Qualitative approximation to Dynamic Time Warping similarity between time

series data

Blaž Strle, Martin Možina, Ivan BratkoFaculty of Computer and Information Science

University of LjubljanaSlovenia

Dynamic Time Warping (1/4)

• Dynamic time warping (DTW) is a method for measuring similarity between two time series

• Time series is a sequence of observations, measured at successive times, spaced at (often uniform) time intervals

• Time series ex.: TS = (9.47, 9.50, 9.48, 9.41, 9.32, 9.26, 9.21, 9.11, 9.01, 8.83, … )

Dynamic Time Warping (2/4)

Euclidean distance• does not align values• both time series need to be

of the same size

DTW• aligns two time series in the

way some distance measure is minimized

• time series sizes may vary

Dynamic Time Warping (3/4)• DTW can be efficiently calculated using dynamic

programming:D(i, j) = min(

D(i-1, j),D(i, j-1),D(i-1, j-1) ) + d(ai, bj)

D(i, j) = DTW( A(1..i), B(1..j))

d(ai, bj) is a distance betweentwo values of time series

Dynamic Time Warping (4/4)

Large amout of data Portable devices

Drawback – time complexity O(N2)

Improvements of Dynamic Time Warping

• Constraints– limit a minimum distance warp path search space by reducing allowed

warp along time axis

• Data abstraction– reduce the size of the input time series

QDTW (1/3)• Idea: reduce time series size by removing information that is

irrelevant for DTW

• Theorem: If two sequences A and B, |A| = n, |B| = m are qualitatively equal then:

DTW(A,B) ≤ ε, where ε = min(n × maxdiff(A)/2, m × maxdif(B)/2).

• two sequences are qualitatively equal if both sequences are monotonic and their start and end values are equal

• Term maxdiff (S) is the maximal absolute difference between two adjacent elements in a time series S.

QDTW (2/3)

1. Transform time series into qualitative representation (QING)

2. Use DTW on extreme points

QING

QDTW (3/3)

• If two sequences A = (a1, a2, … an), B = (b1, b2, …, bm), are qualitatively equal than :– QDTW(A,B) = DTW( (a1, an), (b1, bm) ) = 0

– |DTW(A,B) - QDTW(A,B)| ≤ ε – from Theorem

Violatins of conditions for the applicability of TheoremExtreme points do not coincide Sequences are not monotonic

DTW(A,D) > DTW(A,C) > DTW(A,B)QDTW(A,D) = QDTW(A,C) > QDTW(A,B)

1. Non monotonic part of B is not detected:QDTW(A, B) = 0, DTW(A, B) > 0

2. Monotonic part is detected:QDTW(A, B) is larger than it should be

Experimental Evaluation (1/3)

• Datasets:– Australian Sign Language signs (UCI)– Character Trajectories (UCI)– Character Recognition

• Accuracy– Classification using k-nn (k=3)– Leave one out

Experimental Evaluation (2/3)

• Efficiency– Estimated by size of the martix D– Size D = |A| x |B|, where

A and B are time series we are comparing

Experimental Evaluation (3/3)

Conclusion• DTW is a method for measuring similarity between two time

series• DTW’s time complexity O(N2) makes it useful only for

relatively short time series• QDTW is qualitative approximation to DTW• QDTW can be up to 1000 faster than DTW• Improvements in efficiency are often obtained at acceptable

loss in classification accuracy

• Future work: improve QDTW accuracy by reducing errors due to violations of the conditions for the applicability of the theorem