 filter result candidates anticipatory pruning
Experiments demonstrate substantially reduced runtime AP can be flexibly combined with existing and future DTW lower bounds AP is orthogonal to speed-up via indexing etc. Thank you for your attention. Questions? Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
21 / 21
RWTH Aachen University, Germany
◦
Ralph Krieger
Aalborg University, Denmark
VLDB 2009 Lyon, France
•
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Overview
1
Introduction
2
Dynamic Time Warping
3
Anticipatory pruning
4
Experiments
5
Conclusion
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
2 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Time series similarity search
time series from EEG data set 3000 2800
Time series Sequence of time related values Stock data, sensor data, EEG measurements, climate data, ...
value (ti) [[mV]
2600 2400 2200 2000 1800 1600 1400 1200 1000 800 0
128
256
384
512
point in time (i)
Similarity search Find time series with similar patterns over time
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
3 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Euclidean Distance Dynamic Time Warping (DTW)
Most widely used distance functions: Euclidean distance and Dynamic Time Warping Dynamic Time Warping allows scaling and stretching for better alignment, but computationally costly Euclidean Distance
Dynamic Time Warping
Figure: Euclidean distance (left) and Dynamic Time Warping (right)
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
4 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW definition k-band DTW DTW ([s1 , ..., sn ], [t1 , ..., tm ]) = DTW ([s1 , ..., sn−1 ], [t1 , ..., tm−1 ]) distband (sn , tm ) + min DTW ([s1 , ..., sn ], [t1 , ..., tm−1 ]) DTW ([s1 , ..., sn−1 ], [t1 , ..., tm ]) with ( distband (si , tj ) = DTW (∅, ∅) = 0,
Assent et al.
l m dist(si , tj ) |i − j·n m |≤k ∞ else
DTW (x, ∅) = ∞,
DTW (∅, y ) = ∞
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
5 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW Computation
tj
si
dist(si,tj) + min{ci-1,j-1,ci,j-1,ci-1,j} Figure: Cumulative warping matrix Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
6 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Existing DTW algorithms DTW is computationally expensive Many approaches use multistep filter-and-refine architecture If filter lower bounds DTW ⇒ lossless Different filters have been proposed and achieve substantial speed-ups
query index (filter) candidates
refinement
database (exact)
result
Figure: Multistep filter-and-refine architecture Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
7 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW properties for speed-up DTW is incremental. For any cumulative DTW matrix C = [ci,j ], the column minima are monotonically non-decreasing: mini=1,...,n {ci,x } ≤ mini=1,...,n {ci,y } for x < y . → Existing approach: early stopping /early abandon compute DTW cumulative matrix after each filled column (band), check threshold for pruning Not as tight as it could be ⇒ new anticipatory pruning
tj
si
dist(si,tj) + min{ci-1,j-1,ci,j-1,ci-1,j} Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Anticpatory pruning
DTW
query q
data
AP2 … APn AP1 filter result candidates anticipatory pruning
Figure: Anticipatory pruning in multistep filter-and-refine
Multistep with anticipatory pruning Benefit from work already done in filter step Low overhead, substantial speed-up
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
9 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
General idea Anticipatory pruning: As in early stopping, compute DTW incrementally Additionally: re-use filter information to anticipate full DTW (no additional computation necessary!) Requires: filter for remainder of time series
1..5
Assent et al.
1..6
es
...
TW
es
D
TW D
6..15
tim
at e tim
at e tim es TW D
...
at e
We characterize a class of filters to construct such anticipation: piecewise DTW property: reversible
7..15
1..7
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8..15
10 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Piecewise filter Piecewise DTW lower bound. A piecewise lower bounding filter for the DTW distance is a set f = {f0 , ..., fm } with the following property: j = 0 : fj (s, t) = 0 ∀j > 0 : fj (s, t) ≤
min
(i,j)∈bandj
DTW ([s1 , ..., si ], [t1 , ..., tj ])
1..5
Assent et al.
