StepPattern

class dtw.StepPattern(mx, hint='NA')

Bases: object

Step patterns for DTW

A stepPattern object lists the transitions allowed while searching for the minimum-distance path. DTW variants are implemented by passing one of the objects described in this page to the stepPattern argument of the [dtw()] call.

Details

A step pattern characterizes the matching model and slope constraint specific of a DTW variant. They also known as local- or slope-constraints, transition types, production or recursion rules (GiorginoJSS).

Pre-defined step patterns

## Well-known step patterns
symmetric1
symmetric2
asymmetric

## Step patterns classified according to Rabiner-Juang (Rabiner1993)
rabinerJuangStepPattern(type,slope_weighting="d",smoothed=False)

## Slope-constrained step patterns from Sakoe-Chiba (Sakoe1978)
symmetricP0;  asymmetricP0
symmetricP05; asymmetricP05
symmetricP1;  asymmetricP1
symmetricP2;  asymmetricP2

## Step patterns classified according to Rabiner-Myers (Myers1980)
typeIa;   typeIb;   typeIc;   typeId;
typeIas;  typeIbs;  typeIcs;  typeIds;  # smoothed
typeIIa;  typeIIb;  typeIIc;  typeIId;
typeIIIc; typeIVc;

## Miscellaneous
mori2006;
rigid;

A variety of classification schemes have been proposed for step patterns, including Sakoe-Chiba (Sakoe1978); Rabiner-Juang (Rabiner1993); and Rabiner-Myers (Myers1980). The dtw package implements all of the transition types found in those papers, with the exception of Itakura’s and Velichko-Zagoruyko’s steps, which require subtly different algorithms (this may be rectified in the future). Itakura recursion is almost, but not quite, equivalent to typeIIIc.

For convenience, we shall review pre-defined step patterns grouped by classification. Note that the same pattern may be listed under different names. Refer to paper (GiorginoJSS) for full details.

1. Well-known step patterns

Common DTW implementations are based on one of the following transition types.

symmetric2 is the normalizable, symmetric, with no local slope constraints. Since one diagonal step costs as much as the two equivalent steps along the sides, it can be normalized dividing by N+M (query+reference lengths). It is widely used and the default.

asymmetric is asymmetric, slope constrained between 0 and 2. Matches each element of the query time series exactly once, so the warping path index2~index1 is guaranteed to be single-valued. Normalized by N (length of query).

symmetric1 (or White-Neely) is quasi-symmetric, no local constraint, non-normalizable. It is biased in favor of oblique steps.

2. The Rabiner-Juang set

A comprehensive table of step patterns is proposed in Rabiner-Juang’s book (Rabiner1993), tab. 4.5. All of them can be constructed through the rabinerJuangStepPattern(type,slope_weighting,smoothed) function.

The classification foresees seven families, labelled with Roman numerals I-VII; here, they are selected through the integer argument type. Each family has four slope weighting sub-types, named in sec. 4.7.2.5 as “Type (a)” to “Type (d)”; they are selected passing a character argument slope_weighting, as in the table below. Furthermore, each subtype can be either plain or smoothed (figure 4.44); smoothing is enabled setting the logical argument smoothed. (Not all combinations of arguments make sense.)

Subtype | Rule       | Norm | Unbiased
--------|------------|------|---------
   a    | min step   |  --  |   NO
   b    | max step   |  --  |   NO
   c    | Di step    |   N  |  YES
   d    | Di+Dj step | N+M  |  YES

3. The Sakoe-Chiba set

Sakoe-Chiba (Sakoe1978) discuss a family of slope-constrained patterns; they are implemented as shown in page 47, table I. Here, they are called symmetricP<x> and asymmetricP<x>, where <x> corresponds to Sakoe’s integer slope parameter P. Values available are accordingly: 0 (no constraint), 1, 05 (one half) and 2. See (Sakoe1978) for details.

4. The Rabiner-Myers set

The type<XX><y> step patterns follow the older Rabiner-Myers’ classification proposed in (Myers1980) and (MRR1980). Note that this is a subset of the Rabiner-Juang set (Rabiner1993), and the latter should be preferred in order to avoid confusion. <XX> is a Roman numeral specifying the shape of the transitions; <y> is a letter in the range a-d specifying the weighting used per step, as above; typeIIx patterns also have a version ending in s, meaning the smoothing is used (which does not permit skipping points). The typeId, typeIId and typeIIds are unbiased and symmetric.

