Multiscale Cross-Entropies

Functions for estimating the multiscale cross-entropy between two univariate time series.

Just as one can calculate multiscale entropy using any Base entropy, the same functionality is possible with multiscale cross-entropy using any of the Cross Entropies: XApEn, XSampEn, XK2En, XCondEn, XPermEn, XSpecEn, XDistEn, XFuzzEn

To do so, we again use the MSobject function to pass a multiscale object (Mobj) to the multiscale cross-entropy functions.

Important

Multiscale cross-entropy functions have three positional arguments:

  1. the first data sequence, Sig1 (a vector > 10 elements),

  2. the second data sequence, Sig2 (a vector > 10 elements),

  3. the multiscale entropy object, Mobj.

MSobject(EnType='SampEn', **kwargs)

MSobject creates an object to store multiscale entropy parameters.

[Mobj] = MSobject()

Returns a multiscale entropy object (Mobj) based on that originally proposed by Costa et al. using the following default parameters: EnType = ‘SampEn’, embedding dimension = 2, time delay = 1, radius = 0.2*SD(Sig), logarithm = natural

[Mobj] = MSobject(EnType)

Returns a multiscale entropy object using the specified entropy method (EnType) and the default parameters for that entropy method. To see the default parameters for a particular entropy method, type: help(EnType) (e.g. help(SampEn))

[Mobj] = MSobject(EnType, keyword = value, ...)

Returns a multiscale entropy object using the specified entropy method (EnType) and the name/value parameters for that particular method. To see the default parameters for a particular entropy method, type: help(EnType) (e.g. help(SampEn))

EnType can be any of the following (case sensitive) string names:

Base Entropies:
'ApEn':

  • Approximate Entropy

'SampEn':

  • Sample Entropy

'FuzzEn':

  • Fuzzy Entropy

'K2En':

  • Kolmogorov Entropy

'PermEn':

  • Permutation Entropy

'CondEn':

  • Conditional Entropy

'DistEn':

  • Distribution Entropy

'DispEn':

  • Dispersion Entropy

'SpecEn':

  • Spectral Entropy

'SyDyEn':

  • Symbolic Dynamic Entropy

'IncrEn':

  • Increment Entropy

'CoSiEn':

  • Cosine Similarity Entropy

'PhasEn':

  • Phase Entropy

'SlopEn':

  • Slope Entropy

'BubbEn':

  • Bubble Entropy

'GridEn':

  • Grid Distribution Entropy

'EnofEn':

  • Entropy of Entropy

'AttnEn':

  • Attention Entropy

'DivEn':

  • Diversity Entropy

'RangEn':

  • Range Entropy

Cross Entropies:
'XApEn':

  • Cross-Approximate Entropy

'XSampEn':

  • Cross-Sample Entropy

'XFuzzEn':

  • Cross-Fuzzy Entropy

'XK2En':

  • Cross-Kolmogorov Entropy

'XPermEn':

  • Cross-Permutation Entropy

'XCondEn':

  • Cross-Conditional Entropy (corrected)

'XDistEn':

  • Cross-Distribution Entropy

'XSpecEn':

  • Cross-Spectral Entropy

Multivariate Entropies:
'MvSampEn':

  • Multivariate Sample Entropy

'MvFuzzEn':

  • Multivariate Fuzzy Entropy

'MvDispEn':

  • Multivariate Dispersion Entropy

'MvCoSiEn':

  • Multivariate Cosine Similarity Entropy

'MvPermEn':

  • Multivariate Permutation Entropy

See also:

MSEn, MvMSEn, cMSEn, cMvMSEn, rMSEn, hMSEn, XMSEn, rXMSEn, cXMSEn, hXMSEn

The following functions use the multiscale entropy object shown above.

XMSEn(Sig1, Sig2, Mbjx, Scales=3, Methodx='coarse', RadNew=0, Plotx=False)

XMSEn returns the multiscale cross-entropy between two univariate data sequences.

MSx, CI = XMSEn(Sig1, Sig2, Mobj)

Returns a vector of multiscale cross-entropy values (MSx) and the complexity index (CI) between the data sequences contained in Sig1 and Sig2 using the parameters specified by the multiscale object (Mobj) over 3 temporal scales with coarse-graining (default).

MSx, CI = XMSEn(Sig1, Sig2, Mobj, keyword = value, ...)

