Multiscale Entropies

Functions for estimating the multiscale entropy of a univariate time series.

Multiscale entropy can be calculated using any of the Base Entropies: ApEn, AttnEn, BubbEn, CondEn, CoSiEn, DistEn, DispEn, DivEn, EnofEn, FuzzEn, GridEn, IncrEn, K2En, PermEn, PhasEn, RangEn, SampEn, SlopEn, SpecEn, SyDyEn.

Important

Multiscale cross-entropy functions have two positional arguments:

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

  2. 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.

MSEn(Sig, Mbjx, Scales=3, Methodx='coarse', RadNew=0, Plotx=False)

MSEn Returns the multiscale entropy of a univariate data sequence.

MSx,CI = MSEn(Sig, Mobj)

Returns a vector of multiscale entropy values (MSx) and the complexity index (CI) of the data sequence Sig using the parameters specified by the multiscale object (Mobj) over 3 temporal scales with coarse- graining (default).

MSx,CI = MSEn(Sig, Mobj, keyword = value, ...)

Returns a vector of multiscale entropy values (MSx) and the complexity index (CI) of the data sequence Sig 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 entropy specified by Mobj is SampEn or ApEn, 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] Standard Deviation - r*std(Ykj)

  • [2] 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]

For further info on these graining procedures see the EntropyHub guide.

See also:

MSobject, rMSEn, cMSEn, hMSEn, XMSEn, cXMSEn, hXMSEn, SampEn

References:
[1] Madalena Costa, Ary Goldberger, and C-K. Peng,

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

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

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

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

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

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

“Multiscale entropy analysis of biological signals.” Physical review E 71.2 (2005): 021906.

[5] Ranjit A. Thuraisingham and Georg A. Gottwald,

“On multiscale entropy analysis for physiological data.” Physica A: Statistical Mechanics and its Applications 366 (2006): 323-332.

[6] Meng Hu and Hualou Liang,

“Intrinsic mode entropy based on multivariate empirical mode decomposition and its application to neural data analysis.” Cognitive neurodynamics 5.3 (2011): 277-284.

[7] Anne Humeau-Heurtier

“The multiscale entropy algorithm and its variants: A review.” Entropy 17.5 (2015): 3110-3123.

[8] Jianbo Gao, et al.,

“Multiscale entropy analysis of biological signals: a fundamental bi-scaling law.” Frontiers in computational neuroscience 9 (2015): 64.

[9] Paolo Castiglioni, et al.,

“Multiscale Sample Entropy of cardiovascular signals: Does the choice between fixed-or varying-tolerance among scales influence its evaluation and interpretation?.” Entropy 19.11 (2017): 590.

[10] Tuan D Pham,

“Time-shift multiscale entropy analysis of physiological signals.” Entropy 19.6 (2017): 257.

[11] Hamed Azami and Javier Escudero,

“Coarse-graining approaches in univariate multiscale sample and dispersion entropy.” Entropy 20.2 (2018): 138.

[12] Madalena Costa and Ary L. Goldberger,

“Generalized multiscale entropy analysis: Application to quantifying the complex volatility of human heartbeat time series.” Entropy 17.3 (2015): 1197-1203.

cMSEn(Sig, Mbjx, Scales=3, RadNew=0, Refined=False, Plotx=False)

cMSEn Returns the composite (or refined-composite) multiscale entropy of a univariate data sequence.

MSx, CI = cMSEn(Sig, Mobj)

Returns a vector of composite multiscale entropy values (MSx) for the data sequence (Sig) using the parameters specified by the multiscale object (Mobj) using the composite multiscale entropy method (cMSE) over 3 temporal scales.

MSx, CI = cMSEn(Sig, Mobj, Refined = True)

Returns a vector of refined-composite multiscale entropy values (MSx) for the data sequence (Sig) 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 SampEn or FuzzEn. If the entropy method is SampEn, cMSEn employs the method described in [5]. If the entropy method is FuzzEn, cMSEn employs the method described in [6].

MSx, CI = cMSEn(Sig, Mobj, keyword = value, ...)

Returns a vector of composite multiscale entropy values (MSx) of the data sequence (Sig) 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 entropy specified by Mobj is SampEn or ApEn, 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] Standard Deviation - r*std(Ykj)

  • [2] Variance - r*var(Ykj)

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

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

Refined:

  • Refined-composite MSEn method. When Refined == True and the entropy function specified by Mobj is SampEn or FuzzEn,

cMSEn returns the refined-composite multiscale entropy (rcMSEn) [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, MSEn, rMSEn, hMSEn, XMSEn, cXMSEn, SampEn, ApEn

References:
[1] Madalena Costa, Ary Goldberger, and C-K. Peng,

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

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

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

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

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

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

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

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

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

[6] Hamed Azami et al.,

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

hMSEn(Sig, Mbjx, Scales=3, RadNew=0, Plotx=False)

hMSEn returns the hierarchical entropy of a univariate data sequence.

MSx, Sn, CI = hMSEn(Sig, Mobj)

Returns a vector of entropy values (MSx) calculated at each node in the hierarchical tree, the average entropy value across all nodes at each scale (Sn), and the complexity index (CI) of the hierarchical tree (i.e. sum(Sn)) for the data sequence (Sig) 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 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 = hMSEn(Sig, Mobj, keyword = value, ...)

Returns a vector of entropy values (MSx) calculated at each node in the hierarchical tree, the average entropy value across all nodes at each scale (Sn), and the complexity index (CI) of the entire hierarchical tree for the data sequence (Sig) using the following ‘keyword’ arguments:

Scales:

  • Number of temporal scales, an integer > 1 (default = 3) At each scale (T), entropy is estimated for 2^(T-1) nodes.

RadNew:

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

When the entropy specified by Mobj is SampEn or ApEn, 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] Standard Deviation - r*std(Ykj)

  • [2] 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 entropy value at each time scale (i.e. the multiscale entropy curve) and a network graph showing the entropy value of each node in the hierarchical tree decomposition. (default: False)

See also:

MSobject, MSEn, rMSEn, cMSEn, XMSEn, hXMSEn, rXMSEn, cXMSEn

References:
[1] 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.

rMSEn(Sig, Mbjx, Scales=3, F_Order=6, F_Num=0.5, RadNew=0, Plotx=False)

rMSEn returns the refined multiscale entropy of a univariate data sequence.

MSx, CI = rMSEn(Sig, Mobj)

Returns a vector of refined multiscale entropy values (MSx) and the complexity index (CI) of the data sequence (Sig) 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 SampEn or ApEn, rMSEn updates the threshold radius of the data sequence (Xt) at each scale to 0.2*std(Xt) if no r value is provided by Mobj, or r*std(Xt) if r is specified.

MSx, CI = rMSEn(Sig, Mobj, keyword = value, ...)

Returns a vector of refined multiscale entropy values (MSx) and the complexity index (CI) of the data sequence (Sig) using the parameters specified by the multiscale object (Mobj) and the following ‘keyword’ arguments:

Scales:

  • Number of temporal scales, a positive integer (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 entropy specified by Mobj is SampEn or ApEn, 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] Standard Deviation - r*std(Ykj)

  • [2] 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, MSEn, cMSEn, hMSEn, XMSEn, rXMSEn, SampEn, ApEn

References:
[1] Madalena Costa, Ary Goldberger, and C-K. Peng,

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

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

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

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

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

[4] 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.

[5] 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.