將時間規整與距離測度結合起來，采用動態規划技術，比較兩個大小不同的模式，解決語音識別中語速多變的難題；

一種非線性時間規整模式匹配算法；

DTW ( Dynamic Time Warping )，即「動態時間扭曲」或是「動態時間規整」。這是一套根基於「動態規划」（Dynamic Programming，簡稱DP）的方法，可以有效地將搜尋比對的時間大幅降低。

DTW 的目標就是要找出兩個向量之間的最短距離。一般而言，對於兩個 n 維空間中的向量 x 和 y，它們之間的距離可以定義為兩點之間的直線距離，稱為尤拉距離（Euclidean Distance）。

dist(x, y) = |x – y| ，

但是如果向量的長度不同，那它們之間的距離，就無法使用上述的數學式來計算。一般而言，假設這兩個向量的元素位置都是代表時間，由於我們必須容忍在時間軸的偏差，因此我們並不知道兩個向量的元素對應關系，因此我們必須靠着一套有效的運算方法，才可以找到最佳的對應

關系。

是

動態規划算法總體思想

動態規划基本步驟

v找出最優解的性質，並刻划其結構特征。

v遞歸地定義最優值。

v以自底向上的方式計算出最優值。

v根據計算最優值時得到的信息，構造最優解

https://en.wikipedia.org/wiki/Dynamic_time_warping

Dynamic time warping

From Wikipedia, the free encyclopedia

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.

In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences which may vary in time or speed. For instance, similarities in walking patterns could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data which can be turned into a linear sequence can be analyzed with DTW. A well known application has been automaticspeech recognition, to cope with different speaking speeds. Other applications include speaker recognition and onlinesignature recognition. Also it is seen that it can be used in partial shape matching application.

In general, DTW is a method that calculates 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 time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold.

Implementation[edit]

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

int DTWDistance(s: array [1..n], t: array [1..m]) {
    DTW := array [0..n, 0..m]

    for i := 1 to n
        DTW[i, 0] := infinity
    for i := 1 to m
        DTW[0, i] := 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(s: array [1..n], t: array [1..m], w: int) {
    DTW := array [0..n, 0..m]
 
    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]

Fast computation[edit]

Computing the DTW requires $O(N^2)$ in general. Fast techniques for computing DTW include SparseDTW^[1] and the FastDTW.^[2] A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh^[3] or LB_Improved.^[4] In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient.^[5]

Average sequence[edit]

Averaging for Dynamic Time Warping is the problem of finding an average sequence for a set of sequences. The average sequence is the sequence that minimizes the sum of the squares to the set of objects. NLAAF^[6] is the exact method for two sequences. For more than two sequences, the problem is related to the one of the Multiple alignment and requires heuristics. DBA^[7] is currently the reference method to average a set of sequences consistently with DTW. COMASA^[8] efficiently randomizes the search for the average sequence, using DBA as a local optimization process.

Supervised Learning[edit]

A Nearest Neighbour Classifier can achieve state-of-the-art performance when using Dynamic Time Warping as a distance measure.^[9]

Alternative approach[edit]

An alternative technique for DTW is based on functional data analysis, in which the time series are regarded as discretizations of smooth (differentiable) functions of time and therefore continuous mathematics is applied.^[10] Optimal nonlinear time warping functions are computed by minimizing a measure of distance of the set of functions to their warped average. Roughness penalty terms for the warping functions may be added, e.g., by constraining the size of their curvature. The resultant warping functions are smooth, which facilitates further processing. This approach has been successfully applied to analyze patterns and variability of speech movements.^[11]^[12]

Open Source software[edit]

The lbimproved C++ library implements Fast Nearest-Neighbor Retrieval algorithms under the GNU General Public License (GPL). It also provides a C++ implementation of Dynamic Time Warping as well as various lower bounds.
The FastDTW library is a Java implementation of DTW and a FastDTW implementation that provides optimal or near-optimal alignments with an O(N) time and memory complexity, in contrast to the O(N^2) requirement for the standard DTW algorithm. FastDTW uses a multilevel approach that recursively projects a solution from a coarser resolution and refines the projected solution..
FastDTW fork (Java) published to Maven Central
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.
The pydtw C++/Python library implements the Manhattan and Euclidean flavoured DTW measures including the LB_Keogh lower bounds.
What about the dtw python library?
The cudadtw C++/CUDA library implements subsequence alignment of Euclidean-flavoured DTW and z-normalized Euclidean Distance similar to the popular UCR-Suite on CUDA-enabled accelerators.
The JavaML machine learning library implements DTW.
The ndtw C# library implements DTW with various options.
Sketch-a-Char uses Greedy DTW (implemented in JavaScript) as part of LaTeX symbol classifier program.
The MatchBox implements DTW to match Mel-Frequency Cepstral Coefficients of audio signals.
Sequence averaging: a GPL Java implementation of DBA.^[7]
C/Python library implements DTW with some variations(distance functions, step patterns and windows)

