Cross Entropies
Functions for estimating the entropy between two univariate time series.
The following functions also form the cross-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. We are currently working to enable different signal lengths for cross-entropy estimation.
- XApEn(Sig, varargin)
XApEn estimates the cross-approximate entropy between two univariate data sequences.
[XAp, Phi] = XApEn(Sig)
Returns the cross-approximate entropy estimates (
XAp
) and the log-average number of matched vectors (Phi
) form
= [0,1,2], estimated for the data sequences contained inSig
using the default parameters: embedding dimension = 2, time delay = 1, radius distance threshold = 0.2*SD(Sig
), logarithm = naturalNOTE:
XApEn
is direction-dependent. Thus, the first row/column ofSig
is used as the template data sequence, and the second row/column is the matching sequence.
[XAp, Phi] = XApEn(Sig, name, value, …)
Returns the cross-approximate entropy estimates (
XAp
) between the data sequences contained inSig
using the specified name/value pair 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, varargin)
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
) form
= [1,2] estimated for the data sequences contained inSig
using the default parameters: embedding dimension = 2, time delay = 1, number of symbols = 6, logarithm = naturalNote:
XCondEn
is direction-dependent. Therefore, the order of the data sequences inSig
matters. If the first row/column ofSig
is sequence ‘y’, and the second row/column is sequence ‘u’, thenXCond
is the amount of information carried by y(i) when the pattern u(i) is found.
[XCond, SEw, SEz] = XCondEn(Sig, name, value, …)
Returns the corrected cross-conditional entropy estimates (
XCond
) for the data sequences contained inSig
using the specified name/value pair 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 ofXCond
value, a boolean value:[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, varargin)
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 inSig
using the default parameters: embedding dimension = 2, time delay = 1, binning method ='Sturges'
, logarithm = base 2, normalisation = w.r.t number of histogram bins[XDist, Pi] = XDistEn(Sig, name, value, …)
Returns the cross-distribution entropy estimate (
XDist
) estimated beween the data sequences contained inSig
using the specified name/value pair 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 logNorm
- Normalisation ofXDist
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, varargin)
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
) form
= [1,2] estimated for the data sequences contained inSig
, using the default parameters: embedding dimension = 2, time delay = 1, fuzzy function (Fx
) ='default'
, fuzzy function paramters (r
) = [0.2, 2], logarithm = natural[XFuzz, Ps1, Ps2] = XFuzzEn(Sig, name, value, …)
Returns the cross-fuzzy entropy estimates (
XFuzz
) for dimensions = [1, …,m
] estimated for the data sequences inSig
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'
,'linear'
}r
- Fuzzy function parameters, a 1 element scalar or a 2 element vector of positive values. Ther
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 name/value paire 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, varargin)
XK2En estimates the cross-Kolmogorov (K2) entropy between two univariate data sequences.
[XK2, Ci] = XK2En(Sig)
Returns the cross-Kolmogorov entropy estimates (
XK2
) and the correlation integrals (Ci
) form
= [1,2] estimated between the data sequences contained inSig
using the default parameters: embedding dimension = 2, time delay = 1, distance threshold (r
) = 0.2*SD(Sig
), logarithm = natural[XK2, Ci] = XK2En(Sig, name, value, …)
Returns the cross-Kolmogorov entropy estimates (
XK2
) estimated between the data sequences contained inSig
using the specified name/value pair 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, varargin)
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 inSig
using the default parameters: embedding dimension = 3, time delay = 1, logarithm = base 2,[XPerm] = XPermEn(Sig, name, value, …)
Returns the permutation entropy estimates (
XPerm
) for the data sequences contained inSig
using the specified name/value pair 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, varargin)
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 inSig
using the default parameters: embedding dimension = 2, time delay = 1, radius distance threshold = 0.2*SD(Sig
), logarithm = natural[XSamp, A, B] = XSampEn(Sig, name, value, …)
Returns the cross-sample entropy estimates (
XSamp
) for dimensions [0,1, …,m
] estimated between the data sequences inSig
using the specified name/value pair 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:
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, varargin)
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 between the data sequences contained inSig
using the default parameters: N-point FFT = length ofSig
, normalised band edge frequencies = [0 1], logarithm = natural, normalisation = w.r.t # of spectrum/band frequency values.[XSpec, BandEn] = XSpecEn(Sig, name, value, …)
Returns the cross-spectral entropy (
XSpec
) and the within-band entropy (BandEn
) estimate between the data sequences contained inSig
using the following specified name/value pair arguments:N
- Resolution of spectrum (N-point FFT), an integer > 1Freqs
- Normalised band 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 ofXSpec
value, a boolean:[false] no normalisation.
[true] normalises w.r.t # of frequency values within the spectrum/band [default]
- See also:
SpecEn, fft, XDistEn, periodogram, XSampEn, XApEn
- References:
- [1] Matthew W. Flood,
“XSpecEn - EntropyHub Project” (2021) https://github.com/MattWillFlood/EntropyHub