Dynamic Time Warping

This package allows to measure the similarity between two-time sequences, i.e., it finds the optimal alignment between two time-dependent sequences. It will enable the calculation of univariate and multivariate time series. Any distance available in scipy.spatial.distance and gower distance can be used. Extra functionality has been incorporated to transform the resulting DTW matrix into an exponential kernel.

Common functionalities for 2-time series (TS):

• It incorporates the possibility of visualizing the cost matrix and the path to reach the DTW distance value between two TS. This will allow its use in a didactic way, providing a better understanding of the method used.
• It is possible to calculate TS with the same and different lengths.

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 not only the calculation of distances but also similarities (based on an exponential kernel).

Multivariate TS functionalities:

• Calculation of dependent DTW and independent DTW is available.

Novelties:

• Two variants of DTW have been included: the Itakura parallelogram and the Sakoe-Chiba band.
• Calculation of DTW for irregular time series, both for the case of dependent and independent DTW.
• Visualization of the alignment of two time series.
• Computational time optimization for norm 1 and norm 2.
• Availability of input data format: numpy, pandas and tensors.

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

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

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 2.5 4.3 6.6 8.0 1 0 0 1 5.5 15.2 -y 1 0 0 1 -t "sakoe_chiba"

[out]: 22.38

dtwParallel -x 2.5 4.3 6.6 8.0 1 0 0 1 5.5 15.2 -y 1 0 0 1 -t "itakura"

[out]: 21.53

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

dtwParallel -x 2.5 4.3 6.6 8.0 1 0 0 1 5.5 15.2 -y 1 0 0 1 -d "norm1"

[out]: 19.11

dtwParallel -x 2.5 4.3 6.6 8.0 1 0 0 1 5.5 15.2 -y 1 0 0 1 -d "norm2"

[out]: 45.4

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]: 26.99

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.

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

[out]: [[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, distance, 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/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,dist=d.euclidean)

[out]: 5.0

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]: dependence with random

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))
y = pd.DataFrame(np.random.randn(10))

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

[out]: dependence with random

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

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

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

[out]: dependence with random

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]

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

[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]

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

[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, "d", 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, "d", 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, "d", 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.distance = "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.distance = "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.distance = "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.distance = 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.distance = 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
distance = 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/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/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/E2_FinanceData).

Reference

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

[To be done]

License

Licensed under the BSD 2-Clause License.