Cross Entropies

Functions for estimating the entropy between two univariate time series.

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

Attention

For cross-entropy and multiscale cross-entropy functions, the two time series signals are passed as a two-column or two-row matrix. At present, it is not possible to pass signals of different lengths separately

These functions are directly available when EntropyHub is imported:

import EntropyHub as EH

dir(EH)
XApEn(Sig, m=2, tau=1, r=None, Logx=numpy.exp)

XApEn estimates the cross-approximate entropy between two univariate data sequences.

XAp, Phi = XApEn(Sig)

Returns the cross-approximate entropy estimates (XAp) and the average number of matched vectors (Phi) for m = [0,1,2], estimated for the data sequences contained in ‘Sig’ using the default parameters: embedding dimension = 2, time delay = 1, radius distance threshold = 0.2*SD(Sig), logarithm = natural

**NOTE: XApEn is direction-dependent. Thus, the first row/column of Sig is used as the template data sequence, and the second row/column is the matching sequence.

XAp, Phi = XApEn(Sig, keyword = value, ...)

Returns the cross-approximate entropy estimates (XAp) between the data sequences contained in Sig using the specified ‘keyword’ arguments:

m

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

tau

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

r

  • Radius Distance Threshold, a positive scalar [default: 0.2*SD(Sig)]

Logx

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

See also

XSampEn, XFuzzEn, XMSEn, ApEn, SampEn, MSEn

References
[1] Steven Pincus and Burton H. Singer,

“Randomness and degrees of irregularity.” Proceedings of the National Academy of Sciences 93.5 (1996): 2083-2088.

[2] Steven Pincus,

“Assessing serial irregularity and its implications for health.” Annals of the New York Academy of Sciences 954.1 (2001): 245-267.

XCondEn(Sig, m=2, tau=1, c=6, Logx=numpy.exp, Norm=False)

XCondEn estimates the corrected cross-conditional entropy between two univariate data sequences.

XCond, SEw, SEz = XCondEn(Sig)

Returns the corrected cross-conditional entropy estimates (XCond) and the corresponding Shannon entropies (m: SEw, m+1: SEz) for m = [1,2] estimated for the data sequences contained in Sig using the default parameters: embedding dimension = 2, time delay = 1, number of symbols = 6, logarithm = natural ** Note: XCondEn is direction-dependent. Therefore, the order of the data sequences in Sig matters. If the first row/column of Sig is the sequence ‘y’, and the second row/column is the sequence ‘u’, then XCond is the amount of information carried by y(i) when the pattern u(i) is found.

XCond, SEw, SEz = XCondEn(Sig, keyword = value, ...)

Returns the corrected cross-conditional entropy estimates (XCond) for the data sequences contained in Sig using the specified ‘keyword’ arguments:

m

  • Embedding Dimension, an integer > 1 [default: 2]

tau

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

c

  • Number of symbols, an integer > 1 [default: 6]

Logx

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

Norm

  • Normalisation of XCond value, one of the following integers:

  • [False] no normalisation [default]

  • [True] normalises w.r.t cross-Shannon entropy.

See also

XFuzzEn, XSampEn, XApEn, XPermEn, CondEn, XMSEn

References
[1] Alberto Porta, et al.,

“Conditional entropy approach for the evaluation of the coupling strength.” Biological cybernetics 81.2 (1999): 119-129.

XDistEn(Sig, m=2, tau=1, Bins='Sturges', Logx=2, Norm=True)

XDistEn estimates the cross-distribution entropy between two univariate data sequences.

XDist, Ppi = XDistEn(Sig)

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

XDist, Ppi = XDistEn(Sig, keyword = value, ...)

Returns the cross-distribution entropy estimate (XDist) estimated between the data sequences contained in ‘Sig’ using the specified ‘keyword’ = arguments:

m

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

tau

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

Bins

  • Histogram bin selection method for distance distribution,

  • 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 [default: 2] (enter 0 for natural log)

Norm

  • Normalisation of DistEn value, a boolean value:

  • [False] no normalisation.

  • [True] normalises w.r.t # of histogram bins [default]

See also

XSampEn, XApEn, XPermEn, XCondEn, DistEn, DistEn2D, XMSEn

References
[1] Yuanyuan Wang and Pengjian Shang,

“Analysis of financial stock markets through the multiscale cross-distribution entropy based on the Tsallis entropy.” Nonlinear Dynamics 94.2 (2018): 1361-1376.

XFuzzEn(Sig, m=2, tau=1, r=(0.2, 2), Fx='default', Logx=numpy.exp)

XFuzzEn estimates the cross-fuzzy entropy between two univariate data sequences.

XFuzz, Ps1, Ps2 = XFuzzEn(Sig)

Returns the cross-fuzzy entropy estimates (XFuzz) and the average fuzzy distances (m: Ps1, m+1: Ps2) for m = [1,2] estimated for the data sequences contained in Sig, using the default parameters: embedding dimension = 2, time delay = 1, fuzzy function (Fx) = ‘default’, fuzzy function parameters (r) = (0.2, 2), logarithm = natural

XFuzz, Ps1, Ps2 = XFuzzEn(Sig, keyword = value, ...)

