Dynamic time warping

Not to be confused with the Time Warp mechanism for discrete event simulation, or the Time Warp Operating System that used this mechanism.

Dynamic time warping (DTW) 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 monotonicity 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 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.

Example of one of the many forms of the algorithm

This example illustrates the implementation of dynamic time warping when the two sequences are strings of discrete symbols. d(x, y) is a distance between symbols, e.g. d(x, y) = | x - y |.

int DTWDistance(char s[1..n], char t[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] := cost + minimum(DTW[i-1, j  ],    // insertion
                                        DTW[i  , j-1],    // deletion
                                        DTW[i-1, j-1])    // match

    return DTW[n, m]
}

We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i - j | is no larger than w, a window parameter.

We can easily modify the above algorithm to add a locality constraint (differences marked in bold italic). However, the above given modification works only if |n - m| is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that |n - m ≤ w| (see the line marked with (*) in the code).

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

    w := max(w, abs(n-m)) // adapt window size (*)

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

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

    return DTW[n, m]
}

Open Source software

• The lbimproved C++ library implements Fast Nearest-Neighbor Retrieval algorithms under the Dynamic Time Warping (GPL). It also provides a C++ implementation of Dynamic Time Warping as well as various lower bounds.
• The R package dtw implements most known variants of the DTW algorithm family, including a variety of recursion rules (also called step patterns), constraints, and substring matching.
• The mlpy Python library implements DTW.

References

• Sakoe, H. and Chiba, S., Dynamic programming algorithm optimization for spoken word recognition, IEEE Transactions on Acoustics, Speech and Signal Processing, 26(1) pp. 43- 49, 1978, ISSN: 0096-3518
• 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.
• L. R. Rabiner and B. Juang.
  Fundamentals of speech recognition.
  Prentice-Hall, Inc., 1993 (Chapter 4)

See also

• Levenshtein distance
• Elastic matching