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WarpingDistance
✖
WarpingDistance

✖

gives the dynamic time warping (DTW) distance between sequences s1 and s2.

✖
WarpingDistance[s1,s2,win]

uses a window specified by win for local search.

Details and Options

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Find the time warping distance between two sequences of values:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-2y9m73
Out[1]=1

Show the correspondence between the sequences:

In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-gr4ap9
Out[2]=2

Scope  (10)Survey of the scope of standard use cases

Data  (7)

Find the time warping distance between two real-valued vectors:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-k639lv
Out[1]=1

Find the time warping distance between two sequences of 2D points:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-vftgtj
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-nzbrvi
Out[2]=2

Show the correspondence between the sequences:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-kkfpw7
Out[3]=3

Find the time warping distance between two Boolean vectors:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-uxok45
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-vw8iqp
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-by8gpk
Out[3]=3

Sequences of Boolean-valued vectors:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-xz644a
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-w482wo
Out[2]=2

Sequences of quantities with compatible units:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-mqt072
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-g4ussc
Out[2]=2

All units are converted to base SI units:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-bulhww
Out[3]=3

Two-dimensional quantity arrays:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-9hl3wz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-3zzrt4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-lyzqtd
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-t63o31
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-sa985d
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-7868fq
Out[3]=3

Search Window  (3)

By default, the search is not locally constrained:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-sg6rii
In[5]:=5
✖
https://wolfram.com/xid/0bc63utsee9zifm-0ssudd
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bc63utsee9zifm-oo5tn9
Out[6]=6

Specify a radius for the search window:

In[7]:=7
✖
https://wolfram.com/xid/0bc63utsee9zifm-l3ll0
Out[7]=7

Increase the radius to find a more optimal alignment:

In[8]:=8
✖
https://wolfram.com/xid/0bc63utsee9zifm-lheg5a
Out[8]=8

A window size is interpreted as the radius of a slanted band window:

In[9]:=9
✖
https://wolfram.com/xid/0bc63utsee9zifm-cyze5m
Out[9]=9

Use a band of radius 1:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-i4tkpv
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-c3cgs
Out[2]=2

Use a parallelogram of slope 3:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-b83fy6
Out[3]=3

For signals of equal length, the "Band" window is equivalent to the "SlantedBand" window:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-ec6w55
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-hht6a8
Out[2]=2

Generalizations & Extensions  (1)Generalized and extended use cases

Compute DTW-Delta (or DTW-D) by normalizing DTW by the Euclidean distance of sequences:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-e9gays

Comparing a sine wave, a distorted sine wave, and a random noise using DTW:

In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-3obxg0
Out[4]=4

Using regular DTW, the sine wave seems to be more similar to the random noise:

In[30]:=30
✖
https://wolfram.com/xid/0bc63utsee9zifm-zvjwze
In[26]:=26
✖
https://wolfram.com/xid/0bc63utsee9zifm-5hscww
Out[26]=26

Using DTW-Delta, the sine wave is more similar to the distorted sine wave:

In[32]:=32
✖
https://wolfram.com/xid/0bc63utsee9zifm-ntzf1d

Options  (6)Common values & functionality for each option

DistanceFunction  (5)

With Boolean sequences, WarpingDistance uses MatchingDissimilarity:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-1ofaka
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-6teu7z
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-dh6s7c
Out[4]=4

Compare two sequences using different distance functions:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-ikif28
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-9y0te
Out[2]=2

In case of one-dimensional signals, some distance functions are equivalent:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-ta5zbe
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-w8fyrf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-f79u0n
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-cb00zk
In[5]:=5
✖
https://wolfram.com/xid/0bc63utsee9zifm-oemoud
Out[5]=5

In case of one-dimensional signals, NormalizedSquaredEuclideanDistance and CorrelationDistance always return 0:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-dcxpfl
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-bx6sw
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-dmruwr
Out[3]=3

For signals containing one element, time warping distance is equal to the distance between elements:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-b68o6i
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-3vp5nc
Out[2]=2

Method  (1)

By default, the query sequence will be matched with the entire reference sequence:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-rjldr5
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-zqjyih
Out[3]=3

With "MatchingInterval""FlexibleEnd", any suffix of reference sequence can be omitted:

In[17]:=17
✖
https://wolfram.com/xid/0bc63utsee9zifm-zmawxq
Out[17]=17

With "MatchingInterval""Flexible", any suffix and prefix of reference can be omitted:

In[18]:=18
✖
https://wolfram.com/xid/0bc63utsee9zifm-zp8lao
Out[18]=18

Applications  (6)Sample problems that can be solved with this function

Cluster time series of normalized stock values:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-8dvqyv
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-0zmwng
Out[3]=3

Find the nearest stock based on normalized stock values:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-3ykokt
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-jz29t3
Out[2]=2

