dtwParallel 0.9.25

Python implementation of Dynamic Time Warping (DTW), which allows computing the dtw distance between one-dimensional and multidimensional time series, with the possibility of visualisation (one-dimensional case) and parallelisation (multidimensional case).

Dynamic Time Warping

This package allows to measurement of the similarity between two-time sequences, i.e., it finds the optimal alignment between two time-dependent sequences. It allows working with univariate (UTS) and multivariate (MTS) time series, regular (same time length), or irregular (different time length).

One of the parameters available for this method is the method used to calculate the local similarity. For this case, it is possible to use any distance available in scipy.spatial.distance (it does not allow to work with variables of different nature, i.e., discrete, continuous, and categorical), gower distance (it allows to work with variables of different nature).

At this point, using local dissimilarities such as norma1, norma2, or square euclidean distance provides optimization in terms of computational time with respect to the rest of the available local dissimilarities.

The available variants of DTW are detailed below:

1. dependent DTW ("d").
2. Independent DTW ("i").
3. Itakura parallelogram.
4. Sakoe-Chiba.

Extra functionality has been incorporated to transform the resulting DTW matrix into an exponential kernel, given the sigma value (default 1).

Display for 2-time series (TS):

• It incorporates the possibility of visualizing the cost matrix, the optimal path to reach the DTW distance value between two TS, and the alignment between two TS. This will allow its use in a didactic way, providing a better understanding of the method used.

Common functionalities for N (> 2) time series (TS):

• The calculation can be parallelized by the CPU by selecting the number of threads. As a result, we will obtain the distance matrix.
• It is possible to perform distance computation and similarity computation (based on an exponential kernel).

The input data types via API are (1) CSV; (2) array; (3) pandas (pd.DataFrame or pd.Series); (4) npy (for tensors).

Table of content

Section	Description
Installation	Installing the dependencies and dtwParallel
Getting started	Packages necessary to work with dtwParallel
Available parameters	Modifiable parameters in terminal and API with their possible values
Usage	Different examples for terminal and API
Configuration	Composition of configuration.ini file
Examples with public data	Examples with real financial data
Reference	References to cite
License	Package license

Package structure

Installation

dtwParallel can be installed using pip, a tool for installing Python packages. To do it, run the following command:

pip install dtwParallel

Current version: 0.9.20

Requirements

dtwParallel requires Python >= 3.6.1 or later to run. For other Python dependencies, please check the pyproject.toml file included on this repository.

Note that you should have also the following packages installed in your system:

• numpy
• pandas
• matplotlib
• seaborn
• gower
• setuptools
• scipy
• joblib
• numba
• tslearn*
• h5py*

Available parameters

The different parameters available with their possible values are listed below:

Parameter description	Terminal usage	API usage	Possible values
Check errors	--check_errors or -ce	check_errors	True or False
Type of DTW variant	--type_dtw or -t	type_dtw	"d", "i", "itakura" or "sakoe_chiba"
Time series introduced: univariate or multivariate	MTS	MTS	True or False
Value used to complete irregular MTS. This value is removed transparently to the user	--regular_flag or -rf	regular_flag	int
Local dissimilarity value	--local_dissimilarity or -d	local_dissimilarity	any distance available in scipy.spatial.distance, "norm1", "norm2", "gower" or "square_euclidean_distance"
Number of threads used for multiple MTS parallelization	--n_threads or -n	n_threads	int
Visualization	--visualization or -vis	get_visualization	True or False
Obtain the result in a file	--output_file or -of	not possible	True or False
Name for the output file	--name_file or -nf	not possible	string
Transformation of the DTW distance matrix to an exponential kernel	--DTW_to_kernel or -k	DTW_to_kernel	True or False
Sigma value for kernel transformation	--sigma_kernel or -s	sigma_kernel	float
Maximum slope for the Itakura parallelogram	--itakura_max_slope or -imx	itakura_max_slope	float or None
Radius to be used for Sakoe-Chiba band	--sakoe_chiba_radius or -scr	sakoe_chiba_radius	int or None

Usage

Based on the previous scheme, this package can be used in three different contexts:

1) Calculation of the DTW distance with input from the terminal.

The generic example is shown below:

  dtwParallel -x <floats> -y <floats> -d <str> -ce <bool> -of <bool>

Note that only the x and y values need to be set. If not indicated, the rest of the values will be selected from the file containing the default values, configuration.ini.