Returns a vector of multiscale cross-entropy values (MSx) and the complexity index (CI) between the data sequences contained in Sig1 and Sig2 using the parameters specified by the multiscale object (Mobj) and the following ‘keyword’ arguments:

Scales:

  • Number of temporal scales, an integer > 1 [default: 3]

Methodx:

  • Graining method, one of the following: [default: 'coarse'] {'coarse', 'modified', 'imf' , 'timeshift', 'generalized'}

RadNew:

  • Radius rescaling method, an integer in the range [1 4].

When the cross-entropy specified by Mobj is XSampEn or XApEn, RadNew rescales the radius threshold in each sub-sequence at each time scale (Ykj). If a radius value (r) is specified by Mobj, this becomes the rescaling coefficient, otherwise it is set to 0.2 (default). The value of RadNew specifies one of the following methods:

  • [1] Pooled Standard Deviation - r*std(Ykj)

  • [2] Pooled Variance - r*var(Ykj)

  • [3] Mean Absolute Deviation - r*mad(Ykj)

  • [4] Median Absolute Deviation - r*mad(Ykj,1)

Plotx:

  • When Plotx == True, returns a plot of the entropy value at each time scale (i.e. the multiscale entropy curve) [default: False]

See also:

MSobject, XSampEn, XApEn, rXMSEn, cXMSEn, hXMSEn, MSEn

References:
[1] Rui Yan, Zhuo Yang, and Tao Zhang,

“Multiscale cross entropy: a novel algorithm for analyzing two time series.” 5th International Conference on Natural Computation. Vol. 1, pp: 411-413 IEEE, 2009.

[2] Madalena Costa, Ary Goldberger, and C-K. Peng,

“Multiscale entropy analysis of complex physiologic time series.” Physical review letters 89.6 (2002): 068102.

[3] Vadim V. Nikulin, and Tom Brismar,

“Comment on “Multiscale entropy analysis of complex physiologic time series”.” Physical review letters 92.8 (2004): 089803.

[4] Madalena Costa, Ary L. Goldberger, and C-K. Peng.

“Costa, Goldberger, and Peng reply.” Physical Review Letters 92.8 (2004): 089804.

[5] Antoine Jamin, et al,

“A novel multiscale cross-entropy method applied to navigation data acquired with a bike simulator.” 41st annual international conference of the IEEE EMBC IEEE, 2019.

[6] Antoine Jamin and Anne Humeau-Heurtier.

“(Multiscale) Cross-Entropy Methods: A Review.” Entropy 22.1 (2020): 45.

cXMSEn(Sig1, Sig2, Mbjx, Scales=3, RadNew=0, Refined=False, Plotx=False)

cXMSEn returns the composite (or refined-composite) multiscale cross-entropy between two univariate data sequences.

MSx, CI = cXMSEn(Sig1, Sig2, Mobj)

Returns a vector of composite multiscale cross-entropy values (MSx) between two univariate data sequences contained in Sig1 and Sig2 using the parameters specified by the multiscale object (Mobj) using the composite multiscale method (cMSE) over 3 temporal scales.

MSx, CI = cXMSEn(Sig1, Sig2, Mobj, Refined = True)

Returns a vector of refined-composite multiscale cross-entropy values (MSx) for the data sequences (Sig1, Sig2) using the parameters specified by the multiscale object (Mobj) using the refined-composite multiscale entropy method (rcMSE) over 3 temporal scales. When Refined == True, the base entropy method must be XSampEn or XFuzzEn. If the entropy method is XSampEn, cXMSEn employs the method described in [7]. If the entropy method is XFuzzEn, cXMSEn employs the method described in [8].

MSx, CI = cXMSEn(Sig1, Sig2, Mobj, keyword = value, ...)

Returns a vector of composite multiscale cross-entropy values (MSx) between the data sequences contained in Sig1 and Sig2 using the parameters specified by the multiscale object (Mobj) and the following ‘keyword’ arguments:

Scales:

  • Number of temporal scales, an integer > 1 (default: 3)

RadNew:

  • Radius rescaling method, an integer in the range [1 4].