5. Others

The rigid pattern enforces a fixed unitary slope. It only makes sense in combination with open_begin=True, open_end=True to find gapless subsequences. It may be seen as the P->inf limiting case in Sakoe’s classification.

mori2006 is Mori’s asymmetric step-constrained pattern (Mori2006). It is normalized by the matched reference length.

[mvmStepPattern()] implements Latecki’s Minimum Variance Matching algorithm, and it is described in its own page.

Methods

print_stepPattern prints an user-readable description of the recurrence equation defined by the given pattern.

plot_stepPattern graphically displays the step patterns productions which can lead to element (0,0). Weights are shown along the step leading to the corresponding element.

t_stepPattern transposes the productions and normalization hint so that roles of query and reference become reversed.

Parameters:

  • x – a step pattern object

  • type – path specification, integer 1..7 (see (Rabiner1993), table 4.5)

  • slope_weighting – slope weighting rule: character “a” to “d” (see (Rabiner1993), sec. 4.7.2.5)

  • smoothed – logical, whether to use smoothing (see (Rabiner1993), fig. 4.44)

  • ... – additional arguments to [print()].

Notes

Constructing stepPattern objects is tricky and thus undocumented. For a commented example please see source code for symmetricP1.

References

  • (GiorginoJSS) Toni Giorgino. Computing and Visualizing Dynamic Time Warping Alignments in R: The dtw Package. Journal of Statistical Software, 31(7), 1-24. doi:10.18637/jss_v031.i07

  • (Itakura1975) Itakura, F., Minimum prediction residual principle applied to speech recognition, Acoustics, Speech, and Signal Processing, IEEE Transactions on , vol.23, no.1, pp. 67-72, Feb 1975. doi:10.1109/TASSP.1975.1162641

  • (MRR1980) Myers, C.; Rabiner, L. & Rosenberg, A. Performance tradeoffs in dynamic time warping algorithms for isolated word recognition, IEEE Trans. Acoust., Speech, Signal Process., 1980, 28, 623-635. doi:10.1109/TASSP.1980.1163491

  • (Mori2006) 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. doi:10.1109/ICPR.2006.467

  • (Myers1980) Myers, Cory S. A Comparative Study Of Several Dynamic Time Warping Algorithms For Speech Recognition, MS and BS thesis, Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, archived Jun 20 1980, https://hdl_handle_net/1721.1/27909

  • (Rabiner1993) Rabiner, L. R., & Juang, B.-H. (1993). Fundamentals of speech recognition. Englewood Cliffs, NJ: Prentice Hall.

  • (Sakoe1978) 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 doi:10.1109/TASSP.1978.1163055

Examples

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

The usual (normalizable) symmetric step pattern Step pattern recursion, defined as:

g[i,j] = min(

g[i,j-1] + d[i,j] , g[i-1,j-1] + 2 * d[i,j] , g[i-1,j] + d[i,j] ,

)

>>> print(symmetric2)            
Step pattern recursion:
 g[i,j] = min(
     g[i-1,j-1] + 2 * d[i  ,j  ] ,
     g[i  ,j-1] +     d[i  ,j  ] ,
     g[i-1,j  ] +     d[i  ,j  ] ,
 )

Normalization hint: N+M

The well-known plotting style for step patterns

>>> import matplotlib.pyplot as plt;            
... symmetricP2.plot().set_title("Sakoe's Symmetric P=2 recursion")

Same example seen in ?dtw , now with asymmetric step pattern

>>> (query, reference) = dtw_test_data.sin_cos()

Do the computation

>>> asy = dtw(query, reference, keep_internals=True,
...                  step_pattern=asymmetric);

>>> dtwPlot(asy,type="density"                  
...         ).set_title("Sine and cosine, asymmetric step")

Hand-checkable example given in [Myers1980] p 61 - see JSS paper

>>> tm = numpy.reshape( [1, 3, 4, 4, 5, 2, 2, 3, 3, 4, 3, 1, 1, 1, 3, 4, 2,
...                      3, 3, 2, 5, 3, 4, 4, 1], (5,5), "F" )

Methods Summary

T()

Transpose a step pattern.

get_n_patterns()

Number of rules in the recursion.

get_n_rows()

Total number of steps in the recursion.

plot()

Provide a visual description of a StepPattern object

Methods Documentation

T()

Transpose a step pattern.

get_n_patterns()

Number of rules in the recursion.

get_n_rows()

Total number of steps in the recursion.

plot()

Provide a visual description of a StepPattern object