Next, different uses are shown by modifying the parameters of the function:

a) Example 1. Considers the Euclidean distance, activates the option -ce to check for errors, and uses as input two UTS (denoted as x and y) with the same length (T=10). Firstly, the input TS and distance are checked as valid entries. Secondly, the dtwParallel package computes the DTW distance. Thirdly, the output is shown in the terminal (while in most cases, the outputs are forwarded to a file, we selected this option to facilitate exposition).

dtwParallel -x 2 4 6 8 5 3 6 8 9 15 -y 12 0 0 3 5 6 30 1 2 4 -ce True

[out]: 65,0

b) Example 2. Considering the CityBlock distance.

dtwParallel -x 2.5 4.3 6.6 8.0 1 0 0 1 5.5 15.2  -y 12.1 0 0 1 1 6.4 3.5 1 0 0  -d cityblock

[out]: 45,4

c) Example 3. This examples are , respectively, counterparts to Example 1 and Example 2.

dtwParallel -x 2 4 6 8 5 3 -y 12 0 0 3 5 6 30 1 2 4 

[out]: 44,0

dtwParallel -x 2.5 4.3 6.6 8.0 1 0 0 1 5.5 15.2 -y 1 0 0 1 -d cityblock

[out]: 36,1

d) Example 4. Novelty: It has been included the possibility to calculate the Itakura parallelogram and the Sakoe-Chiba band.

dtwParallel -x 2 4 6 8 1 0 0 1 5 15 -y 1 0 0 1 -t "sakoe_chiba"

[out]: 17.20

dtwParallel -x 2 4 6 8 1 0 0 1 5 15 -y 1 0 0 1 2 4 7 1 9 10 -t "itakura"

[out]: 13.0

e) Example e. Novelty: A straightforward and optimal way to calculate norm 1, norm 2 and square euclidean distance is included.

dtwParallel -x 2 4 6 8 1 0 0 1 5 15 -y 1 0 0 1 5 6 1 0 9 11 -d "norm1"

[out]: 12.24

dtwParallel -x 2 4 6 8 1 0 0 1 5 15 -y 1 0 0 1 5 6 1 0 9 11 -d "norm2"

[out]: 18.0

dtwParallel -x 2 4 6 8 1 0 0 1 5 15 -y 1 0 0 1 5 6 1 0 9 11 -d "square_euclidean_distance"

[out]: 48.0

We include another example with differents lengths of the time series:

dtwParallel -x 2 4 6 8 1 0 0 -y 1 0 0 1 5 6 1 0 9 11 -d "norm1"

[out]: 13.55

dtwParallel -x 2 4 6 8 1 0 0 -y 1 0 0 1 5 6 1 0 9 11 -d "norm2"

[out]: 29.0

dtwParallel -x 2 4 6 8 1 0 0 -y 1 0 0 1 5 6 1 0 9 11 -d "square_euclidean_distance"

[out]: 

Remarks: The calculation of the DTW distance from the command line is limited to simple examples that allow a quick understanding due to the complexity of the terminal handling:

• Univariate time series.
• Dependent DTW and two DTW variants: the Itakura parallelogram and the Sakoe-Chiba band.
• Include a straightforward and optimal way to calculate norm 1 and norm 2.
• To visualize the cost matrix, routing and the alignment between a pair of series, it will be necessary to use an integrated development environment.

2) Calculation of the DTW distance with input from a file, haciendo uso de terminal.

The generic example of univariate time series entered by means of csv files is shown below:

dtwParallel <file_X> -d <str> -ce <bool> -of <bool>

If you want to modify any of the possible values, it is necessary to modify the configuration.ini file. The possible values are those shown in Configuration.

a) Example 1. Calculation of univariate time series taking as input a csv file containing x and y.

dtwParallel exampleData/Data/E1_SyntheticData/example_1.csv

[out]: 40.6

dtwParallel exampleData/Data/E1_SyntheticData/example_1.csv -d "gower"

[out]: 10.00

dtwParallel exampleData/Data/E1_SyntheticData/example_1.csv -d "norm1"

[out]: 18.46

The generic example of multivariate time series entered by means of csv files is shown below:

dtwParallel <file_X> -d <str> -t <str> -ce <bool> -of <bool> -n <int> -k <bool> -s <float>

b) Example 2. Multivariate time series computation using a csv file containing x and y as input.

