# Base Entropies

## Functions for estimating the entropy of a single univariate time series.

The following functions also form the base entropy method used by multiscale entropy functions.

ApEn(Sig, varargin)

ApEn estimates the approximate entropy of a univariate data sequence.

[Ap, Phi] = ApEn(Sig)

Returns the approximate entropy estimates (Ap) and the log-average number of matched vectors (Phi) for m = [0, 1, 2], estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, radius distance threshold = 0.2*SD(Sig), logarithm = natural

[Ap, Phi] = ApEn(Sig, name, value, …)

Returns the approximate entropy estimates (Ap) of the data sequence (Sig) for dimensions = [0, 1, …, m] using the specified name/value pair arguments:

• m - Embedding Dimension, a positive integer

• tau - Time Delay, a positive integer

• r - Radius Distance Threshold, a positive scalar

• Logx - Logarithm base, a positive scalar

XApEn, SampEn, MSEn, FuzzEn, PermEn, CondEn, DispEn

References:
[1] Steven M. Pincus,

“Approximate entropy as a measure of system complexity.” Proceedings of the National Academy of Sciences 88.6 (1991): 2297-2301.

AttnEn(Sig, varargin)

AttnEn estimates the attention entropy of a univariate data sequence.

[Attn] = AttnEn(Sig)

Returns the attention entropy (Attn) calculated as the average of the sub-entropies (Hxx, Hxn, Hnn, Hnx) estimated from the data sequence (Sig) using a base-2 logarithm.

[Attn, Hxx, Hnn, Hxn, Hnx] = AttnEn(Sig, ‘Logx’, value)

Returns the attention entropy (Attn) and the sub-entropies (Hxx, Hxn, Hnn, Hnx) from the data sequence (Sig) where,

• Hxx - entropy of local-maxima intervals

• Hnn - entropy of local minima intervals

• Hxn - entropy of intervals between local maxima and subsequent minima

• Hnx - entropy of intervals between local minima and subsequent maxima with the following name/value pair argument:

• Logx - Logarithm base, a positive scalar (enter 0 for natural log)

EnofEn, SpecEn, XSpecEn, PermEn, MSEn

References:
[1] Jiawei Yang, et al.,

“Classification of Interbeat Interval Time-series Using Attention Entropy.” IEEE Transactions on Affective Computing (2020)

BubbEn(Sig, varargin)

BubbEn estimates the bubble entropy of a univariate data sequence.

[Bubb, H] = BubbEn(Sig)

Returns the bubble entropy (Bubb) and the conditional Renyi entropy (H) estimates from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, logarithm = natural

[Bubb, H] = BubbEn(Sig, name, value, …)

Returns the bubble entropy (Bubb) estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, an integer > 1. BubbEn returns estimates for each dimension [2, …, m]

• tau - Time Delay, a positive integer

• Logx - Logarithm base, a positive scalar

PhasEn, MSEn

References:
[1] George Manis, M.D. Aktaruzzaman and Roberto Sassi,

“Bubble entropy: An entropy almost free of parameters.” IEEE Transactions on Biomedical Engineering 64.11 (2017): 2711-2718.

CondEn(Sig, varargin)

CondEn estimates the corrected conditional entropy of a univariate data sequence.

[Cond, SEw, SEz] = CondEn(Sig)

Returns the corrected conditional entropy estimates (Cond) and the corresponding Shannon entropies (m: SEw, m+1: SEz) for m = [1,2] estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, symbols = 6, logarithm = natural normalisation = false.

• Note: CondEn(m=1) returns the Shannon entropy of Sig.

[Cond, SEw, SEz] = CondEn(Sig, name, value, …)

Returns the corrected conditional entropy estimates (Cond) from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, an integer > 1

• tau - Time Delay, a positive integer

• c - Number of symbols, an integer > 1

• Logx - Logarithm base, a positive scalar

• Norm - Normalisation of Cond value, a boolean.
• [false] no normalisation - default

• [true] normalises w.r.t Shannon entropy of data sequence Sig

XCondEn, MSEn, PermEn, DistEn, XPermEn

References:
[1] Alberto Porta, et al.,

“Measuring regularity by means of a corrected conditional entropy in sympathetic outflow.” Biological cybernetics 78.1 (1998): 71-78.

