# Bidimensional Entropies

## Functions for estimating the entropy of a two-dimensional univariate matrix.

While EntropyHub functions primarily apply to time series data, with the following bidimensional entropy functions one can estimate the entropy of two-dimensional (2D) matrices. Hence, bidimensional entropy functions are useful for applications such as image/texture analysis.

Danger

`IMPORTANT: Locked Matrix Size`

Each bidimensional entropy function (`SampEn2D`, `FuzzEn2D`, `DistEn2D`, `DispEn2D`, `PermEn2D`, `EspEn2D`) has an important keyword argument - `Lock`. Bidimensional entropy functions are “locked” by default (`Lock == true`) to only permit matrices with a maximum size of 128 x 128.

The reason for this is because there are hundreds of millions of pairwise calculations performed in the estimation of bidimensional entropy, so memory errors often occur when storing data on RAM.

e.g. For a matrix of size [200 x 200], an embedding dimension (`m`) = 3, and a time delay (`tau`) = 1, there are 753,049,836 pairwise matrix comparisons (6,777,448,524 elemental subtractions). To pass matrices with sizes greater than [128 x 128], set Lock = false.

`CAUTION: unlocking the permitted matrix size may cause your MatLab application to crash.`

SampEn2D(Mat, varargin)

SampEn2D estimates the bidimensional sample entropy of a data matrix.

[SE2D, Phi1, Phi2] = SampEn2D(Mat)

Returns the bidimensional sample entropy estimate (`SE2D`) and the number of matched sub-matricess (`m: Phi1`, `m+1: Phi2`) estimated for the data matrix (`Mat`) using the default parameters: time delay = 1, radius distance threshold = 0.2*SD(`Mat`), logarithm = natural matrix template size = [floor(H/10) floor(W/10)] (where H and W represent the height (rows) and width (columns) of the data matrix `Mat`) ** The minimum number of rows and columns of `Mat` must be > 10.

[SE2D, Phi1, Phi2] = SampEn2D(Mat, name, value, …)

Returns the bidimensional sample entropy (`SE2D`) estimates for the data matrix (`Mat`) using the specified name/value pair arguments:

• `m` - Template submatrix dimensions, an integer scalar (for sub- matrix with same height and width) or a two-element vector of integers [height, width] with a minimum value > 1. (default: [floor(H/10) floor(W/10)])

• `tau` - Time Delay, a positive integer (default: 1)

• `r` - Distance Threshold, a positive scalar (default: 0.2*SD(`Mat`))

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

• `Lock` - By default, `SampEn2D` only permits matrices with a maximum size of 128 x 128 to prevent RAM overload. e.g. For `Mat` = [200 x 200], `m = 3`, and `tau = 1`, `SampEn2D` creates a vector of 753049836 elements. To enable matrices greater than [128 x 128] elements, set `'Lock' = false`. (default: true)

WARNING: unlocking the permitted matrix size may cause memory errors that could lead Matlab to crash.

SampEn, FuzzEn2D, DistEn2D, XSampEn, MSEn.

References:
 Luiz Eduardo Virgili Silva, et al.,

“Two-dimensional sample entropy: Assessing image texture through irregularity.” Biomedical Physics & Engineering Express 2.4 (2016): 045002.

FuzzEn2D(Mat, varargin)

FuzzEn2D estimates the bidimensional fuzzy entropy of a data matrix.

[Fuzz2D] = FuzzEn2D(Mat)

Returns the bidimensional fuzzy entropy estimate (`Fuzz2D`) estimated for the data matrix (`Mat`) using the default parameters: time delay = 1, fuzzy function (`Fx`) = `'default'`, fuzzy function parameters (`r`) = [0.2,2], logarithm = natural, template matrix size = [floor(H/10) floor(W/10)] (where H and W represent the height (rows) and width (columns) of the data matrix `Mat`) ** The minimum number of rows and columns of `Mat` must be > 10.

[Fuzz2D] = FuzzEn2D(Mat, name, value, …)

Returns the bidimensional fuzzy entropy (`Fuzz2D`) estimates for the data matrix (`Mat`) using the specified name/value pair arguments:

• `m` - Template submatrix dimensions, an integer scalar (for sub- matrix with same height and width) or a two-element vector of integers [height, width] with a minimum value > 1. (default: [floor(H/10) floor(W/10)])

• `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]

• `Lock` - By default, `FuzzEn2D` only permits matrices with a maximum size of 128 x 128 to prevent RAM overload. e.g. For `Mat` = [200 x 200], `m = 3`, and `tau = 1`, `FuzzEn2D` creates a vector of 753049836 elements. To enable matrices greater than [128 x 128] elements, set `'Lock' = false`. (default: true)

WARNING: unlocking the permitted matrix size may cause memory errors that could lead Matlab to crash.