dtwParallel exampleData/Data/E1_SyntheticData/example_2.csv

[out]: 81.99

dtwParallel exampleData/Data/E1_SyntheticData/example_2.csv -d gower -t i 

[out]: 28.99

The generic example for npy files is shown below:

dtwParallel <file_X> <file_Y> -d <str> -t <str> -ce <bool> -of <bool> -n <int> -k <bool> -s <float>

c) Example 3. It computes the distance to itself. With differents types of DTW.

dtwParallel exampleData/Data/E0/X_train.npy 

[out]: [[0.00000000e+00 6.36756028e+17 2.94977907e+16 9.96457616e+17]
       [6.36756028e+17 0.00000000e+00 6.07258237e+17 1.63321364e+18]
       [2.94977907e+16 6.07258237e+17 0.00000000e+00 1.02595541e+18]
       [9.96457616e+17 1.63321364e+18 1.02595541e+18 0.00000000e+00]]

dtwParallel exampleData/Data/E0/X_train.npy -t "i"

[out]: [[0.00000000e+00 1.68469810e+18 7.80438184e+16 2.63637904e+18]
        [1.68469810e+18 0.00000000e+00 1.60665428e+18 4.32107714e+18]
        [7.80438184e+16 1.60665428e+18 0.00000000e+00 2.71442286e+18]
        [2.63637904e+18 4.32107714e+18 2.71442286e+18 0.00000000e+00]]

dtwParallel exampleData/Data/E0/X_train.npy -t "itakura"

[out]: [[0.00000000e+00 2.40671156e+17 1.11491169e+16 3.76625578e+17]
        [2.40671156e+17 0.00000000e+00 2.29522040e+17 6.17296734e+17]
        [1.11491169e+16 2.29522040e+17 0.00000000e+00 3.87774695e+17]
        [3.76625578e+17 6.17296734e+17 3.87774695e+17 0.00000000e+00]]

d) Example 4. Compute the distance between X and Y.

dtwParallel exampleData/Data/E0/X_train.npy exampleData/Data/E0/X_test.npy

[out]: [[2.47396197e+16 9.07388652e+17 2.23522660e+17 1.68210525e+18]
       [6.12016408e+17 1.54414468e+18 8.60278687e+17 2.31886127e+18]
       [4.75817098e+15 9.36886443e+17 2.53020450e+17 1.71160304e+18]
       [1.02119724e+18 8.90689643e+16 7.72934957e+17 6.85647630e+17]]

e) Example 5. Compute the gower distance between X and Y.

dtwParallel exampleData/Data/E0/X_train.npy exampleData/Data/E0/X_test.npy -d "gower"

[out]: [[1.7200027  2.16000016 1.92000033 2.53999992]
       [1.59999973 1.79999978 1.83999987 2.27999987]
       [0.5399895  1.52000002 1.04000024 1.66      ]
       [0.70000006 1.57999993 1.10000018 1.69999999]]

dtwParallel exampleData/Data/E0/X_train.npy exampleData/Data/E0/X_test.npy -d "norm1"

[out]: [[4.16145813e+08 2.52026200e+09 1.25086315e+09 3.43143363e+09]
        [2.06981034e+09 3.28770631e+09 2.45396634e+09 4.02889922e+09]
        [1.82502594e+08 2.56089928e+09 1.33084302e+09 3.46139008e+09]
        [2.67364557e+09 7.89609239e+08 2.32605776e+09 2.19078374e+09]]

f) Example 6. Compute the gower distance between X and Y and we vary the number of threads.

dtwParallel exampleData/Data/E0/X_train.npy exampleData/Data/E0/X_test.npy -d "gower" -n 12

[out]: [[1.7200027  2.16000016 1.92000033 2.53999992]
       [1.59999973 1.79999978 1.83999987 2.27999987]
       [0.5399895  1.52000002 1.04000024 1.66      ]
       [0.70000006 1.57999993 1.10000018 1.69999999]]

g) Example 7. Compute the gower distance between X and Y and we obtain the output per file.

dtwParallel exampleData/Data/E0/X_train.npy exampleData/Data/E0/X_test.npy -d "gower" -n 12 -of True

[out]: output.csv

h) Example 8. We calculate the distance between X and Y and transform to Gaussian kernel with sigma_kernel=0.5. We return the distance matrix and the kernel.

dtwParallel exampleData/Data/E0/X_train.npy -k True -s 1000000000

[out]: (array([[0.00000000e+00, 6.36756028e+17, 2.94977907e+16, 9.96457616e+17],
               [6.36756028e+17, 0.00000000e+00, 6.07258237e+17, 1.63321364e+18],
               [2.94977907e+16, 6.07258237e+17, 0.00000000e+00, 1.02595541e+18],
               [9.96457616e+17, 1.63321364e+18, 1.02595541e+18, 0.00000000e+00]]),
        array([[1.        , 0.7273278 , 0.98535934, 0.60760589],
               [0.7273278 , 1.        , 0.73813458, 0.44192866],
               [0.98535934, 0.73813458, 1.        , 0.59871014],
               [0.60760589, 0.44192866, 0.59871014, 1.        ]]))

Remarks:

• You can run from any repository, but be careful! The .npy file must be found.