When the cross-entropy specified by Mobj is XSampEn or XApEn, RadNew rescales the radius threshold in each sub-sequence at each time scale (Ykj). If a radius value (r) is specified by Mobj, this becomes the rescaling coefficient, otherwise it is set to 0.2 (default). The value of RadNew specifies one of the following methods:

  • [1] Pooled Standard Deviation - r*std(Ykj)

  • [2] Pooled Variance - r*var(Ykj)

  • [3] Mean Absolute Deviation - r*mad(Ykj)

  • [4] Median Absolute Deviation - r*mad(Ykj,1)

Refined:

  • Refined-composite XMSEn method. When Refined == True and the cross-entropy function specified by

Mobj is XSampEn or XFuzzEn, cXMSEn returns the refined-composite multiscale entropy (rcXMSEn) [default: False]

Plotx:

  • When Plotx == True, returns a plot of the entropy value at each time scale (i.e. the multiscale entropy curve) [default: False]

See also:

MSobject, XMSEn, rXMSEn, hXMSEn, XSampEn, XApEn, MSEn, cMSEn, rMSEn

References:
[1] Rui Yan, Zhuo Yang, and Tao Zhang,

“Multiscale cross entropy: a novel algorithm for analyzing two time series.” 5th International Conference on Natural Computation. Vol. 1, pp: 411-413 IEEE, 2009.

[2] Yi Yin, Pengjian Shang, and Guochen Feng,

“Modified multiscale cross-sample entropy for complex time series.” Applied Mathematics and Computation 289 (2016): 98-110.

[3] Madalena Costa, Ary Goldberger, and C-K. Peng,

“Multiscale entropy analysis of complex physiologic time series.” Physical review letters 89.6 (2002): 068102.

[4] Antoine Jamin, et al,

“A novel multiscale cross-entropy method applied to navigation data acquired with a bike simulator.” 41st annual international conference of the IEEE EMBC IEEE, 2019.

[5] Antoine Jamin and Anne Humeau-Heurtier.

“(Multiscale) Cross-Entropy Methods: A Review.” Entropy 22.1 (2020): 45.

[6] Shuen-De Wu, et al.,

“Time series analysis using composite multiscale entropy.” Entropy 15.3 (2013): 1069-1084.

[7] Shuen-De Wu, et al.,

“Analysis of complex time series using refined composite multiscale entropy.” Physics Letters A 378.20 (2014): 1369-1374.

[8] Hamed Azami et al.,

“Refined multiscale fuzzy entropy based on standard deviation for biomedical signal analysis” Med Biol Eng Comput 55 (2017):2037–2052

hXMSEn(Sig1, Sig2, Mbjx, Scales=3, RadNew=0, Plotx=False)

hXMSEn returns the hierarchical cross-entropy between two univariate data sequences.

MSx, Sn, CI = hXMSEn(Sig1, Sig2, Mobj)

Returns a vector of cross-entropy values (MSx) calculated at each node in the hierarchical tree, the average cross-entropy value across all nodes at each scale (Sn), and the complexity index (CI) of the hierarchical tree (i.e. sum(Sn)) between the data sequences contained in Sig1 and``Sig2`` using the parameters specified by the multiscale object (Mobj) over 3 temporal scales (default). The entropy values in MSx are ordered from the root node (S_00) to the Nth subnode at scale T (S_TN): i.e. S_00, S_10, S_11, S_20, S_21, S_22, S_23, S_30, S_31, S_32, S_33, S_34, S_35, S_36, S_37, S_40, … , S_TN. The average cross-entropy values in Sn are ordered in the same way, with the value of the root node given first: i.e. S0, S1, S2, …, ST

MSx, Sn, CI = hXMSEn(Sig1, Sig2, Mobj, Keyword = value, ...)

Returns a vector of cross-entropy values (MSx) calculated at each node in the hierarchical tree, the average cross-entropy value across all nodes at each scale (Sn), and the complexity index (CI) of the entire hierarchical tree between the data sequences contained in Sig1 and Sig2 using the following keyword arguments:

Scales:

  • Number of temporal scales, an integer > 1 (default: 3)

RadNew:

  • Radius rescaling method, an integer in the range [1 4].

When the cross-entropy specified by Mobj is XSampEn or XApEn, RadNew rescales the radius threshold in each sub-sequence at each time scale (Ykj). If a radius value (r) is specified by Mobj, this becomes the rescaling coefficient, otherwise it is set to 0.2 (default). The value of RadNew specifies one of the following methods:

  • [1] Pooled Standard Deviation - r*std(Ykj)

  • [2] Pooled Variance - r*var(Ykj)

  • [3] Mean Absolute Deviation - r*mad(Ykj)

  • [4] Median Absolute Deviation - r*mad(Ykj,1)

Plotx:

  • When Plotx == True, returns a plot of the average cross-entropy value at each time scale (i.e. the multiscale cross-entropy curve) and a hierarchical graph showing the cross-entropy value of each node in the hierarchical tree decomposition. (default: False)

See also:

MSobject, XMSEn, rXMSEn, cXMSEn, XSampEn, XApEn, MSEn, hMSEn, rMSEn, cMSEn

References:
[1] Matthew W. Flood,

“hXMSEn - EntropyHub Project” 2021, https://github.com/MattWillFlood/EntropyHub

[2] Rui Yan, Zhuo Yang, and Tao Zhang,

“Multiscale cross entropy: a novel algorithm for analyzing two time series.” 5th International Conference on Natural Computation. Vol. 1, pp: 411-413 IEEE, 2009.

