# 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 analysis.

IMPORTANT: Locked Matrix Size

Each bidimensional entropy function (SampEn2D, FuzzEn2D, DistEn2D, Disp2DEn) 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 Julia IDE to crash.

EntropyHub._SampEn2D.SampEn2DFunction
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 dimension size of Mat must be > 10.**

SE2D, Phi1, Phi2 = SampEn2D(Mat::AbstractArray{T,2} where T<:Real; m::Union{Int,Tuple{Int,Int}}=floor.(Int, size(Mat)./10),
tau::Int=1, r::Real=0.2*std(Mat,corrected=false), Logx::Real=exp(1), Lock::Bool=true)

Returns the bidimensional sample entropy (SE2D) estimates for the data matrix (Mat) using the specified 'keyword' arguments:

Arguments:

m - Template submatrix dimensions, an integer scalar (i.e. the 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 memory errors when storing data on RAM. 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 to false. (default: true)

      WARNING: unlocking the permitted matrix size may cause your Julia
IDE to crash.

See also SampEn, FuzzEn2D, XSampEn, MSEn

References:

[1] Luiz Eduardo Virgili Silva, et al.,
"Two-dimensional sample entropy: Assessing image texture
through irregularity."
Biomedical Physics & Engineering Express
2.4 (2016): 045002.
EntropyHub._FuzzEn2D.FuzzEn2DFunction
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 and width of the data matrix 'Mat')

** The minimum dimension size of Mat must be > 10.**

Fuzz2D = FuzzEn2D(Mat::AbstractArray{T,2} where T<:Real; m::Union{Int,Tuple{Int,Int}}=floor.(Int, size(Mat)./10),
tau::Int=1, r::Union{Real,Tuple{Real,Real}}=(.2*std(Mat, corrected=false),2),
Fx::String="default", Logx::Real=exp(1), Lock::Bool=true)

Returns the bidimensional fuzzy entropy (Fuzz2D) estimates for the data matrix (Mat) using the specified 'keyword' arguments:

Arguments:

m - Template submatrix dimensions, an integer scalar (i.e. the 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 funtion name, one of the following: {"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:

      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))
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 memory errors when storing data on RAM. 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 to false. (default: true)

       WARNING: unlocking the permitted matrix size may cause
your Julia IDE to crash.

See also SampEn2D, FuzzEn, XFuzzEn

References:

[1] 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.

[2] Mirvana Hilal and Anne Humeau-Heurtier,
"Bidimensional fuzzy entropy: Principle analysis and biomedical
applications."
41st Annual International Conference of the IEEE (EMBC) Society
2019.
EntropyHub._DistEn2D.DistEn2DFunction
Dist2D = DistEn2D(Mat)

Returns the bidimensional distribution entropy estimate (Dist2D) estimated for the data matrix (Mat) using the default parameters: time delay = 1, histogram binning method = "sturges", 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.**

Dist2D = DistEn2D(Mat::AbstractArray{T,2} where T<:Real; m::Union{Int,Tuple{Int,Int}}=floor.(Int, size(Mat)./10), tau::Int=1,
Bins::Union{Int,String}="Sturges", Logx::Real=2, Norm::Int=2, Lock::Bool=true)

Returns the bidimensional distribution entropy (Dist2D) estimate for the data matrix (Mat) using the specified 'keyword' arguments:

Arguments:

m - Template submatrix dimensions, an integer scalar (i.e. the same height and width) or a two-element tuple 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]

      ** enter 0 for natural logarithm.**

Norm - Normalisation of Dist2D value, one of the following integers: [0] no normalisation. [1] normalises values of data matrix (Mat) to range [0 1]. [2] 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] [3] normalises the distribution entropy value 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 memory errors when storing data on RAM. 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 to false. [default: 'true'] WARNING: unlocking the permitted matrix size may cause your Julia IDE to crash.

See also DistEn, XDistEn, SampEn2D, FuzzEn2D, MSEn

References:

[1] 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.
EntropyHub._DispEn2D.DispEn2DFunction
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'), 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.**

Disp2D, RDE = DispEn2D(Mat::AbstractArray{T,2} where T<:Real;
m::Union{Int,Tuple{Int,Int}}=floor.(Int, size(Mat)./10), tau::Int=1,
c::Int=3, Typex::String="ncdf", Logx::Real=exp(1), Norm::Bool=false, Lock::Bool=true)

Returns the bidimensional dispersion entropy (Disp2D) and reverse bidimensional distribution entropy (RDE) estimate for the data matrix (Mat) using the specified 'keyword' arguments:

Arguments:

m - Template submatrix dimensions, an integer scalar (i.e. the same height and width) or a two-element tuple 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 [default: natural]

      ** enter 0 for natural logarithm.**

Norm - Normalisation of Disp2D value, a boolean: - [false] no normalisation - default - [true] normalises w.r.t number of possible dispersion patterns. Lock - By default, DispEn2D only permits matrices with a maximum size of 128 x 128 to prevent memory errors when storing data on RAM. 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 to false. [default: 'true'] WARNING: unlocking the permitted matrix size may cause your Julia IDE to crash.

See also DispEn, DistEn2D, SampEn2D, FuzzEn2D, MSEn

References:

[1] 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.`