3) Making use of the API

The generic example is shown below:

from dtwParallel import dtw_functions
 
# For Univariate Time Series
dtw_functions.dtw(x, y, type_dtw, local_dissimilarity, MTS, get_visualization, check_errors)

# For Multivariate Time Series
dtw_functions.dtw_tensor_3d(X_1, X_2, object)

The examples shown below are executed in jupyter-notebook. Code available in exampleData/CodeExamples/E1_SyntheticData (https://github.com/oscarescuderoarnanz/dtwParallel/tree/main/exampleData/CodeExamples_new/E1_SyntheticData). These examples can be executed in any Integrated Development Environment.

Example 1. For univariate time series.

from dtwParallel import dtw_functions
from scipy.spatial import distance as d

# For Univariate Time Series
x = [1,2,3]
y = [0,0,1]

dtw_functions.dtw(x,y,local_dissimilarity=d.euclidean)

[out]: 5.0

import pandas as pd
import numpy as np
from dtwParallel import dtw_functions 

# Use of dataframes with 1D (UTS) as entry data
x = pd.DataFrame(np.random.randint(0,10, size=(1,8)))
y = pd.DataFrame(np.random.randint(0,10, size=(1,8)))
dtw_functions.dtw(x, y, n_threads=8)

[out]: Result with random dependency.

import pandas as pd
import numpy as np
from dtwParallel import dtw_functions 

# Use of dataframes with 2D (MTS) as entry data
x = pd.DataFrame([np.random.randn(10), np.random.randn(10)])
y = pd.DataFrame([np.random.randn(10), np.random.randn(10)])
dtw_functions.dtw(x, y, MTS=True, n_threads=8)

[out]: Result with random dependency.

import pandas as pd 
import numpy as np
from dtwParallel import dtw_functions 

# Use of dataframes with 1D (UTS) as entry data
x = pd.DataFrame(np.random.randn(10)).T
y = pd.DataFrame(np.random.randn(10)).T

dtw_functions.dtw(x,y,n_threads=8)

[out]: Result with random dependency.

import pandas as pd
import numpy as np
from dtwParallel import dtw_functions 

# Use of pd.Series as entry data
x = pd.Series(np.random.randn(10))
y = pd.Series(np.random.randn(10))

dtw_functions.dtw(x,y,local_dissimilarity="norm2")

[out]: Result with random dependency.

Example 2. For univariate time series with different lengths.

from dtwParallel import dtw_functions
from scipy.spatial import distance as d

# For Univariate Time Series
x = [1,2,3,5,8,9,5,4,2]
y = [1,0,1,0,1,1]

dtw_functions.dtw(x, y, local_dissimilarity=d.euclidean)

[out]: 32.0

Example 3. For univariate time series with visualization (cost matrix, path and alignment between a pair of time series).

from dtwParallel import dtw_functions
from scipy.spatial import distance as d

# For Univariate Time Series
x = [4,2,8,4,5]
y = [0,1,0,8,9]

dtw_functions.dtw(x, y, local_dissimilarity=d.euclidean, get_visualization=True)

[out]: 15.0

Example 4. For multivariate time series.

from dtwParallel import dtw_functions
from scipy.spatial import distance as d
import numpy as np

X = np.array([[3,5,8], 
             [5, 1,9]])

Y = np.array([[2, 0,8],
             [4, 3,8]])
            
dtw_functions.dtw(X, Y, type_dtw="d", local_dissimilarity=d.euclidean, MTS=True)

[out]: 7.548509256375962

Example 5. For multivariate time series with different lengths.

from dtwParallel import dtw_functions
from scipy.spatial import distance as d
import numpy as np

X = np.array([[3, 5, 8], 
              [5, 1, 9],
              [0, 1, 1], 
              [1, 4, 2]])