1..6
es
...
TW
es
D
TW D
6..15
tim
at e tim
at e tim es TW D
...
at e
Piecewise is the property of the filter that complements the incremental nature of DTW.
7..15
1..7
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8..15
11 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
DTW is reversible DTW is reversible. For any two time series [s1 , ..., sn ] and [t1 , ..., tm ] their DTW distance is the same as for the reversed time series: DTW ([s1 , ..., sn ],[t1 , ..., tm ]) = DTW ([sn , ..., s1 ],[tm , ..., t1 ])
1..5
Assent et al.
1..6
es
...
TW
es
D
TW D
6..15
tim
at e tim
at e tim es TW D
...
at e
Reversible is the DTW property that allows alteration of the time series direction between filter and DTW.
7..15
1..7
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
8..15
12 / 21
[5 , 3 , 4 , 5 , 3 , 4 , 8 , 3 , 9 ,
Introduction
Dynamic Time Warping
Anticipatory pruning
23 27 28
Experiments
Conclusion
19 18 21
Anticipatory pruning
16 17 21
Anticipatory Pruning qDistance.
14 15 17 12 13 16
9 13 n13and m, a cumulative distance matrix Given two time series s and t of length 6 8 10 C = [ci,j ], a piecewise lower bounding filter f for reversed time series, the j th 2 7 6 anticipatory pruning is 2 5 1 2 min 3 4 {c 5 } 6 + 7 f 8 (s 9 ← 10, t ← ). 0 := 6 10 13i,j 15 17 m−j 18 21 28 0 2 5i=1,...,n f : 18 18 15 14 11 10 10 10 10 7 0 AP: 18 20 20 20 21 23 25 27 28 28 28
stept) j: APj (s, min{}:
40
DTW
35
AP=min{} + f
distance
30 25
pruning threshold
20 15 10
f: anticipatory part on inverted time series min{}: early stopping part
5 0
step j: 0 Assent et al.
1
2
3
4
5
6
7
8
9 10
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
13 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Example
q
step j: min{}: f: AP:
[5 , 3 , 4 , 5 , 3 , 4 , 8 , 3 , 9 , 10]
t [3 , 8 , 2 , 0 , 2 , 6 , 6 , 4 , 0 , 2]
0 0 18 18
33 35 23 27 28 19 18 21 16 17 21 14 15 17 12 13 16 9 13 13 2
6
8 10
7
6
2 5 1 2 3 4 5 6 2 5 6 10 13 15 18 15 14 11 10 10 20 20 20 21 23 25
7 17 10 27
8 9 10 18 21 28 10 7 0 28 28 28
40 35 Figure: 30 Assent et al.
DTW
Anticipatory pruning example AP=min{} + f
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
14 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Anticipatory pruning is lossless Theorem: Anticipatory Pruning lower bounds DTW. Anticipatory pruning between two time series s, t with respect to a bandwidth k and a lower bounding filter f lower bounds the DTW: APj (s, t) ≤ DTW (s, t) ∀ j ∈ {1, . . . , n} Proof sketch Anticipatory pruning: series of partial DTW paths plus lower bound estimate of the remainder from the previous filter step (1) For each possible step r , 1 ≤ r ≤ n, column minima lower bound the true path (2) DTW is reversible (3) Combine (1),(2): lower bound of DTW Lower bounding means: lossless pruning, i.e. speed-up + no loss of accuracy Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
15 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Existing piecewise lower bounds
Linearization LBKeogh : Euclidean distance to envelopes (upper, lower bound of segments) [Keogh, VLDB 2002; Zhu/Shasha, SIGMOD 2003] Corner boundaries LBHybrid : piecewise corner-like shapes in the warping matrix through which every warping path has to pass [Zhou/Wong, ICDE 2007] Path Approximation FTW (Fast search method for Dynamic Time Warping): go from coarser DTW (less costly) to finer as needed: [Sakurai/Yoshikawa/Faloutsos, PODS 2005] → are all piecewise lower bounds as required for anticipatory pruning (details in paper)
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
16 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Experimental setup Synthetic and real world data sets SignLanguage: 1, 400 multivariate (11 attributes) of sign language finger tracking data[1] , length 64 to 512 TRECVid: 650 to 2, 000 benchmark data [2] NEWSVid: 2, 000 to 8, 000 TV news recorded at 30 fps (20 attributes), length 64 to 2048 Random walk RW1/RW2: zero normalized time series of length 512 and of cardinality 10,time000 (1 to 50 attributes) time series generated via RW2 series generated via RW1 40
6000
30
4000
20
value (tti)
value (tti)
2000 10 0
0 -2000
-10 -4000
-20 -30
-6000 0
[1]
(a)
128
256
384
RW1 t(i+1) ti + point = in time (i)N(0, 1) j j
www.cse.unsw.edu.au/ waleed/tml/data/ campaigns and TRECVid, MIR 2006
Assent et al.
