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
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a step pattern object
additional arguments to
path specification, integer 1..7 (see (Rabiner1993), table 4.5)
slope weighting rule: character
logical, whether to use smoothing (see (Rabiner1993), fig. 4.44)
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
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## 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
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,
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
(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
The classification foresees seven families, labelled with Roman numerals
I-VII; here, they are selected through the integer argument
Each family has four slope weighting sub-types, named in sec. 184.108.40.206 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
smoothed. (Not all combinations of arguments make
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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
to Sakoe's integer slope parameter P. Values available are
0 (no constraint),
05 (one half) and
2. See (Sakoe1978) for details.
4. The Rabiner-Myers set
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;
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.
rigid pattern enforces a fixed unitary slope. It only makes sense
in combination with
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.
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.
stepPattern objects is tricky and thus undocumented. For a commented example please see source code for
(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
mvmStepPattern(), implementing Latecki's Minimal
Variance Matching algorithm.
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######### ## ## 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) # or just "symmetric2" ######### ## ## The well-known plotting style for step patterns plot(symmetricP2,main="Sakoe's Symmetric P=2 recursion") ######### ## ## Same example seen in ?dtw , now with asymmetric step pattern idx<-seq(0,6.28,len=100); query<-sin(idx)+runif(100)/10; reference<-cos(idx); ## Do the computation asy<-dtw(query,reference,keep=TRUE,step=asymmetric); dtwPlot(asy,type="density",main="Sine and cosine, asymmetric step") ######### ## ## Hand-checkable example given in [Myers1980] p 61 ## `tm` <- structure(c(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), .Dim = c(5L, 5L))
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