Y = np.array([[2, 0,8],
              [4, 3,8]])

dtw_functions.dtw(X, Y, type_dtw="d", local_dissimilarity=d.euclidean, MTS=True)

[out]: 22.546443515422986

Example 5. For multivariate time series with visualization.

from dtwParallel import dtw_functions
from scipy.spatial import distance as d
import numpy as np

X = np.array([[3, 5, 8], 
              [0, 1, 3],
              [1, 2, 3]])

Y = np.array([[2, 0, 8],
              [1, 3, 8],
              [4, 8, 12]])

dtw_functions.dtw(X, Y, type_dtw="d", local_dissimilarity=d.euclidean, MTS=True)

[out]: 21.801217248966267

Example 7. For a tensor formed by N x T x F, where N is the number of observations, T the time instants and F the characteristics.

 import numpy as np
 from dtwParallel import dtw_functions as dtw

 x = np.load('../../Data/E0/X_train.npy')
 y = np.load('../../Data/E0/X_test.npy')

 class Input:
     def __init__(self):
         self.check_errors = False 
         self.type_dtw = "d"
         self.MTS = True
         self.regular_flag = False
         self.n_threads = -1
         self.local_dissimilarity = "gower"
         self.visualization = False
         self.output_file = True
         self.DTW_to_kernel = False
         self.sigma_kernel = 1
         self.itakura_max_slope = None
         self.sakoe_chiba_radius = None

 input_obj = Input()
 # API call. 
 dtw_functions.dtw_tensor_3d(x, y, input_obj)

[out]: 
array([[1.7200027 , 2.16000016, 1.92000033, 2.53999992],
    [1.59999973, 1.79999978, 1.83999987, 2.27999987],
    [0.5399895 , 1.52000002, 1.04000024, 1.66      ],
    [0.70000006, 1.57999993, 1.10000018, 1.69999999]])

 import numpy as np
 from dtwParallel import dtw_functions as dtw

 x = np.load('../../Data/E0/X_train.npy')
 y = np.load('../../Data/E0/X_test.npy')

 class Input:
     def __init__(self):
         self.check_errors = False 
         self.type_dtw = "i"
         self.MTS = True
         self.regular_flag = False
         self.n_threads = -1
         self.local_dissimilarity = "gower"
         self.visualization = False
         self.output_file = True
         self.DTW_to_kernel = False
         self.sigma_kernel = 1
         self.itakura_max_slope = None
         self.sakoe_chiba_radius = None

 input_obj = Input()
 # API call. 
 dtw_functions.dtw_tensor_3d(x, y, input_obj)

[out]: 
array([[ 86.0001335 , 108.00000931,  96.00001545, 126.99999504],
    [ 79.99998522,  89.99999088,  91.99999337, 113.9999931 ],
    [ 26.99947403,  76.0000011 ,  52.00001171,  82.99999923],
    [ 35.00000282,  78.99999571,  55.00000872,  84.99999817]]))

 import numpy as np
 from dtwParallel import dtw_functions as dtw

 x = np.load('../../Data/E0/X_train.npy')
 y = np.load('../../Data/E0/X_test.npy')

 class Input:
     def __init__(self):
         self.check_errors = False 
         self.type_dtw = "i"
         self.MTS = True
         self.regular_flag = False
         self.n_threads = -1
         self.local_dissimilarity = "gower"
         self.visualization = False
         self.output_file = True
         self.DTW_to_kernel = False
         self.sigma_kernel = 1
         self.itakura_max_slope = None
         self.sakoe_chiba_radius = None

 input_obj = Input()
 # API call. 
 dtw_functions.dtw_tensor_3d(x, y, input_obj)

[out]: 
array([[2.47396197e+16, 9.07388652e+17, 2.23522660e+17, 1.68210525e+18],
    [6.12016408e+17, 1.54414468e+18, 8.60278687e+17, 2.31886127e+18],
    [4.75817098e+15, 9.36886443e+17, 2.53020450e+17, 1.71160304e+18],
    [1.02119724e+18, 8.90689643e+16, 7.72934957e+17, 6.85647630e+17]])

 import numpy as np
 from dtwParallel import dtw_functions as dtw

 x = np.load('../../Data/E0/X_train.npy')
 y = np.load('../../Data/E0/X_test.npy')

 class Input:
     def __init__(self):
         self.check_errors = False 
         self.type_dtw = "itakura"
         self.MTS = True
         self.regular_flag = False
         self.n_threads = -1
         self.local_dissimilarity = None
         self.visualization = False
         self.output_file = True
         self.DTW_to_kernel = False
         self.sigma_kernel = 1
         self.itakura_max_slope = None
         self.sakoe_chiba_radius = None

 input_obj = Input()
 # API call. 
 dtw_functions.dtw_tensor_3d(x, y, input_obj)

