dtw

dtw.dtw(x, y=None, dist_method='euclidean', step_pattern='symmetric2', window_type=None, window_args={}, keep_internals=False, distance_only=False, open_end=False, open_begin=False)

Compute Dynamic Time Warp and find optimal alignment between two time series.

Details

The function performs Dynamic Time Warp (DTW) and computes the optimal alignment between two time series x and y, given as numeric vectors. The “optimal” alignment minimizes the sum of distances between aligned elements. Lengths of x and y may differ.

The local distance between elements of x (query) and y (reference) can be computed in one of the following ways:

  1. if dist_method is a string, x and y are passed to the scipy.spatial.distance.cdist function with the method given;

  2. multivariate time series and arbitrary distance metrics can be handled by supplying a local-distance matrix. Element [i,j] of the local-distance matrix is understood as the distance between element x[i] and y[j]. The distance matrix has therefore n=length(x) rows and m=length(y) columns (see note below).

Several common variants of the DTW recursion are supported via the step_pattern argument, which defaults to symmetric2. Step patterns are commonly used to locally constrain the slope of the alignment function. See [stepPattern()] for details.

Windowing enforces a global constraint on the envelope of the warping path. It is selected by passing a string or function to the window_type argument. Commonly used windows are (abbreviations allowed):

  • "none" No windowing (default)

  • "sakoechiba" A band around main diagonal

  • "slantedband" A band around slanted diagonal

  • "itakura" So-called Itakura parallelogram

window_type can also be an user-defined windowing function. See [dtwWindowingFunctions()] for all available windowing functions, details on user-defined windowing, and a discussion of the (mis)naming of the “Itakura” parallelogram as a global constraint. Some windowing functions may require parameters, such as the window_size argument.

Open-ended alignment, i_e. semi-unconstrained alignment, can be selected via the open_end switch. Open-end DTW computes the alignment which best matches all of the query with a leading part of the reference. This is proposed e_g. by Mori (2006), Sakoe (1979) and others. Similarly, open-begin is enabled via open_begin; it makes sense when open_end is also enabled (subsequence finding). Subsequence alignments are similar e_g. to UE2-1 algorithm by Rabiner (1978) and others. Please find a review in Tormene et al. (2009).

If the warping function is not required, computation can be sped up enabling the distance_only=True switch, which skips the backtracking step. The output object will then lack the index{1,2,1s,2s} and stepsTaken fields.

Parameters:

  • x – query vector or local cost matrix

  • y – reference vector, unused if x given as cost matrix

  • dist_method – pointwise (local) distance function to use.

  • step_pattern – a stepPattern object describing the local warping steps allowed with their cost (see [stepPattern()])

  • window_type – windowing function. Character: “none”, “itakura”, “sakoechiba”, “slantedband”, or a function (see details).

  • open_begin – perform open-ended alignments

  • open_end – perform open-ended alignments

  • keep_internals – preserve the cumulative cost matrix, inputs, and other internal structures

  • distance_only – only compute distance (no backtrack, faster)

  • window_args – additional arguments, passed to the windowing function

Return type:

An object of class DTW. See docs for the corresponding properties.

Notes

Cost matrices (both input and output) have query elements arranged row-wise (first index), and reference elements column-wise (second index). They print according to the usual convention, with indexes increasing down- and rightwards. Many DTW papers and tutorials show matrices according to plot-like conventions, i_e. reference index growing upwards. This may be confusing.

A fast compiled version of the function is normally used. Should it be unavailable, the interpreted equivalent will be used as a fall-back with a warning.

References

  1. Toni Giorgino. Computing and Visualizing Dynamic Time Warping Alignments in R: The dtw Package. Journal of Statistical Software, 31(7), 1-24. http://www.jstatsoft.org/v31/i07/

  2. Tormene, P.; Giorgino, T.; Quaglini, S. & Stefanelli, M. Matching incomplete time series with dynamic time warping: an algorithm and an application to post-stroke rehabilitation. Artif Intell Med, 2009, 45, 11-34. http://dx.doi.org/10.1016/j.artmed.2008.11.007

  3. Sakoe, H.; Chiba, S., Dynamic programming algorithm optimization for spoken word recognition, Acoustics, Speech, and Signal Processing, IEEE Transactions on , vol.26, no.1, pp. 43-49, Feb 1978. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1163055

  4. Mori, A.; Uchida, S.; Kurazume, R.; Taniguchi, R.; Hasegawa, T. & Sakoe, H. Early Recognition and Prediction of Gestures Proc. 18th International Conference on Pattern Recognition ICPR 2006, 2006, 3, 560-563

  5. Sakoe, H. Two-level DP-matching–A dynamic programming-based pattern matching algorithm for connected word recognition Acoustics, Speech, and Signal Processing, IEEE Transactions on, 1979, 27, 588-595

  6. Rabiner L, Rosenberg A, Levinson S (1978). Considerations in dynamic time warping algorithms for discrete word recognition. IEEE Trans. Acoust., Speech, Signal Process., 26(6), 575-582. ISSN 0096-3518.

  7. Muller M. Dynamic Time Warping in Information Retrieval for Music and Motion. Springer Berlin Heidelberg; 2007. p. 69-84. http://link.springer.com/chapter/10.1007/978-3-540-74048-3_4

Examples

>>> import numpy as np
>>> from dtw import *

A noisy sine wave as query

>>> idx = np.linspace(0,6.28,num=100)
>>> query = np.sin(idx) + np.random.uniform(size=100)/10.0

A cosine is for reference; sin and cos are offset by 25 samples

>>> reference = np.cos(idx)

Find the best match

>>> alignment = dtw(query,reference)

Display the mapping, AKA warping function - may be multiple-valued Equivalent to: plot(alignment,type=”alignment”)

>>> import matplotlib.pyplot as plt;
... plt.plot(alignment.index1, alignment.index2)

Partial alignments are allowed.

>>> alignmentOBE = dtw(query[44:88], reference,
...                      keep_internals=True,
...                      step_pattern=asymmetric,
...                      open_end=True,open_begin=True)

>>> alignmentOBE.plot(type="twoway",offset=1)

Subsetting allows warping and unwarping of timeseries according to the warping curve. See first example below.

Most useful: plot the warped query along with reference

>>> plt.plot(reference);
... plt.plot(alignment.index2,query[alignment.index1])

Plot the (unwarped) query and the inverse-warped reference

>>> plt.plot(query)                                     
... plt.plot(alignment.index1,reference[alignment.index2])

A hand-checkable example

>>> ldist = np.ones((6,6))                    # Matrix of ones
>>> ldist[1,:] = 0; ldist[:,4] = 0;           # Mark a clear path of zeroes
>>> ldist[1,4] = .01;                         # Forcely cut the corner

>>> ds = dtw(ldist);                          # DTW with user-supplied local

>>> da = dtw(ldist,step_pattern=asymmetric)   # Also compute the asymmetric

Symmetric: alignment follows the low-distance marked path

>>> plt.plot(ds.index1,ds.index2)

Asymmetric: visiting 1 is required twice

>>> plt.plot(da.index1,da.index2,'ro')        

>>> float(ds.distance)
2.0
>>> float(da.distance)
2.0