1. Time Series Filtering Time Series 1 5 9 2 6 10
Matches Q11
Time Series 1 5 9 2 6 10
Given a Time Series T, a set of Candidates C and a distance threshold r, find all subsequences in T that are within r distance to any of the candidates in C. 3 7 11 4 8 12
Candidates

2. Filtering vs. Querying Query Database Database 1 5 9 6 1 2 6 10 2 7 8
(template)
Database
Database
Matches Q11
Best match 1 5 9 6 1 2 6 10 2 7 8 3 3 7 11 9 4 4 8 12 5 10
Queries
Database

3. Euclidean Distance Metric
Given two time series Q = q1…qn and C = c1…cn ,
their Euclidean distance is defined as: 10 20 30 40 50 60 70 80 90
100 Q C

4. Early Abandon
During the computation, if current sum of the squared differences between each pair of corresponding data points exceeds r 2, we can safely stop the calculation. 10 20 30 40 50 60 70 80 90
100
calculation abandoned at this point Q C

5. Every possible warping between two time series, is a path though the matrix. We want the best one…
How is DTW Calculated? Q C C Q
This recursive function gives us the minimum cost path
(i,j) = d(qi,cj) + min{ (i-1,j-1), (i-1,j ), (i,j-1) }
Warping path w

6. Classic Approach Time Series1 5 9 2 6 10
Individually compare each candidate sequence to the query using the early abandoning algorithm. 3 7 11 4 8 12
Candidates

7. Euclidean Distance Lower Bound
Having candidate sequences C1, .. , Ck , we can form two new sequences U and L :
Ui = max(C1i , .. , Cki )
Li = min(C1i , .. , Cki )
They form the smallest possible bounding envelope that encloses sequences C1, .. ,Ck .
We call the combination of U and L a wedge, and denote a wedge as W.
W = {U, L}
A lower bounding measure for Euclidean distance between an arbitrary query Q and the entire set of candidate sequences contained in a wedge W: C1 C2 U W L U L W Q

8. DTW Distance Lower Bound2 1 U L W
DTW_U
DTW_L Q A B D
Based on the wedge W and the allowed warping range R, we define two new sequences, DTW_U and DTW_L:
DTW_Ui = max(Ui-R : Ui+R )
DTW_Li = min(Li-R : Li+R )
They form an additional envelope above and below the wedge, as illustrated in left figure.
We can now define a lower bounding measure for DTW distance between an arbitrary query Q and the entire set of candidate sequences contained in a wedge W :

9. Generalized Wedge
Use W(1,2) to denote that a wedge is built from sequences C1 and C2 .
Wedges can be hierarchally nested. For example, W((1,2),3) consists of W(1,2) and C3 .
C1 (or W1 )
C2 (or W2 )
C3 (or W3 )
W(1, 2)
W((1, 2), 3)

10. H-Merge
Time Series 1 5 9
Compare the query to the wedge using LB_Keogh
If the LB_Keogh function early abandons, we are done
Otherwise individually compare each candidate sequences to the query using the early abandoning algorithm 2 6 10 3 7 11 4 8 12
Candidates

11. Hierarchal ClusteringW3 W2 W5 W1 W4
W(2,5)
W(1,4)
W((2,5),3)
W(((2,5),3), (1,4))
K = 5
K = 4
K = 3
K = 2
K = 1
C3 (or W3)
C5 (or W5)
C2 (or W2)
C4 (or W4)
C1 (or W1)
Which wedge set to choose ?

12. Which Wedge Set to Choose ?
Test all k wedge sets on a representative sample of data
Choose the wedge set which performs the best

13. Upper Bound on H-Merge Worst case
Wedge based approach seems to be efficient when comparing a set of time series to a large batch dataset.
But, what about streaming time series ?
Streaming algorithms are limited by their worst case.
Being efficient on average does not help.
Worst case
C1 (or W1 )
C2 (or W2 )
C3 (or W3 )
W(1, 2)
Subsequence
W((1, 2), 3)

14. ? Triangle Inequality If dist(W((2,5),3), W(1,4)) >= 2 r < r
Subsequence W3 W2 W5 W1 W4 W3
W(2,5) W1 W4 W3
W(2,5)
W(1,4)
W((2,5),3)
W((2,5),3)
< r
W(((2,5),3), (1,4))
>= 2r ?
W(1,4)
K = 5
K = 4
K = 3
K = 2
K = 1
W(1,4)
cannot fail on both wedges
fails

15. Euclidean Distance: ECG Dataset
Batch time series
650,000 data points (half an hour’s ECG signals)
Candidate set
200 time series of length 40
r = 0.5
Algorithm
Number of Steps
brute force
5,199,688,000
classic
210,190,006
H-Merge
8,853,008
H-Merge-R
29,480,264
x 10 9 6
brute force 5 4
Number of Steps 3 2 1
classic
H-Merge
H-Merge-R
Algorithms

16. Euclidean Distance: Stock Dataset
Batch time series
2,119,415 data points
Candidate set
337 time series with length 128
r = 4.3
Algorithm
Number of Steps
brute force
91,417,607,168
classic
13,028,000,000
H-Merge
3,204,100,000
H-Merge-R
10,064,000,000
brute force
x 10 10 10 9 8 7
Number of Steps 6 5 4 3
classic 2
H-Merge-R
H-Merge 1
Algorithms

17. Euclidean Distance: Audio Dataset
Batch time series
46,143,488 data points (one hour’s sound)
Candidate set
68 time series with length 101
r = 4.14
Sliding window
11,025 (1 second)
Step
5,512 (0.5 second)
Algorithm
Number of Steps
brute force
57,485,160
classic
1,844,997
H-Merge
1,144,778
H-Merge-R
2,655,816
brute force
x 10 7 6 5 4
Number of Steps 3 2 1
H-Merge-R
classic
H-Merge
Algorithms

18. DTW Distance: Gun Dataset
Batch time series
18,750 data points
Candidate set
80 time series of length 150
r = 1.23
offset = 15
warping window size = 3%
Class
Number of Interesting Segments
Number of Hits (Euclidean)
Number of Hits (DTW)
Female-Gun 37 35
Female-Point 30 22
Male-Gun 13 19
Male-Point 28 14 17
Total
125 84
103
Accuracy
67.2%
82.4%

19. DTW Distance: ECG Dataset
Batch time series
200,000 data points
Candidate set
200 time series of length 40
r = 0.5
offset = 20
warping window size = 3%
Class
Number of Interesting Segments
Number of Hits (Euclidean)
Number of Hits (DTW) A
107
104 E
105 67
101 L 74 62 72 R 86 79 84
Total
373
312
364
Accuracy
83.87%
97.85%

20. Speedup by Sorting Wedge Random walk time series with length 1,000
Sorted
95,025
151,723
345,226
778,367
Unsorted
1,906,244
2,174,994
2,699,885
3,286,213

21. Semi Supervised Time Series Classification10 30 50 70 90
110
130
150
170
190
210
0.65
0.7
0.75
0.8
0.85
0.9
0.95 1
Number of examples in P
Precision-recall breakeven point
Training Set
Testing Set
ECG Dataset
Positive
(Abnormal)
208
312
520
Negative
(Normal)
602
904
1,506
Total
810
1,216
2,026