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. 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

6. Wedge
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 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

7. 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)

8. 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

9. 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 ?

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

11. 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)

12. ? 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

13. Experimental Setup Datasets
ECG Dataset
Stock Dataset
Audio Dataset
We measure the number of computational steps used by the following methods:
Brute force
Brute force with early abandoning (classic)
Our approach (H-Merge)
Our approach with random wedge set (H-Merge-R)

14. Experimental Results: 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

15. Experimental Results: 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

16. Experimental Results: 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

17. Experimental Results: Sorting
Wedge
with length 1,000
Random walk time series
with length 65,536
r = 0.5
r = 1
r = 2
r = 3
Sorted
95,025
151,723
345,226
778,367
Unsorted
1,906,244
2,174,994
2,699,885
3,286,213