[3] Ying Jiang, C-K. Peng and Yuesheng Xu,

“Hierarchical entropy analysis for biological signals.” Journal of Computational and Applied Mathematics 236.5 (2011): 728-742.

rXMSEn(Sig1, Sig2, Mbjx, Scales=3, F_Order=6, F_Num=0.5, RadNew=0, Plotx=False)

rXMSEn returns the refined multiscale cross-entropy between two univariate data sequences.

MSx, CI = rXMSEn(Sig1, Sig2, Mobj)

Returns a vector of refined multiscale cross-entropy values (MSx) and the complexity index (CI) between the data sequences contained in Sig1 and Sig2 using the parameters specified by the multiscale object (Mobj) and the following default parameters: Scales = 3, Butterworth LPF Order = 6, Butterworth LPF cutoff frequency at scale (T): Fc = 0.5/T. If the entropy function specified by Mobj is XSampEn or XApEn, rXMSEn updates the threshold radius of the data sequences at each scale to 0.2*SDpooled(Sig1,Sig2) if no r value is provided by Mobj, or r*SDpooled(Sig1,Sig2) if r is specified.

MSx, CI = rXMSEn(Sig1, Sig2, Mobj, keyword = value, ...)

Returns a vector of refined multiscale cross-entropy values (MSx) and the complexity index (CI) between the data sequences contained in Sig1 and Sig2 using the parameters specified by the multiscale object (Mobj) and the following ‘keyword’ arguments:

Scales:

  • Number of temporal scales, an integer > 1 (default: 3)

F_Order:

  • Butterworth low-pass filter order, a positive integer (default: 6)

F_Num:

  • Numerator of Butterworth low-pass filter cutoff frequency, a scalar value in range [0 < F_Num < 1]. The cutoff frequency at each scale (T) becomes: Fc = F_Num/T. (default: 0.5)

RadNew:

  • Radius rescaling method, an integer in the range [1 4].

When the cross-entropy specified by Mobj is XSampEn or XApEn, RadNew rescales the radius threshold in each sub-sequence at each time scale (Ykj). If a radius value (r) is specified by Mobj, this becomes the rescaling coefficient, otherwise it is set to 0.2 (default). The value of RadNew specifies one of the following methods:

  • [1] Pooled Standard Deviation - r*std(Ykj)

  • [2] Pooled Variance - r*var(Ykj)

  • [3] Mean Absolute Deviation - r*mad(Ykj)

  • [4] Median Absolute Deviation - r*mad(Ykj,1)

Plotx:

  • When Plotx == True, returns a plot of the entropy value at each time scale (i.e. the multiscale entropy curve) [default: False]

See also:

MSobject, XMSEn, cXMSEn, hXMSEn, XSampEn, XApEn, MSEn, rMSEn

References:
[1] Matthew W. Flood,

“rXMSEn - EntropyHub Project” 2024, https://github.com/MattWillFlood/EntropyHub

[2] Rui Yan, Zhuo Yang, and Tao Zhang,

“Multiscale cross entropy: a novel algorithm for analyzing two time series.” 5th International Conference on Natural Computation. Vol. 1, pp: 411-413 IEEE, 2009.

[3] José Fernando Valencia, et al.,

“Refined multiscale entropy: Application to 24-h holter recordings of heart period variability in healthy and aortic stenosis subjects.” IEEE Transactions on Biomedical Engineering 56.9 (2009): 2202-2213.

[4] Puneeta Marwaha and Ramesh Kumar Sunkaria,

“Optimal selection of threshold value ‘r’ for refined multiscale entropy.” Cardiovascular engineering and technology 6.4 (2015): 557-576.

[5] Yi Yin, Pengjian Shang, and Guochen Feng,

“Modified multiscale cross-sample entropy for complex time series.” Applied Mathematics and Computation 289 (2016): 98-110.

[6] Antoine Jamin, et al,

“A novel multiscale cross-entropy method applied to navigation data acquired with a bike simulator.” 41st annual international conference of the IEEE EMBC IEEE, 2019.

[7] Antoine Jamin and Anne Humeau-Heurtier.

“(Multiscale) Cross-Entropy Methods: A Review.” Entropy 22.1 (2020): 45.