CoSiEn(Sig, varargin)

CoSiEn estimates the cosine similarity entropy of a univariate data sequence.

[CoSi, Bm] = CoSiEn(Sig)

Returns the cosine similarity entropy (CoSi) and the corresponding global probabilities (Bm) estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, angular threshold = .1, logarithm = base 2,

[CoSi, Bm] = CoSiEn(Sig, name, value, …)

Returns the cosine similarity entropy (CoSi) estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, an integer > 1

• tau - Time Delay, a positive integer

• r - Angular threshold, a value in range [0 < r < 1]

• Logx - Logarithm base, a positive scalar (enter 0 for natural log)

• Norm - Normalisation of Sig, one of the following integers:
• [0] no normalisation - default

• [1] remove median(Sig) to get zero-median series

• [2] remove mean(Sig) to get zero-mean series

• [3] normalises Sig w.r.t. SD(Sig)

• [4] normalises Sig values to range [-1 1]

PhasEn, SlopEn, GridEn, MSEn, hMSEn

References:
[1] Theerasak Chanwimalueang and Danilo Mandic,

“Cosine similarity entropy: Self-correlation-based complexity analysis of dynamical systems.” Entropy 19.12 (2017): 652.

DistEn(Sig, varargin)

DistEn estimates the distribution entropy of a univariate data sequence.

[Dist, Ppi] = DistEn(Sig)

Returns the distribution entropy estimate (Dist) and the corresponding distribution probabilities (Ppi) estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, binning method = 'Sturges', logarithm = base 2, normalisation = w.r.t # of histogram bins

[Dist, Ppi] = DistEn(Sig, name, value, …)

Returns the distribution entropy estimate (Dist) estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, a positive integer

• tau - Time Delay, a positive integer

• Bins - Histogram bin selection method for distance distribution, either an integer > 1 indicating the number of bins, or one of the following strings {'sturges', 'sqrt', 'rice', 'doanes'} [default: 'sturges']

• Logx - Logarithm base, a positive scalar (enter 0 for natural log)

• Norm - Normalisation of Dist value, a boolean:
• [false] no normalisation.

• [true] normalises w.r.t # of histogram bins (default)

XDistEn, DistEn2D, MSEn, K2En

References:
[1] Li, Peng, et al.,

“Assessing the complexity of short-term heartbeat interval series by distribution entropy.” Medical & biological engineering & computing 53.1 (2015): 77-87.

DispEn(Sig, varargin)

DispEn estimates the dispersion entropy of a univariate data sequence.

[Dispx, RDE] = DispEn(Sig)

Returns the dispersion entropy (Dispx) and the reverse dispersion entropy (RDE) estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, symbols = 3, logarithm = natural, data transform = normalised cumulative density function (ncdf)

[Dispx, RDE] = DispEn(Sig, name, value, …)

Returns the dispersion entropy (Dispx) and the reverse dispersion entropy (RDE) estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, a positive integer

• tau - Time Delay, a positive integer

• c - Number of symbols, an integer > 1

• Typex - Type of data-to-symbolic sequence transform, one of the following:

{linear, kmeans, ncdf, finesort, equal}

• Logx - Logarithm base, a positive scalar

• Fluct - When Fluct == true, DispEn returns the fluctuation-based Dispersion entropy. [default: false]

• Norm - Normalisation of Dispx and RDE values, a boolean:
• [false] no normalisation - default

• [true] normalises w.r.t number of possible dispersion patterns (c^m or (2c -1)^m-1 if Fluct == true).

• rho - *If Typex == 'finesort', rho is the tuning parameter (default: 1)

DispEn2D, MvDispEn, PermEn, SyDyEn, MSEn.

References:
[1] Mostafa Rostaghi and Hamed Azami,

“Dispersion entropy: A measure for time-series analysis.” IEEE Signal Processing Letters 23.5 (2016): 610-614.