[out]: 
array([[9.35069732e+15, 3.42960674e+17, 8.44836242e+16, 6.35776023e+17],
    [2.31320459e+17, 5.83631830e+17, 3.25154781e+17, 8.76447180e+17],
    [1.79841959e+15, 3.54109791e+17, 9.56327411e+16, 6.46925140e+17],
    [3.85976275e+17, 3.36649042e+16, 2.92141954e+17, 2.59150445e+17]])

Example 8. For a tensor formed by N x T x F, where N is the number of observations, T the time instants and F the characteristics. Novelty: irregular multivariate time series. You indicate the value of the flag. This value will be searched and removed from each of the MTS entered (this process is carried out transparently to the user). For the case of DTW (d), we normalize the value of the distance dtw obtained by the square root of the product of the length of the time series 1 by the time series 2. For the case of DTW (i), we replicate the values of the last time instant until completing the time series.

  import numpy as np
  from dtwParallel import dtw_functions as dtw

  x = np.load('../../Data/E0/X_train.npy')
  y = np.load('../../Data/E0/X_test.npy')

  class Input:
      def __init__(self):
          self.check_errors = False 
          self.type_dtw = "d"
          self.MTS = True
          self.regular_flag = False
          self.n_threads = -1
          self.local_dissimilarity = None
          self.visualization = False
          self.output_file = True
          self.DTW_to_kernel = False
          self.sigma_kernel = 1
          self.itakura_max_slope = None
          self.sakoe_chiba_radius = None

  input_obj = Input()
  # API call. 
  dtw_functions.dtw_tensor_3d(x, y, input_obj)

Configuration

For any modification of the default parameters, the configuration.ini file can be edited.

The default values are:

[DEFAULT]
check_errors = False
type_dtw = d
mts = False
regular_flag = 0
local_dissimilarity = euclidean
n_threads = -1
visualization = False
output_file = False
name_file = output
dtw_to_kernel = False
sigma_kernel = 1
itakura_max_slope = None
sakoe_chiba_radius = None

Examples with public data

I have used data from yahoo finance (https://finance.yahoo.com/) of 505 companies, available in a .zip file. The folder where the data is located is exampleData/Data/E2_FinanceData (https://github.com/oscarescuderoarnanz/dtwParallel/tree/main/exampleData/Data/E2_FinanceData). The code needed to process the information of each of the 505 companies, obtaining the tensor input to the package is located in exampleData/CodeExamples/E2_FinanceData/tensorGenerator (https://github.com/oscarescuderoarnanz/dtwParallel/tree/main/exampleData/CodeExamples_new/E2_FinanceData).

Experiment 1. Computational time as a function of the number of threads.

The computation of the distance matrix has been carried out using dependent and independent DTW varying the number of threads. Code of this example is available at exampleData/Code/E2_FinanceData (https://github.com/oscarescuderoarnanz/dtwParallel/tree/main/exampleData/CodeExamples_new/E2_FinanceData).

DTW dependent

DTW independent

Experiment 2. Comparison of computational time with other packages to calculate dependent DTW.

Code available for this example at exampleData/Code/E2_FinanceData (https://github.com/oscarescuderoarnanz/dtwParallel/tree/main/exampleData/CodeExamples_new/E2_FinanceData).

Reference

If you use dtwParallel in your research papers, please refer to ...

[To be done]

In case of using the variants itakura parrallelogram or sakoe_chiba band refer to the following work:

@article{JMLR:v21:20-091,
  author  = {Romain Tavenard and Johann Faouzi and Gilles Vandewiele and 
             Felix Divo and Guillaume Androz and Chester Holtz and 
             Marie Payne and Roman Yurchak and Marc Ru{\ss}wurm and 
             Kushal Kolar and Eli Woods},
  title   = {Tslearn, A Machine Learning Toolkit for Time Series Data},
  journal = {Journal of Machine Learning Research},
  year    = {2020},
  volume  = {21},
  number  = {118},
  pages   = {1-6},
  url     = {http://jmlr.org/papers/v21/20-091.html}
}

License

Licensed under the BSD 2-Clause License.