1. FastDTW: Toward Accurate Dynamic Time Warping in Linear Time and Space
Department of Computer Sciences
Florida Institute of Technology
Stan Salvador and Philip Chan

2. Outline Dynamic Time Warping (DTW) Problem Statement
Related Work for Speeding up DTW
FastDTW Algorithm
Evaluation of FastDTW
Contributions
Limitations and Future Work

3. Dynamic Time Warping (DTW)
Aligns two time series by warping the time dimension
Warping - expanding/contracting the time dimension

4. The Dynamic Time Warping Algorithm
A dynamic programming approach
Solutions to slightly smaller problems used to find larger solutions

5. The DTW Cost Matrix

6. Distance of Min-Cost Warp Path

7. Finding Min-Cost Warp Path

8. Advantages of DTW DTW is optimal An intuitive distance measurement
Local variation in the time axis is common
Handwriting
Speech
“Events” that start after varying delays

9. Disadvantages of DTW O(N2) time and space complexity
Only practical for small data sets (<3,000)
Time series are often very long
Data mining requires a scalable DTW algorithm

10. Problem Statement
We desire an efficient Dynamic Time Warping algorithm
Linear time complexity
Linear space complexity
Warp path is needed in addition to warp distance
Warp path must be nearly optimal

11. Does DTW Need to be Faster?
“Myth 3: There is a need (and room) for improvements in the speed of DTW for data mining applications.”
(Keogh today-9:45am)
Keogh: many time series
FastDTW: Long time series

12. Existing Methods to Speed Up DTW
Constraints – only fill in part of the cost matrix
Abstraction – sample the data before time warping

13. Constraints Sakoe-Chiba Band (Sakoe & Chiba 1978)
Itakura Parallelogram (Itakura 1975)
Still O(N2) if the window width is a function of input size (linear if the width is constant)
Assumes a near-optimal warp path stays near the i=j axis
Accuracy depends on the domain

14. (Keogh & Pazzani 2000), (Chu et al. 2002)
Abstraction
(Keogh & Pazzani 2000), (Chu et al. 2002)
O(N) if N pts are sampled down to ≤
Assumptions
Sampling preserves time series structure
Small deviations from the optimal path cause little increase in warp-path distance

15. Our FastDTW Algorithm
A multi-resolution approach inspired by a multi-level graph bisection algorithm (Karypis 1997)
3 key operations
Coarsening – reduce the resolution of a time series
Projection – use a low-res warp path as an initial solution at a higher resolution
Refinement – Refine a projected warp path locally adjusting the path

16. Sample Run of FastDTW

17. FastDTW Algorithm Set the resolution to be the coarsest
Find the initial path using regular DTW
Repeat
Double the resolution
Project the path onto the finer resolution
Find a path through the projected area (plus a small radius around the projected area)
Until the original resolution is reached

18. Complexity
O(N) time
O(N) space
Details in the paper

19. Evaluation Criteria Accuracy Efficiency
The error of an approximate Time Warping algorithm:
% error =
where:
approxDist – the warp path distance of the approximate algorithm
optimalDist – the warp path distance of the DTW algorithm
Efficiency
Runtime (measured in seconds)

20. Evaluation Procedure (Accuracy)
Data Sets – UCR Time Series Data Mining Archive (Keogh & Folias 2002), 3 groups used:
Random – 45 unrelated time series (earthquakes, random walk, eeg, speech, etc.)
Trace – 200 time series simulating nuclear power plant failure (4 classes)
Gun – 200 time series of a gun being drawn and pointed (2 classes)
Procedure
Run FastDTW, Constraints (Sakoe-Chiba Band), and Data Abstraction on all pairs within a data set group, also vary the radius
Record the average error of all three methods for a group of data and a radius

21. Average % Error (Accuracy)
Radius 1 10 20 30
FastDTW
19.2%
8.6%
1.5%
0.8%
0.6%
Abstraction
983.3%
547.9%
6.5%
2.8%
1.8%
Band
2749.2%
2385.7%
794.1%
136.8%
9.3%

22. Error in Different Data Sets

23. Evaluation Procedure (Execution-time)
Data Sets
Synthetic sine waves with Gaussian noise
10 to 180,000 data points
Procedure
Run FastDTW and DTW on each data set, vary the radius for FastDTW
Compare the Execution times

24. Execution Time

25. Summary of Contributions
FastDTW – an approximation of DTW
O(N) time and space complexity
Scales well to long time series
Accurate, 8.6% error if radius=1, 0.8% error if radius=20

26. Limitations and Future Work
FastDTW does not always find an optimal solution
Future Work
Examine using different step sizes between resolutions
Investigate search algorithms to help improve refinement
Examine # of cells evaluated vs. accuracy between the FastDTW, Abstraction, and Band algorithms.

27. Questions? Thanks to those who helped with this research:
Matt Mahoney (Florida Institute of Technology),
Brian Buckley, Walter Schiefele (Interface & Control Systems)
This research is partially supported by NASA

28. FastDTW Pseudocode Input: X, Y, radius
Output: 1) A minimum distance warp path between X and Y
2) The warped path distance between X and Y
1| // The min size of the coarsest resolution.
2| Integer minTSsize = radius+2 3|
4| IF (|X|≤ minTSsize OR |Y|≤ minTSsize)
5| {
6| // Base Case: for a very small time series run the full DTW algorithm
7| RETURN DTW(X, Y)
8| }
9| ELSE
10| {
11| // Recursive Case: Project the warp path from a coarser resolution onto the current current resolution.
12| // Run DTW only along the projected path (and also radius cells from the projected path).
13| TimeSeries shrunkX = X.reduceByHalf() // Coarsening
14| TimeSeries shrunkY = Y.reduceByHalf() // Coarsening
15|
16| WarpPath lowResPath = FastDTW(shrunkX, shrunkY, radius)
17|
18| SearchWindow window = ExpandedResWindow(lowResPath, X, Y, radius) // Projection
19|
20| RETURN DTW(X, Y, window) // Refinement
21| }