[2] Hamed Azami and Javier Escudero,

“Amplitude-and fluctuation-based dispersion entropy.” Entropy 20.3 (2018): 210.

[3] Li Yuxing, Xiang Gao and Long Wang,

“Reverse dispersion entropy: A new complexity measure for sensor signal.” Sensors 19.23 (2019): 5203.

[4] Wenlong Fu, et al.,

“Fault diagnosis for rolling bearings based on fine-sorted dispersion entropy and SVM optimized with mutation SCA-PSO.” Entropy 21.4 (2019): 404.

DivEn(Sig, varargin)

DivEn estimates the diversity entropy of a univariate data sequence.

[Div, CDEn, Bm] = DivEn(Sig)

Returns the diversity entropy (Div), the cumulative diversity entropy (CDEn), and the corresponding probabilities (Bm) estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, # bins = 5, logarithm = natural,

[Div, CDEn, Bm] = DivEn(Sig, name, value, …)

Returns the diversity entropy (Div) estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, an integer > 1

• tau - Time Delay, a positive integer

• r - Histogram bins #: either
• an integer [r > 1] representing the number of bins

• a vector array of 3 or more increasing values in range [-1 1] representing the bin edges including the rightmost edge.

• Logx - Logarithm base, a positive scalar (enter 0 for natural log)

CoSiEn, PhasEn, SlopEn, GridEn, MSEn

References:
[1] X. Wang, S. Si and Y. Li,

“Multiscale Diversity Entropy: A Novel Dynamical Measure for Fault Diagnosis of Rotating Machinery,” IEEE Transactions on Industrial Informatics, vol. 17, no. 8, pp. 5419-5429, Aug. 2021

[2] Y. Wang, M. Liu, Y. Guo, F. Shu, C. Chen and W. Chen,

“Cumulative Diversity Pattern Entropy (CDEn): A High-Performance, Almost-Parameter-Free Complexity Estimator for Nonstationary Time Series,” IEEE Transactions on Industrial Informatics vol. 19, no. 9, pp. 9642-9653, Sept. 2023

EnofEn(Sig, varargin)

EnofEn estimates the entropy of entropy from a univariate data sequence.

[EoE, AvEn, S2] = EnofEn(Sig)

Returns the entropy of entropy (EoE), the average Shannon entropy (AvEn), and the number of levels (S2) across all windows estimated from the data sequence (Sig) using the default parameters: window length (samples) = 10, slices = 10, logarithm = natural heartbeat interval range (xmin, xmax) = [min(Sig) max(Sig)]

[EoE, AvEn, S2] = EnofEn(Sig, name, value, …)

Returns the entropy of entropy (EoE) estimated from the data sequence (Sig) using the specified name/value pair arguments:

• tau - Window length, an integer > 1

• S - Number of slices, an integer > 1

• Xrange - The min and max heartbeat interval,

a two-element vector where X(1) < X(2)

• Logx - Logarithm base, a positive scalar

SampEn, MSEn

References:
[1] Chang Francis Hsu, et al.,

“Entropy of entropy: Measurement of dynamical complexity for biological systems.” Entropy 19.10 (2017): 550.

FuzzEn(Sig, varargin)

FuzzEn estimates the fuzzy entropy of a univariate data sequence.

[Fuzz, Ps1, Ps2] = FuzzEn(Sig)

Returns the fuzzy entropy estimates (Fuzz) and the average fuzzy distances (m: Ps1, m+1: Ps2) for m = [1,2] estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, fuzzy function = 'default', fuzzy function parameters = [0.2, 2], logarithm = natural

[Fuzz, Ps1, Ps2] = FuzzEn(Sig, name, value, …)

Returns the fuzzy entropy estimates (Fuzz) for dimensions = [1, …, m] estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, a positive integer [default: 2]

• tau - Time Delay, a positive integer [default: 1]

• Fx - Fuzzy function name, one of the following strings: {'sigmoid', 'modsampen', 'default', 'gudermannian', 'bell', 'triangular', 'trapezoidal1', 'trapezoidal2', 'z_shaped', 'gaussian', 'constgaussian'}

