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
andy
, given as numeric vectors. The “optimal” alignment minimizes the sum of distances between aligned elements. Lengths ofx
andy
may differ.The local distance between elements of
x
(query) andy
(reference) can be computed in one of the following ways:if
dist_method
is a string,x
andy
are passed to the scipy.spatial.distance.cdist function with the method given;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 elementx[i]
andy[j]
. The distance matrix has thereforen=length(x)
rows andm=length(y)
columns (see note below).
Several common variants of the DTW recursion are supported via the
step_pattern
argument, which defaults tosymmetric2
. 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 thewindow_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 viaopen_begin
; it makes sense whenopen_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 theindex{1,2,1s,2s}
andstepsTaken
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
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/
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
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
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
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
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.
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')
>>> ds.distance 2.0 >>> da.distance 2.0