1. Efficient Query Filtering for Streaming Time Series
Li Wei Eamonn Keogh
Helga Van Herle
Agenor Mafra-Neto
Computer Science & Engineering Dept. University of California – Riverside Riverside, CA {wli,
David Geffen School of Medicine University of California – Los Angeles Los Angeles, CA
ISCA Technologies Riverside, CA
ICDM '05

2. Outline of Talk Introduction to time series Time series filtering
Wedge-based approach
Experimental results
Conclusions

3. What are Time Series?
Time series are collections of observations made sequentially in time. .

4. Time Series are Everywhere
ECG Heartbeat
Image
Stock
Video
make subtitle bold

5. Time Series Data Mining Tasks
Clustering
Classification
Rule Discovery
Motif Discovery 50
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150
2000
2500 20 40 60 80
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120
140 A B C 
s = 0.5
c = 0.3
Query by Content 10
Anomaly Detection
Visualization

6. Time Series Filtering Time Series
Matches Q11
Time Series 1 5 9
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. 2 6 10
Say which part is given(candidates and r), which we don’t know in advance(long time series).
Application: ECG monitoring, audio sensor monitoring (say more, say it is motivated by the Aerospace) 3 7 11 4 8 12
Candidates

7. 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

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

9. 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

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

11. 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
Important thing to notice: LB_Keogh lower bounds. U L W Q

12. 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)

13. Wedge Based Approach 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

14. Examples of Wedge MergingQ
W((1,2),3)
C1 (or W1 )
C2 (or W2 )
C3 (or W3 )
W(1, 2)
W((1, 2), 3)
C1 (or W1 )
C2 (or W2 )
W(1, 2)

15. 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 ?

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

17. Upper Bound on Wedge Based Approach
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)

18. ? If dist(W((2,5),3), W(1,4)) >= 2 r Triangular Inequality < r
Subsequence W3 W2 W5 W1 W4 W3
W(2,5) W1 W4 W3
W(2,5)
W(1,4)
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

19. Experimental Setup How to choose r ? 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 (Atomic Wedgie)
Our approach with random wedge set (AWR)
How to choose r ?
A logical value for r would be the average distance from a pattern to its nearest neighbor

20. ECG Dataset Batch time series Candidate set r = 0.5
650,000 data points (half an hour’s ECG signals)
Candidate set
200 time series of length 40
4 types of patterns
left bundle branch block beat
right bundle branch block beat
atrial premature beat
ventricular escape beat
r = 0.5
Upper Bound: 2,120 (8,000 for brute force)
Algorithm
Number of Steps
brute force
5,199,688,000
classic
210,190,006
Atomic Wedgie
8,853,008
AWR
29,480,264

21. Stock Dataset Batch time series Candidate set r = 4.3
2,119,415 data points
Candidate set
337 time series with length 128
3 types of patterns
head and shoulders
reverse head and shoulders
cup and handle
r = 4.3
Upper Bound: 18,048 (43,136 for brute force)
Algorithm
Number of Steps
brute force
91,417,607,168
classic
13,028,000,000
Atomic Wedgie
3,204,100,000
AWR
10,064,000,000

22. Audio Dataset Batch time series Candidate set
37,583,512 data points (one hour’s sound)
Candidate set
68 time series with length 51
3 species of harmful mosquitoes
Culex quinquefasciatus
Aedes aegypti
Culiseta spp
Sliding window: 11,025 (1 second)
Step: 5,512 (0.5 second)
r = 2
Upper Bound: 2,929 (6,868 for brute force)
Algorithm
Number of Steps
brute force
57,485,160
classic
1,844,997
Atomic Wedgie
1,144,778
AWR
2,655,816

23. Conclusions We introduce the problem of time series filtering.
Combining similar sequences into a wedge is a quite promising idea.
We have provided the upper bound of the cost of the algorithm to compute the fastest arrival rate we can guarantee to handle.

24. Future Work Dynamic wedge set choosing for data with concept shifting
Extension to other distance measures, for example, DTW (Dynamic Time Warping) and uniform scaling

25. Questions? All datasets used in this talk can be found at

26. Z-Normalization
Normalize a data sequence C to have mean = 0 and standard deviation = 1
C = (C - mean(C )) / std(C )
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