• r - Fuzzy function parameters, a 1 element scalar or a 2 element vector of positive values. The r parameters for each fuzzy function are defined as follows: (default: [.2 2])

sigmoid:
• r(1) = divisor of the exponential argument

• r(2) = value subtracted from argument (pre-division)

modsampen:
• r(1) = divisor of the exponential argument

• r(2) = value subtracted from argument (pre-division)

default:
• r(1) = divisor of the exponential argument

• r(2) = argument exponent (pre-division)

gudermannian:
• r = a scalar whose value is the numerator of

argument to gudermannian function: GD(x) = atan(tanh(r/x)). GD(x) is normalised to have a maximum value of 1.

[DEPRICATED] linear:
• r = an integer value. When r == 0, the

argument of the exponential function is normalised between [0 1]. When r == 1, the minimuum value of the exponential argument is set to 0.

triangular:
• r = a positive scalar whose value is the threshold (corner point) of the triangular function.

trapezoidal1:
• r = a positive scalar whose value corresponds to the upper (2r) and lower (r) corner points of the trapezoid.

trapezoidal2:
• r(1) = a value corresponding to the upper corner point of the trapezoid.

• r(2) = a value corresponding to the lower corner point of the trapezoid.

z_shaped:
• r = a scalar whose value corresponds to the upper (2r) and lower (r) corner points of the z-shape.

bell:
• r(1) = divisor of the distance value

• r(2) = exponent of generalized bell-shaped function

gaussian:
• r = a scalar whose value scales the slope of the Gaussian curve.

constgaussian:
• r = a scalar whose value defines the lower threshod and shape of the Gaussian curve.

• Logx - Logarithm base, a positive scalar [default: natural]

For further information on the name/value paire arguments, see the EntropyHub guide

SampEn, ApEn, PermEn, DispEn, XFuzzEn, FuzzEn2D, MSEn.

References:
[1] Weiting Chen, et al.

“Characterization of surface EMG signal based on fuzzy entropy.” IEEE Transactions on neural systems and rehabilitation engineering 15.2 (2007): 266-272.

[2] Hong-Bo Xie, Wei-Xing He, and Hui Liu

“Measuring time series regularity using nonlinear similarity-based sample entropy.” Physics Letters A 372.48 (2008): 7140-7146.

[3] Hamed Azami, et al.

“Fuzzy Entropy Metrics for the Analysis of Biomedical Signals: Assessment and Comparison” IEEE Access 7 (2019): 104833-104847

GridEn(Sig, varargin)

GridEn estimates the gridded distribution entropy of a univariate data sequence.

[GDE, GDR] = GridEn(Sig)

Returns the gridded distribution entropy (GDE) and the gridded distribution rate (GDR) estimated from the data sequence (Sig) using the default parameters: grid coarse-grain = 3, time delay = 1, logarithm = natural

[GDE, GDR, PIx, GIx, SIx, AIx] = GridEn(Sig)

In addition to GDE and GDR, GridEn returns the following indices estimated from the data sequence (Sig) using the default parameters:

• PIx - Percentage of points below the line of identity (LI)

• GIx - Proportion of point distances above the LI

• SIx - Ratio of phase angles (w.r.t. LI) of the points above the LI

• AIx - Ratio of the cumulative area of sectors of points above the LI

[GDE, GDR, …,] = GridEn(Sig, name, value, …)

Returns the gridded distribution entropy (GDE) estimate of the data sequence (Sig) using the specified name/value pair arguments:

• m - Grid coarse-grain (m x m sectors), an integer > 1

• tau - Time Delay, a positive integer

• Logx - Logarithm base, a positive scalar

• Plotx - When Plotx == true, returns gridded Poicare plot and a bivariate histogram of the grid point distribution (default: false)

PhasEn, CoSiEn, SlopEn, BubbEn, MSEn

References:
[1] Chang Yan, et al.,

“Novel gridded descriptors of Poincare plot for analyzing heartbeat interval time-series.” Computers in biology and medicine 109 (2019): 280-289.

