# 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.

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]

`XSampEn`, `XFuzzEn`, `XMSEn`, `ApEn`, `SampEn`, `MSEn`

References
 Steven Pincus and Burton H. Singer,

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

 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.

`XFuzzEn`, `XSampEn`, `XApEn`, `XPermEn`, `CondEn`, `XMSEn`

References
 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]

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

References
 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.

`FuzzEn`, `XSampEn`, `XApEn`, `FuzzEn2D`, `XMSEn`, `MSEn`

References
 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]

`XSampEn`, `XFuzzEn`, `XApEn`, `K2En`, `XMSEn`, `XDistEn`

References
 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).

`PermEn`, `XApEn`, `XSampEn`, `XFuzzEn`, `XMSEn`

References
 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]

`XFuzzEn`, `XApEn`, `SampEn`, `SampEn2D`, `XMSEn`, `ApEn`

References
 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]

`SpecEn`, `fft`, `XDistEn`, `periodogram`, `XSampEn`, `XApEn`