Find the capital city in the EU that had the most similar temperature to Chicago over the last year. Use WarpingDistance to determine the similarity between temperature sequences:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-keceww

Get the temperature for all capital cities in the European Union:

In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-v6m2gh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-zt3ldt

Find the capital city that has the most similar temperatures to Chicago:

In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-sz3lwd
Out[4]=4

Compute pairwise distances between temperature time series of all EU capitals:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-vj1tme
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-hleri4
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-57zhti
In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-f9n73v
Out[4]=4

Visualize the temperature similarity using a previously computed distance matrix:

In[5]:=5
✖
https://wolfram.com/xid/0bc63utsee9zifm-47r6ne
Out[5]=5

Compare daily mean humidity for some cities in 2014:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-8fmf8o
Out[1]=1

Extract humidity of the cities:

In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-hqub3b

Compare extracted data:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-dip99x
Out[3]=3

Compare two ECG signals of heartbeats:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-eddq3v
Out[1]=1

Find the distance between the two signals:

In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-e35xj5
Out[2]=2

Properties & Relations  (8)Properties of the function, and connections to other functions

The two sequences need not have the same length:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-rk7mnt
Out[1]=1

The distance between two equal sequences is always 0:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-87tyjr
Out[1]=1

Relation with WarpingCorrespondence:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-rwm58t
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-3ih6b1

Find the time warped sequences using the correspondence:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-z9idpu

WarpingDistance gives the sum of all the distances between corresponding elements:

In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-2xvltj
Out[4]=4

Smaller window radius results in faster computation:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-udwdjx
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-r7xql
Out[2]=2

However, using smaller radius may lead to a less optimal alignment, resulting in higher distance values:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-yod0tl
Out[3]=3

WarpingDistance is a symmetric function:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-cs6o1
Out[1]=1

The "triangle inequality" does not hold:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-xylps8
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-7ogzxe
Out[2]=2

Dynamic time warping is not translation invariant:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-qg9yrx
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-p8v1z0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-uhubw0
Out[3]=3

Canonical time warping is translation invariant:

In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-p6sm2u
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bc63utsee9zifm-w2lssc
Out[5]=5

The time warping distance increases for longer signals:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-w4hiky
Out[1]=1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-52o9kz
Out[2]=2
Out[2]=2

Normalization allows a more reasonable comparison:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-o7llsz
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-3t718w
Out[4]=4

Dividing by the length of the correspondence path is also a common way of normalization:

In[5]:=5
✖
https://wolfram.com/xid/0bc63utsee9zifm-n4rdw8
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bc63utsee9zifm-na3fh6
Out[6]=6

Possible Issues  (3)Common pitfalls and unexpected behavior

The search window specified may be too narrow to contain a correspondence path. In that case it will be automatically widened:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-ybs66a
Out[3]=3

Two long, similar signals could have a higher distance compared to two short, dissimilar signals:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-uep9ps
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-9e2v3u
Out[2]=2

Comparing two short, random sequences, which gives a relatively smaller distance:

In[3]:=3
✖
https://wolfram.com/xid/0bc63utsee9zifm-idwshy
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bc63utsee9zifm-tfo25s
Out[4]=4

Divide by the length of signals:

In[5]:=5
✖
https://wolfram.com/xid/0bc63utsee9zifm-jhk7im
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bc63utsee9zifm-2ico1r
Out[6]=6

Divide by the length of the correspondence path:

In[7]:=7
✖
https://wolfram.com/xid/0bc63utsee9zifm-m3x44c
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0bc63utsee9zifm-k1zgn5
Out[8]=8

Quantities in input sequences must have compatible units:

In[1]:=1
✖
https://wolfram.com/xid/0bc63utsee9zifm-64r3v0
In[2]:=2
✖
https://wolfram.com/xid/0bc63utsee9zifm-h1t42l
Out[2]=2
Wolfram Research (2016), WarpingDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/WarpingDistance.html.
✖
Wolfram Research (2016), WarpingDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/WarpingDistance.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_warpingdistance, author="Wolfram Research", title="{WarpingDistance}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/WarpingDistance.html}", note=[Accessed: 21-June-2025 ]}

✖
@misc{reference.wolfram_2025_warpingdistance, author="Wolfram Research", title="{WarpingDistance}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/WarpingDistance.html}", note=[Accessed: 21-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_warpingdistance, organization={Wolfram Research}, title={WarpingDistance}, year={2016}, url={https://reference.wolfram.com/language/ref/WarpingDistance.html}, note=[Accessed: 21-June-2025 ]}

✖
@online{reference.wolfram_2025_warpingdistance, organization={Wolfram Research}, title={WarpingDistance}, year={2016}, url={https://reference.wolfram.com/language/ref/WarpingDistance.html}, note=[Accessed: 21-June-2025 ]}