[2] Chang Yan, et al.

“Area asymmetry of heart rate variability signal.” Biomedical engineering online 16.1 (2017): 1-14.

[3] Alberto Porta, et al.,

“Temporal asymmetries of short-term heart period variability are linked to autonomic regulation.” American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 295.2 (2008): R550-R557.

[4] C.K. Karmakar, A.H. Khandoker and M. Palaniswami,

“Phase asymmetry of heart rate variability signal.” Physiological measurement 36.2 (2015): 303.

IncrEn(Sig, varargin)

IncrEn estimates the increment entropy of a univariate data sequence.

[Incr] = IncrEn(Sig)

Returns the increment entropy (Incr) estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, quantifying resolution = 4, logarithm = base 2,

[Incr] = IncrEn(Sig, name, value, …)

Returns the increment entropy (Incr) estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, an integer > 1

• tau - Time Delay, a positive integer

• R - Quantifying resolution, a positive integer

• Logx - Logarithm base, a positive scalar (enter 0 for natural log)

• Norm - Normalisation of IncrEn value, a boolean:
• [false] no normalisation - default

• [true] normalises w.r.t embedding dimension (m-1).

PermEn, SyDyEn, MSEn

References:
[1] Xiaofeng Liu, et al.,

“Increment entropy as a measure of complexity for time series.” Entropy 18.1 (2016): 22.1.

*** “Correction on Liu, X.; Jiang, A.; Xu, N.; Xue, J. - Increment

Entropy as a Measure of Complexity for Time Series, Entropy 2016, 18, 22.” Entropy 18.4 (2016): 133.

[2] Xiaofeng Liu, et al.,

“Appropriate use of the increment entropy for electrophysiological time series.” Computers in biology and medicine 95 (2018): 13-23.

K2En(Sig, varargin)

K2En estimates the Kolmogorov (K2) entropy of a univariate data sequence.

[K2, Ci] = K2En(Sig)

Returns the Kolmogorov entropy estimates (K2) and the correlation integrals (Ci) for m = [1,2] estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, distance threshold (r) = 0.2*SD(Sig), logarithm = natural

[K2, Ci] = K2En(Sig, name, value, …)

Returns the Kolmogorov entropy estimates (K2) for dimensions = [1, …, m] estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, a positive integer

• tau - Time Delay, a positive integer

• r - Radius Distance Threshold, a positive scalar

• Logx - Logarithm base, a positive scalar

DistEn, XK2En, MSEn

References:
[1] Peter Grassberger and Itamar Procaccia,

“Estimation of the Kolmogorov entropy from a chaotic signal.” Physical review A 28.4 (1983): 2591.

[2] Lin Gao, Jue Wang and Longwei Chen

“Event-related desynchronization and synchronization quantification in motor-related EEG by Kolmogorov entropy” J Neural Eng. 2013 Jun;10(3):03602

PermEn(Sig, varargin)

PermEn estimates the permutation entropy of a univariate data sequence.

[Perm, Pnorm, cPE] = PermEn(Sig)

Returns the permuation entropy estimates (Perm), the normalised permutation entropy (Pnorm) and the conditional permutation entropy (cPE) for m = [1,2] estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, logarithm = base 2, normalisation = w.r.t #symbols (m-1) Note: using the standard PermEn estimation, Perm = 0 when m = 1. It is recommeneded that signal length, N > 5m! (see [8] and Amigo et al., Europhys. Lett. 83:60005, 2008)

[Perm, Pnorm, cPE] = PermEn(Sig, m)

Returns the permutation entropy estimates (Perm) estimated from the data sequence (Sig) using the specified embedding dimensions = [1, …, m] with other default parameters as listed above.