SampEn2D, DistEn2D, FuzzEn, XFuzzEn, XMSEn

References:
 Luiz Fernando Segato Dos Santos, et al.,

“Multidimensional and fuzzy sample entropy (SampEnMF) for quantifying H&E histological images of colorectal cancer.” Computers in biology and medicine 103 (2018): 148-160.

 Mirvana Hilal and Anne Humeau-Heurtier,

“Bidimensional fuzzy entropy: Principle analysis and biomedical applications.” 41st Annual International Conference of the IEEE (EMBC) Society 2019.

DistEn2D(Mat, varargin)

DistEn2D estimates the bidimensional distribution entropy of a data matrix.

[Dist2D] = DistEn2D(Mat)

Returns the bidimensional distribution entropy estimate (`Dist2D`) estimated for the data matrix (`Mat`) using the default parameters: time delay = 1, binning method = `'sturges'`, logarithm = natural, matrix template size = [floor(H/10) floor(W/10)] (where H and W represent the height (rows) and width (columns) of the data matrix `Mat`) * The minimum number of rows and columns of `Mat` must be > 10.

[Dist2D] = DistEn2D(Mat, name, value, …)

Returns the bidimensional distribution entropy (`Dist2D`) estimate for the data matrix (`Mat`) using the specified name/value pair arguments:

• `m` - Template submatrix dimensions, an integer scalar (for sub- matrix with same height and width) or a two-element vector of integers [height, width] with a minimum value > 1. (default: [floor(H/10) floor(W/10)])

• `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: natural)

• `Norm` - Normalisation of `Dist2D` value, one of the following integers:
•  no normalisation.

•  normalises values of data matrix (`Mat`) to range [0 1].

•  normalises values of data matrix (`Mat`) to range [0 1], and normalises the distribution entropy value (`Dist2D`) w.r.t the number of histogram bins. [default]

•  normalises the bidimensional distribution entropy value (`Dist2D`) w.r.t the number of histogram bins, without normalising data matrix values.

• `Lock` - By default, `DistEn2D` only permits matrices with a maximum size of 128 x 128 to prevent RAM overload. e.g. For `Mat` = [200 x 200], `m = 3`, and `tau = 1`, `DistEn2D` creates a vector of 753049836 elements. To enable matrices greater than [128 x 128] elements, set `'Lock' = false`. (default: true)

WARNING: unlocking the permitted matrix size may cause memory errors that could lead Matlab to crash.

DistEn, XDistEn, SampEn2D, FuzzEn2D, MSEn

References:
 Hamed Azami, Javier Escudero and Anne Humeau-Heurtier,

“Bidimensional distribution entropy to analyze the irregularity of small-sized textures.” IEEE Signal Processing Letters 24.9 (2017): 1338-1342.

DispEn2D(Mat, varargin)

DispEn2D estimates the bidimensional dispersion entropy of a data matrix.

[Disp2D, RDE] = DispEn2D(Mat)

Returns the bidimensional dispersion entropy estimate (`Disp2D`) and reverse bidimensional dispersion entropy (`RDE`) estimated for the data matrix (`Mat`) using the default parameters: time delay = 1, symbols = 3, logarithm = natural, data transform = normalised cumulative density function (ncdf) matrix template size = [floor(H/10) floor(W/10)] (where H and W represent the height (rows) and width (columns) of the data matrix `Mat`) * The minimum number of rows and columns of Mat must be > 10.

[Disp2D, RDE] = DispEn2D(Mat, name, value, …)

Returns the bidimensional dispersion entropy (`Disp2D`) estimate and reverse bidimensional dispersion entropy (`RDE`) for the data matrix (`Mat`) using the specified name/value pair arguments:

• `m` - Template submatrix dimensions, an integer scalar (for submatrix with same height and width) or a two-element vector of integers [height, width] with a minimum value > 1. (default: [floor(H/10) floor(W/10)])

• `tau` - Time Delay, a positive integer (default: 1)

• `c` - Number of symbols, an integer > 1

• `Typex` - Type of symbolic mapping transform, one of the following: {`linear`, `kmeans`, `ncdf`, `equal`} See the EntropyHub Guide for more info on these transforms.

• `Logx` - Logarithm base, a positive scalar

• `Norm` - Normalisation of `Disp2D` and `RDE` values, a boolean:
• [false] no normalisation - default

• [true] normalises w.r.t number of possible dispersion patterns (`c^m`).

• `Lock` - By default, `DispEn2D` only permits matrices with a maximum size of 128 x 128 to prevent RAM overload. e.g. For `Mat` = [200 x 200], `m = 3`, and `tau = 1`, `DispEn2D` creates a vector of 753049836 elements. To enable matrices greater than [128 x 128] elements, set `'Lock' = false`. (default: true)

WARNING: unlocking the permitted matrix size may cause memory errors that could lead Matlab to crash.

DispEn, SampEn2D, FuzzEn2D, DistEn2D

References:
 Hamed Azami, et al.,

“Two-dimensional dispersion entropy: An information-theoretic method for irregularity analysis of images.” Signal Processing: Image Communication, 75 (2019): 178-187.