Returns the cross-fuzzy entropy estimates (XFuzz) for dimensions = [1, …, m] estimated for the data sequences in ‘Sig’ using the specified ‘keyword’ 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', 'linear'}

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.

  • 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.

Logx

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

For further information on the keyword arguments, see the EntropyHub guide.

See also

FuzzEn, XSampEn, XApEn, FuzzEn2D, XMSEn, MSEn

References
[1] Hong-Bo Xie, et al.,

“Cross-fuzzy entropy: A new method to test pattern synchrony of bivariate time series.” Information Sciences 180.9 (2010): 1715-1724.

XK2En(Sig, m=2, tau=1, r=None, Logx=numpy.exp)

XK2En estimates the cross-Kolmogorov entropy between two univariate data sequences.

XK2, Ci = XK2En(Sig)

Returns the cross-Kolmogorov entropy estimates (XK2) and the correlation integrals (Ci) for m = [1, 2] estimated between the data sequences contained in Sig using the default parameters: embedding dimension = 2, time delay = 1, distance threshold (r) = 0.2*SD(Sig), logarithm = natural

XK2, Ci = XK2En(Sig, keyword = value, ...)

Returns the cross-Kolmogorov entropy estimates (XK2) estimated between the data sequences contained in Sig using the specified ‘keyword’ arguments:

m

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

tau

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

r

  • Radius Distance Threshold, a positive scalar [default: 0.2*SD(Sig)]

Logx

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

See also

XSampEn, XFuzzEn, XApEn, K2En, XMSEn, XDistEn

References
[1] Matthew W. Flood,

“XK2En - EntropyHub Project” (2021) https://github.com/MattWillFlood/EntropyHub

XPermEn(Sig, m=3, tau=1, Logx=numpy.exp)

XPermEn estimates the cross-permutation entropy between two univariate data sequences.

XPerm = XPermEn(Sig)

Returns the cross-permuation entropy estimates (XPerm) estimated betweeen the data sequences contained in Sig using the default parameters: embedding dimension = 3, time delay = 1, logarithm = base 2,

XPerm = XPermEn(Sig, keyword = value, ...)

Returns the permutation entropy estimates (Perm) estimated between the data sequences contained in Sig using the specified ‘keyword’ arguments:

m

  • Embedding Dimension, an integer > 2 [default: 3]

**Note: XPerm is undefined for embedding dimensions < 3.

tau

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

Logx

  • Logarithm base, a positive scalar [default: 2] (enter 0 for natural log).

See also

PermEn, XApEn, XSampEn, XFuzzEn, XMSEn

References
[1] Wenbin Shi, Pengjian Shang, and Aijing Lin,

“The coupling analysis of stock market indices based on cross-permutation entropy.” Nonlinear Dynamics 79.4 (2015): 2439-2447.

XSampEn(Sig, m=2, tau=1, r=None, Logx=numpy.exp)

XSampEn Estimates the cross-sample entropy between two univariate data sequences.

XSamp, A, B = XSampEn(Sig)

Returns the cross-sample entropy estimates (XSamp) and the number of matched vectors (m: B, m+1: A) for m = [0,1,2] estimated for the two univariate data sequences contained in Sig using the default parameters: embedding dimension = 2, time delay = 1, radius = 0.2*SD(Sig), logarithm = natural

XSamp, A, B = XSampEn(Sig, keyword = value, ...)

Returns the cross-sample entropy estimates (XSamp) for dimensions [0,1,…, m] estimated between the data sequences in Sig using the specified ‘keyword’ arguments:

m

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

tau

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

r

  • Radius, a positive scalar [default: 0.2*SD(Sig)]

Logx

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

See also

XFuzzEn, XApEn, SampEn, SampEn2D, XMSEn, ApEn

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)

XSpecEn(Sig, N=None, Freqs=(0, 1), Logx=numpy.exp, Norm=True)

XSpecEn estimates the cross-spectral entropy between two univariate data sequences.

XSpec, BandEn = XSpecEn(Sig)

Returns the cross-spectral entropy estimate (XSpec) of the full cross- spectrum and the within-band entropy (BandEn) estimated for the data sequences contained in Sig using the default parameters: N-point FFT = length of Sig, normalised band edge frequencies = [0 1], logarithm = base 2, normalisation = w.r.t # of spectrum/band frequency values.

XSpec, BandEn = XSpecEn(Sig, keyword = value, ...)

Returns the cross-spectral entropy (XSpec) and the within-band entropy (BandEn) estimate for the data sequences contained in Sig using the following specified ‘keyword’ arguments:

N

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

Freqs

  • Normalised and edge frequencies, a scalar 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]

Norm

  • Normalisation of XSpec value, one of the following integers:

[false] no normalisation. [true] normalises w.r.t # of frequency values within the spectrum/band [default]

See the EntropyHub guide for more info.

See also

SpecEn, fft, XDistEn, periodogram, XSampEn, XApEn

References
[1] Matthew W. Flood,

“XSpecEn - EntropyHub Project” (2021) https://github.com/MattWillFlood/EntropyHub