[Perm, Pnorm, cPE] = PermEn(Sig, name, value, …)

Returns the permutation entropy estimates (Perm) for dimensions = [1, …, m] estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, an integer > 1

• tau - Time Delay, a positive integer

• Logx - Logarithm base, a positive scalar (enter 0 for natural log)

• Norm - Normalisation of Pnorm value, a boolean operator:
• false - normalises w.r.t log(# of permutation symbols [m-1]) - default

• true - normalises w.r.t log(# of all possible permutations [m!])

Note: Normalised permutation entropy is undefined for m = 1. Note: When Typex = 'uniquant' and Norm = true, normalisation is calculated w.r.t. log(tpx^m)

• Typex - Permutation entropy variation, one of the following:

{'uniquant', 'finegrain', 'modified', 'ampaware', 'weighted', 'edge', 'phase'} See the EntropyHub guide for more info on PermEn variants.

• tpx - Tuning parameter for associated permutation entropy variation.
• [uniquant] tpx is the L parameter, an integer > 1 (default =4).

• [finegrain] tpx is the alpha parameter, a positive scalar (default = 1)

• [ampaware] tpx is the A parameter, a value in range [0 1] (default = 0.5)

• [edge] tpx is the r sensitivity parameter, a scalar > 0 (default = 1)

• [phase] tpx is the option to unwrap the phase angle of Hilbert-transformed signal, either [] or 1 (default = 0)

See the EntropyHub guide for more info on these permutation entropy variants.

XPermEn, MSEn, XMSEn, SampEn, ApEn, CondEn

References:
[1] Christoph Bandt and Bernd Pompe,

“Permutation entropy: A natural complexity measure for time series.” Physical Review Letters, 88.17 (2002): 174102.

[2] Xiao-Feng Liu, and Wang Yue,

“Fine-grained permutation entropy as a measure of natural complexity for time series.” Chinese Physics B 18.7 (2009): 2690.

[3] Chunhua Bian, et al.,

“Modified permutation-entropy analysis of heartbeat dynamics.” Physical Review E 85.2 (2012) : 021906

“Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information.” Physical Review E 87.2 (2013): 022911.

[5] Hamed Azami and Javier Escudero,

“Amplitude-aware permutation entropy: Illustration in spike detection and signal segmentation.” Computer methods and programs in biomedicine, 128 (2016): 40-51.

[6] Zhiqiang Huo, et al.,

“Edge Permutation Entropy: An Improved Entropy Measure for Time-Series Analysis,” 45th Annual Conference of the IEEE Industrial Electronics Soc, (2019), 5998-6003

[7] Zhe Chen, et al.

“Improved permutation entropy for measuring complexity of time series under noisy condition.” Complexity 1403829 (2019).

[8] Maik Riedl, Andreas Müller, and Niels Wessel,

“Practical considerations of permutation entropy.” The European Physical Journal Special Topics 222.2 (2013): 249-262.

[9] Kang Huan, Xiaofeng Zhang, and Guangbin Zhang,

“Phase permutation entropy: A complexity measure for nonlinear time series incorporating phase information.” Physica A: Statistical Mechanics and its Applications 568 (2021): 125686.

PhasEn(Sig, varargin)

PhasEn estimates the phase entropy of a univariate data sequence.

[Phas] = PhasEn(Sig)

Returns the phase entropy (Phas) estimate of the data sequence (Sig) using the default parameters: angular partitions = 4, time delay = 1, logarithm = natural, normalisation = true

[Phas] = PhasEn(Sig, name, value, …)

Returns the phase entropy (Phas) estimate of the data sequence (Sig) using the specified name/value pair arguments:

• K - Angular partitions (coarse graining), an integer > 1
• Note: Division of partitions begins along the positive

x-axis. As this point is somewhat arbitrary, it is recommended to use even-numbered (preferably multiples of 4) partitions for sake of symmetry.

• tau - Time Delay, a positive integer

• Logx - Logarithm base, a positive scalar

• Norm - Normalisation of Phas value, a boolean:
• [false] no normalisation

• [true] normalises w.r.t. the # partitions (Log(K)) (Default)

• Plotx - When Plotx == true, returns Poincaré plot (default: false)

SampEn, ApEn, GridEn, MSEn, SlopEn, CoSiEn, BubbEn

References:
[1] Ashish Rohila and Ambalika Sharma,

“Phase entropy: a new complexity measure for heart rate variability.” Physiological measurement 40.10 (2019): 105006.