PermEn2D(Mat, varargin)

m, tau, Logx PermEn2D estimates the bidimensional permutation entropy of a data matrix.

[Perm2D] = PermEn2D(Mat)

Returns the bidimensional permutation entropy estimate (`Perm2D`) estimated for the data matrix (`Mat`) using the default parameters: time delay = 1, logarithm = natural, template matrix size = [floor(H/10) floor(W/10)] (where H and W represent the height (rows) and width (columns) of the data matrix `Mat`) ** The minimum number of rows and columns of `Mat` must be > 10.

[Perm2D] = PermEn2D(Mat, name, value, …)

Returns the bidimensional permutation entropy (`Perm2D`) estimates for the data matrix (`Mat`) using the specified name/value pair arguments:

• `m` - Template submatrix dimensions, an integer scalar (for sub- matrix with same height and width) or a two-element vector of integers [height, width] with a minimum value > 1. (default: [floor(H/10) floor(W/10)])

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

• `Norm` - Normalization of the PermEn2D value by maximum Shannon entropy (Smax = log((mx*my)!) [default: true]

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

• `Lock` - By default, `PermEn2D` only permits matrices with a maximum size of 128 x 128 to prevent RAM overload. e.g. For `Mat` = [200 x 200], `m = 3`, and `tau = 1`, `PermEn2D` creates a vector of 753049836 elements. To enable matrices greater than [128 x 128] elements, set `'Lock' = false`. (default: true)

WARNING: unlocking the permitted matrix size may cause memory errors that could lead Matlab to crash.

NOTE

The original bidimensional permutation entropy algorithms  do not account for equal-valued elements of the embedding matrices. To overcome this, PermEn2D uses the lowest common rank for such instances. For example, given an embedding matrix A where, A =

[3.4 5.5 7.3] [2.1 6 9.9] [7.3 1.1 2.1]

would normally be mapped to an ordinal pattern like so,

[3.4 5.5 7.3 2.1 6 9.9 7.3 1.1 2.1] => [ 8 4 9 1 2 5 3 7 6 ]

However, indices 4 & 9, and 3 & 7 have the same values, 2.1 and 7.3 respectively. Instead, PermEn2D uses the ordinal pattern [ 8 4 4 1 2 5 3 3 6 ] where the lowest rank (4 & 3) are used instead (of 9 & 7). Therefore, the number of possible permutations is no longer (mx*my)!, but (mx*my)^(mx*my). Here, the PermEn2D value is normalized by the maximum Shannon entropy (Smax = log((mx*my)!) ```assuming that no equal values are found in the permutation motif matrices```, as presented in .

SampEn2D, DistEn2D, FuzzEn2D, DispEn2D, PermEn

References:
 Haroldo Ribeiro et al.,

“Complexity-Entropy Causality Plane as a Complexity Measure for Two-Dimensional Patterns” PLoS ONE (2012), 7(8):e40689,

 Luciano Zunino and Haroldo Ribeiro,

“Discriminating image textures with the multiscale two-dimensional complexity-entropy causality plane” Chaos, Solitons and Fractals, 91:679-688 (2016)

 Matthew Flood and Bernd Grimm,

“EntropyHub: An Open-source Toolkit for Entropic Time Series Analysis” PLoS ONE (2021) 16(11): e0259448.

EspEn2D(Mat, varargin)

EspEn2D estimates the bidimensional Espinosa entropy of a data matrix.

[Esp2D] = EspEn2D(Mat)

Returns the bidimensional Espinosa entropy estimate (`Esp2D`) estimated for the data matrix (`Mat`) using the default parameters: time delay = 1, tolerance threshold = 20, percentage similarity = 0.7 logarithm = natural, matrix template size = [floor(H/10) floor(W/10)] (where H and W represent the height (rows) and width (columns) of the data matrix `Mat`) ** The minimum number of rows and columns of `Mat` must be > 10.

[Esp2D] = EspEn2D(Mat, name, value, …)

Returns the bidimensional Espinosa entropy (`Esp2D`) estimates for the data matrix (`Mat`) using the specified name/value pair arguments:

• `m` - Template submatrix dimensions, an integer scalar (for sub- matrix with same height and width) or a two-element vector of integers [height, width] with a minimum value > 1. (default: [floor(H/10) floor(W/10)])

• `tau` - Time Delay, a positive integer (default: 1)

• `r` - Tolerance Threshold, a positive scalar (default: 20)

• `ps` - Percentage similarity, a scalar in range [0 1] (default: 0.7)

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

• `Lock` - By default, `SampEn2D` only permits matrices with a maximum size of 128 x 128 to prevent RAM overload. e.g. For `Mat` = [200 x 200], `m = 3`, and `tau = 1`, `EspEn2D` creates a vector of 753049836 elements. To enable matrices greater than [128 x 128] elements, set `'Lock' = false`. (default: true)

WARNING: unlocking the permitted matrix size may cause memory errors that could lead Matlab to crash.