Applications[edit]

Spoken word recognition[edit]

Due to different speaking rates, a non-linear fluctuation occurs in speech pattern versus time axis which needs to be eliminated.^[13] DP-matching, which is a pattern matching algorithm discussed in paper "Dynamic Programming Algorithm Optimization For Spoken Word Recognition" by Hiroaki Sakoe and Seibi Chiba, uses a time normalisation effect where the fluctuations in the time axis are modeled using a non-linear time-warping function. Considering any two speech patterns, we can get rid off their timing differences by warping the time axis of one so that the maximum coincidence in attained with the other. Moreover, if the warping function is allowed to take any possible value, very less distinction can be made between words belonging to different categories. So, to enhance the distinction between words belonging to different categories, restrictions were imposed on the warping function slope.

References[edit]

Jump up^ Al-Naymat, G., Chawla, S., & Taheri, J. (2012). SparseDTW: A Novel Approach to Speed up Dynamic Time Warping
Jump up^ Stan Salvador & Philip Chan, FastDTW: Toward Accurate Dynamic Time Warping in Linear Time and Space. KDD Workshop on Mining Temporal and Sequential Data, pp. 70-80, 2004
Jump up^ Keogh, E.; Ratanamahatana, C. A. (2005). "Exact indexing of dynamic time warping". Knowledge and Information Systems 7 (3): 358–386.doi:10.1007/s10115-004-0154-9.
Jump up^ Lemire, D. (2009). "Faster Retrieval with a Two-Pass Dynamic-Time-Warping Lower Bound". Pattern Recognition 42 (9): 2169–2180.doi:10.1016/j.patcog.2008.11.030.
Jump up^ Wang, Xiaoyue; et al. "Experimental comparison of representation methods and distance measures for time series data". Data Mining and Knowledge Discovery 2010: 1–35.
Jump up^ Gupta, L.; Molfese, D. L.; Tammana, R.; Simos, P. G. (1996). "Nonlinear alignment and averaging for estimating the evoked potential". IEEE Transactions on Biomedical Engineering 43 (4): 348–356. doi:10.1109/10.486255. PMID 8626184.
^ Jump up to:^a ^b Petitjean, F. O.; Ketterlin, A.; Gançarski, P. (2011). "A global averaging method for dynamic time warping, with applications to clustering".Pattern Recognition 44 (3): 678. doi:10.1016/j.patcog.2010.09.013.
Jump up^ Petitjean, F. O.; Gançarski, P. (2012). "Summarizing a set of time series by averaging: From Steiner sequence to compact multiple alignment".Theoretical Computer Science 414: 76. doi:10.1016/j.tcs.2011.09.029.
Jump up^ Ding, Hui; Trajcevski, Goce; Scheuermann, Peter; Wang, Xiaoyue; Keogh, Eamonn (2008). "Querying and mining of time series data: experimental comparison of representations and distance measures". Proc. VLDB Endow 1 (2): 1542–1552. doi:10.14778/1454159.1454226.
Jump up^ Lucero, J. C.; Munhall, K. G.; Gracco, V. G.; Ramsay, J. O. (1997). "On the Registration of Time and the Patterning of Speech Movements".Journal of Speech, Language, and Hearing Research 40: 1111–1117.
Jump up^ Howell, P.; Anderson, A.; Lucero, J. C. (2010). "Speech motor timing and fluency". In Maassen, B.; van Lieshout, P. Speech Motor Control: New Developments in Basic and Applied Research. Oxford University Press. pp. 215–225. ISBN 978-0199235797.
Jump up^ Koenig, Laura L.; Lucero, Jorge C.; Perlman, Elizabeth (2008). "Speech production variability in fricatives of children and adults: Results of functional data analysis". The Journal of the Acoustical Society of America 124 (5): 3158–3170. doi:10.1121/1.2981639. ISSN 0001-4966.PMC 2677351. PMID 19045800.
Jump up^ Sakoe, Hiroaki; Chiba, Seibi. "Dynamic programming algorithm optimization for spoken word recognition". IEEE Transactions on Acoustics, Speech and Signal Processing 26 (1): 43–49. doi:10.1109/tassp.1978.1163055.

動態時間規整算法