RangEn(Sig, varargin)

RangEn estimates the range entropy of a univariate data sequence.

[Rangx, A, B] = RangEn(Sig)

Returns the range entropy estimate (Rangx) and the number of matched state vectors (m: B, m+1: A) estimated from the data sequence (Sig) using the sample entropy algorithm and the following default parameters: embedding dimension = 2, time delay = 1, radius threshold = 0.2, logarithm = natural.

Rangx, A, B = RangEn(Sig, name, value, …)

Returns the range entropy estimates (Rangx) for dimensions = m estimated for the data sequence (Sig) using the specified name-value arguments:

• m - Embedding Dimension, a positive integer

• tau - Time Delay, a positive integer

• r - Radius Distance Threshold, a positive scalar between 0 and 1

• Methodx - Base entropy method, either ‘SampEn’ [default] or ‘ApEn’

• Logx - Logarithm base, a positive scalar

ApEn, SampEn, FuzzEn, MSEn

References:
[1] Omidvarnia, Amir, et al.

“Range entropy: A bridge between signal complexity and self-similarity” Entropy 20.12 (2018): 962.

[2] Joshua S Richman and J. Randall Moorman.

“Physiological time-series analysis using approximate entropy and sample entropy.” American Journal of Physiology-Heart and Circulatory Physiology 2000

SampEn(Sig, varargin)

SampEn estimates the sample entropy of a univariate data sequence.

[Samp, A, B] = SampEn(Sig)

Returns the sample entropy estimates (Samp) and the number of matched state vectors (m: B, m+1: A) for m = [0,1,2] estimated from the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, radius threshold = 0.2*SD(Sig), logarithm = natural

[Samp, A, B, (Vcp, Ka, Kb)] = SampEn(Sig, …, Vcp = true)

If Vcp == True, an additional vector (Vcp, Ka, Kb) is returned with the sample entropy estimates (Samp) and the number of matched state vectors (m: B, m+1: A). (Vcp, Ka, Kb) contains the variance of the conditional probabilities (Vcp), i.e. CP = A/B, and the number of overlapping matching vector pairs of lengths m+1 (Ka) and m (Kb), respectively. Note Vcp is undefined for the zeroth embedding dimension (m = 0) and due to the computational demand, will take substantially more time to return function outputs. See Appendix B in [2] for more info.

[Samp, A, B] = SampEn(Sig, name, value, …)

Returns the sample entropy estimates (Samp) for dimensions = [0,1,…, m] estimated from the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, a positive integer

• tau - Time Delay, a positive integer

• r - Radius Distance Threshold, a positive scalar

• Logx - Logarithm base, a positive scalar

• Vcp - Option to return variance of conditional probabilities and the number of overlapping matching vector pairs, a boolean [default: false]

ApEn, FuzzEn, PermEn, CondEn, XSampEn, SampEn2D, MSEn.

References:
[1] Joshua S Richman and J. Randall Moorman.

“Physiological time-series analysis using approximate entropy and sample entropy.” American Journal of Physiology-Heart and Circulatory Physiology (2000).

[2] Douglas E Lake, Joshua S Richman, M.P. Griffin, J. Randall Moorman

“Sample entropy analysis of neonatal heart rate variability.” American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 283, no. 3 (2002): R789-R797.

SlopEn(Sig, varargin)

SlopEn estimates the slope entropy of a univariate data sequence.

[Slop] = SlopEn(Sig)

Returns the slope entropy (Slop) estimates for embedding dimensions [2, …, m] of the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, angular thresholds = [5 45], logarithm = base 2

[Slop] = SlopEn(Sig, name, value, …)

Returns the slope entropy (Slop) estimate of the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, an integer > 1. SlopEn returns estimates for each dimension [2, …, m]

• tau - Time Delay, a positive integer

• Lvls - Angular thresolds, a vector of monotonically increasing values in the range [0 90] degrees.

