Dynamic time warping

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Dynamic time warping is an algorithm for measuring similarity between two sequences which may vary in time or speed. For instance, similarities in walking patterns would be detected, even if in one video the person was walking slowly and if in another he or she were walking more quickly, or even if there were accelerations and decelerations during the course of one observation. DTW has been applied to video, audio, and graphics -- indeed, any data which can be turned into a linear representation can be analyzed with DTW. A well known application has been automatic speech recognition, to cope with different speaking speeds.

In general, DTW is a method that allows a computer to find an optimal match between two given sequences (e.g. time series) with certain restrictions. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in the context of hidden Markov models.

One example of the restrictions imposed on the matching of the sequences is on the monotonic of the mapping in the time dimension. Continuity is less important in DTW than in other pattern matching algorithms; DTW is an algorithm particularly suited to matching sequences with missing information, provided there are long enough segments for matching to occur.

The optimization process is performed using dynamic programming, hence the name.

The extension of the problem for two-dimensional "series" like images (planar warping) is NP-complete, while the problem for one-dimensional signals like time series can be solved in polynomial time.

[edit] Example of one of the many forms of the algorithm

int DTWDistance(char s[1..n], char t[1..m], int d[1..n,1..m]) {
    declare int DTW[0..n,0..m]
    declare int i, j, cost

    for i := 1 to m
        DTW[0,i] := infinity
    for i := 1 to n
        DTW[i,0] := infinity
    DTW[0,0] := 0

    for i := 1 to n
        for j := 1 to m
            cost:= d[s[i],t[j]]
            DTW[i,j] := minimum(DTW[i-1,j  ] + cost,    // insertion
                                DTW[i  ,j-1] + cost,    // deletion
                                DTW[i-1,j-1] + cost)    // match

    return DTW[n,m]
}

[edit] Reference

C. S. Myers and L. R. Rabiner.
A comparative study of several dynamic time-warping algorithms for connected word recognition.
The Bell System Technical Journal, 60(7):1389-1409, September 1981.