0
(b) [2]
512
128
256
384
512
RW2 t(i+1) = tpoint in time ij + N(t ij (i)− t(i−1)j , 1) j
Smeaton/Over/Kraaij, Evaluation
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
17 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Experiments
relative Improvement [percent]
100
80
60 AP_FTW ES_FTW
40
AP_LBKeogh ES_LBKeogh
20
AP_LBHybrid ES_LBHybrid 0 1
5
10
20
30
40
50
number of attributes
Figure: Relative improvement (#calc.) for varying number of attributes on RW2
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
18 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Experiments 2
relative Improvement [percent]
65 AP_LBKeogh ES_LBKeogh
55
AP_FTW ES_FTW 45
AP_LBHybrid ES_LBHybrid
35
25 10
30
50
70
90
110
130
150
k
Figure: Efficiency improvement (#calc.) for varying DTW bandwidths on NEWSVid
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
19 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
1
LBKeogh ES_LBKeogh AP_LBKeogh
0.5
0.25
0.125 128
64 32 reduced length average query time [s] log. scale
256
16
8
average query time [s] log. scale
average query time [s] log. scale
Experiments 3 100
10
1
FTW ES_FTW AP_FTW
0.1 256
1
128
64 32 reduced length
16
8
LBHybrid ES_LBHybrid AP_LBHybrid
0.5
0.25
0.125 256
128
64 32 reduced length
16
8
Figure: (log. scale) Absolute improvement (average query time), reduction, RW2 Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
20 / 21
Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Conclusion Anticipatory pruning Speed up DTW (Dynamic Time Warping) Widely used distance function for time series similarity search
Our novel anticipatory pruning makes best use of A family of multistep filter-and-refine approaches Compute an estimated overall DTW distance from already available filter information: series of lower bounds of the DTW DTW
query q
data
AP2 … APn AP1 filter result candidates anticipatory pruning
Experiments demonstrate substantially reduced runtime AP can be flexibly combined with existing and future DTW lower bounds AP is orthogonal to speed-up via indexing etc.
Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
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Introduction
Dynamic Time Warping
Anticipatory pruning
Experiments
Conclusion
Conclusion Anticipatory pruning Speed up DTW (Dynamic Time Warping) Widely used distance function for time series similarity search
Our novel anticipatory pruning makes best use of A family of multistep filter-and-refine approaches Compute an estimated overall DTW distance from already available filter information: series of lower bounds of the DTW DTW
query q
data
AP2 … APn AP1 filter result candidates anticipatory pruning
Experiments demonstrate substantially reduced runtime AP can be flexibly combined with existing and future DTW lower bounds AP is orthogonal to speed-up via indexing etc. Thank you for your attention. Questions? Assent et al.
Anticipatory DTW for Efficient Similarity Search in Time Series Databases
21 / 21