• Logx - Logarithm base, a positive scalar (enter 0 for natural log)

• Norm - Normalisation of Slop value, a boolean:
• [false] no normalisation

• [true] normalises w.r.t. the number of patterns found (default)

PhasEn, GridEn, MSEn, CoSiEn, SampEn, ApEn

References:
[1] David Cuesta-Frau,

“Slope Entropy: A New Time Series Complexity Estimator Based on Both Symbolic Patterns and Amplitude Information.” Entropy 21.12 (2019): 1167.

SpecEn(Sig, varargin)

SpecEn estimates the spectral entropy of a univariate data sequence.

[Spec, BandEn] = SpecEn(Sig)

Returns the spectral entropy estimate of the full spectrum (Spec) and the within-band entropy (BandEn) estimated from the data sequence (Sig) using the default parameters: N-point FFT = length(Sig)*2 + 1, normalised band edge frequencies = [0 1], logarithm = natural, normalisation = w.r.t # of spectrum/band values.

[Spec, BandEn] = SpecEn(Sig, name, value, …)

Returns the spectral entropy (Spec) and the within-band entropy (BandEn) estimate for the data sequence (Sig) using the specified name/value pair arguments:

• N’ - Resolution of spectrum (N-point FFT), an integer > 1

• Freqs - Normalised spectrum band edge-frequencies, a 2 element vector with values in range [0 1] where 1 corresponds to the Nyquist frequency (Fs/2). Note: When no band frequencies are entered, BandEn == SpecEn

• Logx - Logarithm base, a positive scalar (default: natural log)

• Norm - Normalisation of Spec value, a boolean:
• [false] no normalisation.

• [true] normalises w.r.t # of spectrum/band frequency values (default).

XSpecEn, fft, periodogram

References:
[1] G.E. Powell and I.C. Percival,

“A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems.” Journal of Physics A: Mathematical and General 12.11 (1979): 2053.

[2] Tsuyoshi Inouye, et al.,

“Quantification of EEG irregularity by use of the entropy of the power spectrum.” Electroencephalography and clinical neurophysiology 79.3 (1991): 204-210.

SyDyEn(Sig, varargin)

SyDyEn estimates the symbolic dynamic entropy of a univariate data sequence.

[SyDy, Zt] = SyDyEn(Sig)

Returns the symbolic dynamic entropy (SyDy) and the symbolic sequence (Zt) of the data sequence (Sig) using the default parameters: embedding dimension = 2, time delay = 1, symbols = 3, logarithm = natural, symbolic partition type = maximum entropy partitioning (MEP), normalisation = normalises w.r.t # possible vector permutations (c^m)

[SyDy, Zt] = SyDyEn(Sig, name, value, …)

Returns the symbolic dynamic entropy (SyDy) and the symbolic sequence (Zt) estimated of the data sequence (Sig) using the specified name/value pair arguments:

• m - Embedding Dimension, a positive integer

• tau - Time Delay, a positive integer

• c - Number of symbols, an integer > 1

• Typex - Type of symbolic sequence partitioning, one of the following: {'linear', 'uniform', 'MEP' (default), 'kmeans'}

• Logx - Logarithm base, a positive scalar

• Norm - Normalisation of SyDyEn value, a boolean:
• [false] no normalisation

• [true] normalises w.r.t # possible vector permutations (c^m+1) - default

DispEn, PermEn, CondEn, SampEn, MSEn.

References:
[1] Yongbo Li, et al.,

“A fault diagnosis scheme for planetary gearboxes using modified multi-scale symbolic dynamic entropy and mRMR feature selection.” Mechanical Systems and Signal Processing 91 (2017): 295-312.

[2] Jian Wang, et al.,

“Fault feature extraction for multiple electrical faults of aviation electro-mechanical actuator based on symbolic dynamics entropy.” IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), 2015.

[3] Venkatesh Rajagopalan and Asok Ray,

“Symbolic time series analysis via wavelet-based partitioning.” Signal processing 86.11 (2006): 3309-3320.