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CanonicalWarpingDistance
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CanonicalWarpingDistance

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gives the canonical time warping (CTW) distance between sequences s1 and s2.

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uses init as the initial correspondence between the two sequences.

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CanonicalWarpingDistance[s1,s2,init,win]

uses a window win for local search.

Details and Options

  • Canonical time warping (CTW) iteratively performs spatial transformations and dynamic time warping on the reference sequence s1 and the query sequence s2 to find the alignment with minimal distance between sequences.
  • The sequences si can be lists of numeric scalars or vectors. In contrast to dynamic time warping, elements of s1 and s2 can be of different dimensions.
  • The distance is given by , where s1〚ni〛 and s2〚mi〛 are corresponding elements, and α and β are spatial transformation matrices computed using canonical correlation analysis.
  • The correspondence giving the computed distance can be obtained using CanonicalWarpingCorrespondence.
  • Initial correspondence init should be given in the form of {{n1,…,nk}{{m1,…,mk}}, which provides an initial one-to-one correspondence between the elements of s1 and s2.
  • Possible settings for the search window win are:
  • Automatica full search
    ra slanted band window of radius
    {"SlantedBand",r}a slanted band window of radius
    {"Band",r}band window of radius (Sakoe-Chiba)
    {"Parallelogram",a}parallelogram window placed at origin with slopes and (Itakura)
  • The following options are supported:
  • DistanceFunctionAutomaticdistance function used in dynamic time warping
    MaxIterations Automaticmaximal number of iterations
    MethodAutomaticthe variant of CTW to use
  • For possible settings of DistanceFunction, see the reference page for WarpingDistance.
  • The following options are available through Method->opts:
  • "DimensionsToKeep"Automaticdimensions after projection
    "EnergyThreshold"Automaticfraction of "energy" to be kept
    "Lambdas"Automaticregularization values
    "MatchingIntervals"Automaticwhether to match the query to full or part of reference
  • Possible settings for the "MatchingIntervals" option include:
  • Automatica full match
    "Flexible"flexible at both ends
    "FlexibleEnd"flexible only at the end of the interval

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the canonical time warping distance of two scalar sequences:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-c498v6
Out[1]=1

Find the canonical time warping distance between a 3D path and a 2D line segment:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-vojz2u
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Data  (6)

Find the distance between two sequences of numerical scalars with different length:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-4ixvux
Out[1]=1

Find the distance between two sequences of vectors:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-cvospy
Out[1]=1

Find the distance between a sequence of scalars and a sequence of 3D points:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-dippe4
Out[1]=1

Find the distance between two sequences of quantities with compatible units:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-mqt072
In[2]:=2
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-g4ussc
Out[2]=2

All units are converted to base SI units:

In[3]:=3
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-bulhww
Out[3]=3

Find the distance between two sequences of quantity vectors with compatible units:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-9hl3wz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-3zzrt4
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-lyzqtd
Out[3]=3

Find the distance between a sequence of quantities and a sequence of scalars:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-t63o31
In[2]:=2
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-sa985d
Out[2]=2

The scalars are interpreted as if they had a compatible unit from SI base units:

In[3]:=3
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-gxz9x8
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-roclfe
Out[4]=4

Initial Warping  (1)

By default, the shorter sequence is first warped uniformly to match the length of the longer sequence:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-foq6y0
In[3]:=3
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-3ibbdl
Out[3]=3

Construct the initial uniform warping manually:

In[4]:=4
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-27dfdl
Out[4]=4

Search Window  (2)

By default, an unconstrained search is performed:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-75y26h
In[3]:=3
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-erqeni
Out[3]=3

Use a "SlantedBand" search window:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-dk2bqk
In[2]:=2
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-5f7uuh
Out[2]=2

Options  (2)Common values & functionality for each option

MaxIterations  (2)

By default, the maximal number of iterations is 50:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-1im7et
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-rkbltr
Out[4]=4

Decreasing the maximal number of iterations may reduce the timing:

In[5]:=5
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-v65dsl
Out[5]=5

Increasing the maximal number of iterations may yield a smaller distance whenever CTW converges:

In[6]:=6
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-h18cx2
Out[6]=6

Set MaxIterations∞ to find the distance after convergence. CTW may not converge:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-2dgbmz
In[2]:=2
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-79ywmj
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Compare similarity between shapes of different countries:

In[1]:=1
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-hdgm5q

Calculate distances:

In[2]:=2
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-5bkfq

Prepare ticks for the plot:

In[3]:=3
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-qjctac

Show the distance matrix:

In[6]:=6
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-xfdp9
Out[6]=6

Properties & Relations  (1)Properties of the function, and connections to other functions

Canonical time warping is robust to spatial transformations.

Translation invariance:

In[20]:=20
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-2geypg
In[22]:=22
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-bl56vn
Out[22]=22
In[23]:=23
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-ndqcb9
Out[23]=23

Scale invariance:

In[4]:=4
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-ub1s52
Out[6]=6
In[18]:=18
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-iain7s
Out[18]=18

Rotation invariance:

In[24]:=24
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-q2exbo
In[27]:=27
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-gp6zdo
Out[27]=27
In[28]:=28
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https://wolfram.com/xid/08dsoghhr87r0bmo6erki-11odgj
Out[28]=28
Wolfram Research (2016), CanonicalWarpingDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html.
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Wolfram Research (2016), CanonicalWarpingDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html.

Text

Wolfram Research (2016), CanonicalWarpingDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html.

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Wolfram Research (2016), CanonicalWarpingDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html.

CMS

Wolfram Language. 2016. "CanonicalWarpingDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html.

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Wolfram Language. 2016. "CanonicalWarpingDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html.

APA

Wolfram Language. (2016). CanonicalWarpingDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html

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Wolfram Language. (2016). CanonicalWarpingDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html

BibTeX

@misc{reference.wolfram_2025_canonicalwarpingdistance, author="Wolfram Research", title="{CanonicalWarpingDistance}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html}", note=[Accessed: 19-May-2025 ]}

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@misc{reference.wolfram_2025_canonicalwarpingdistance, author="Wolfram Research", title="{CanonicalWarpingDistance}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html}", note=[Accessed: 19-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_canonicalwarpingdistance, organization={Wolfram Research}, title={CanonicalWarpingDistance}, year={2016}, url={https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html}, note=[Accessed: 19-May-2025 ]}

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@online{reference.wolfram_2025_canonicalwarpingdistance, organization={Wolfram Research}, title={CanonicalWarpingDistance}, year={2016}, url={https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.html}, note=[Accessed: 19-May-2025 ]}