wentropy

Wavelet entropy

Syntax

ent = wentropy(X)

ent = wentropy(X,Name=Value)

[ent,re] = wentropy(___)

Description

ent = wentropy(X) returns the normalized Shannon wavelet entropy of X.

ent = wentropy(X,Name=Value) specifies options using one or more name-value arguments.

[ent,re] = wentropy(___) also returns the wavelet relative energies.

Examples

Obtain Wavelet Entropy

Shannon Entropy

Create a signal whose samples are alternating values of 0 and 2.

n = 0:499;
x = 1+(-1).^n;
stem(x)
axis tight
title("Signal")
xlim([0 50])

Obtain the scaled Shannon entropy of the signal. Specify a one-level wavelet transform, use the default wavelet and wavelet transform.

ent = wentropy(x,Level=1);
ent

ent = 2×1

    1.0000
    1.0000

Obtain the unscaled Shannon entropy. Divide the entropy by log(n), where n is the length of the signal. Confirm the result equals the scaled entropy.

ent2 = wentropy(x,Level=1,Scaled=false);
ent2/log(length(x))

ans = 2×1

    1.0000
    1.0000

Create a zero-mean signal from the first signal. Obtain the scaled Shannon entropy of the new signal using a one-level wavelet transform.

x = x-1;
ent = wentropy(x,Level=1);
ent

ent = 2×1

    1.0000
         0

Renyi Entropy

Load the Kobe earthquake data. Obtain the level 4 tunable Q-factor wavelet transform of the data with a quality factor equal to 2.

load kobe
wt = tqwt(kobe,Level=4,QualityFactor=2);

Obtain the Renyi entropy estimates for the tunable Q-factor transform.

ent = wentropy(wt,Entropy="Renyi");
ent

ent = 5×1

    0.8288
    0.8506
    0.8582
    0.8536
    0.7300

Load the ECG data. Obtain the level 5 discrete wavelet transform of the signal using the "db4" wavelet.

load wecg
wv = "db4";
[C,L] = wavedec(wecg,5,wv);

Package the wavelet and approximation coefficients into a cell array suitable for computing the wavelet entropy.

X = detcoef(C,L,"cells");
X{end+1} = appcoef(C,L,wv);

Obtain the Renyi entropy by scale.

ent = wentropy(X,Entropy="Renyi");
ent

ent = 6×1

    0.2412
    0.5239
    0.5459
    0.6520
    0.7661
    0.8547

Tsallis Entropy

Create a Kronecker delta sequence.

N = 512;
seq = zeros(1,N);
seq(N/2) = 1;

Obtain the scaled Shannon entropy of the signal. Specify a level 3 wavelet transform.

ShannonEntropy = wentropy(seq,Level=3);

Obtain the scaled Tsallis entropy of the signal for different values of exponents. Confirm that as the exponent goes to 1, the Tsallis entropy approaches the Shannon entropy.

exps = 3:-1/4:1;
TsallisExponent = zeros(length(exps),1);
TsallisEntropy = zeros(length(exps),4);
ctr = 1;
for k=exps
    ent2 = wentropy(seq,Level=3,Entropy="Tsallis",Exponent=k);
    TsallisExponent(ctr) = k;
    TsallisEntropy(ctr,:) = ent2';
    ctr = ctr+1;
end
TsallisTable = table(TsallisExponent,TsallisEntropy)

TsallisTable=9×2 table
    TsallisExponent                 TsallisEntropy             
    _______________    ________________________________________

            3          0.71454    0.87888    0.97069    0.98285
         2.75          0.67651    0.84955    0.95685    0.97233
          2.5          0.63178    0.81187    0.93596     0.9552
         2.25          0.57852     0.7628    0.90407    0.92718
            2          0.51437    0.69812    0.85499    0.88149
         1.75          0.43679    0.61258    0.77985    0.80825
          1.5          0.34491    0.50213    0.66897    0.69658
         1.25          0.24402    0.37071    0.52076    0.54417
            1           0.1495    0.23839      0.356    0.37278

ShannonEntropy'

ans = 1×4

    0.1495    0.2384    0.3560    0.3728

Input Arguments

X — Input data

real-valued vector | real-valued matrix | cell array

Input data, specified as a real-valued row or column vector, a cell array of real-valued row or column vectors, or a real-valued matrix with at least two rows.

• If X is a row or column vector, X must have at least four samples, and the function assumes X represents time data.

• If X is a cell array, the function assumes X to be a decimated wavelet or wavelet packet transform of a real-valued row or column vector.

• If X is a matrix with at least two rows, the function assumes X to be the maximal overlap discrete wavelet or wavelet packet transform of a real-valued row or column vector.

Example: ent = wentropy(randn(1,1024)) returns the normalized Shannon wavelet entropy. wentropy computes the wavelet coefficients using the default options of modwt.

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: ent = wentropy(X,Wavelet="coif4") uses the "coif4" wavelet to obtain the wavelet transform.

Entropy — Entropy

"Shannon" (default) | "Renyi" | "Tsallis"

Entropy returned by wentropy, specified as one of "Shannon", "Renyi", and "Tsallis". For more information, see Wavelet Entropy.

Exponent — Exponent

2 (default) | real scalar

Exponent to use in the Renyi and Tsallis entropy, specified as a real scalar.

• For the Renyi entropy, the exponent must be nonnegative.

• For the Tsallis entropy, the exponent must be greater than or equal to –1/2.

• For the Renyi and Tsallis entropies, specifying Exponent=1 is a limiting case and produces the Shannon entropy.

Specifying Exponent is valid only when Entropy is "Renyi" or "Tsallis".

Note

When you specify a negative exponent for the Tsallis entropy, entropy computations may become unstable with small changes in the wavelet coefficient energies, resulting in significant changes in the entropy values.

Data Types: single | double

Transform — Transform

"modwt" (default) | "dwt" | "dwpt" | "modwpt"

Transform used to obtain the wavelet or wavelet packet coefficients for the real-valued row or column vector X, specified as one of these:

• "dwt" — Discrete wavelet transform

• "dwpt" — Discrete wavelet packet transform

• "modwt" — Maximal overlap discrete wavelet transform

• "modwpt" — Maximal overlap discrete wavelet packet transform

The default wavelet depends on the value of Transform.

• If Transform is "dwt" or "modwt", wentropy uses the "sym4" wavelet.

• If Transform is "dwpt" or "modwpt", wentropy uses the "fk18" wavelet.

Periodic extension is used for all transforms.

Level — Wavelet decomposition level

positive integer

Wavelet decomposition level if the input X is time data, specified as a positive integer. The wentropy function obtains the wavelet transform down to the specified level. If unspecified, the default level depends on the type of transform and the signal length N.

• If Transform is "dwt" or "modwt", Level defaults to floor(log2(N))-1.

• If Transform is "dwpt" or "modwpt", Level defaults to min(4,floor(log2(N))-1).

Specifying a level is invalid if the input data are wavelet or wavelet packet coefficients.

Data Types: single | double

Wavelet — Wavelet

character vector | string scalar

Wavelet used to obtain the wavelet or wavelet packet transform of a real-valued row or column vector, specified as a character vector or string scalar. If Transform is "modwt" or "modwpt", the wavelet must be orthogonal. For a list of supported orthogonal or biorthogonal wavelets, see wfilters.

Specifying a wavelet name is invalid if the input data are wavelet or wavelet packet coefficients.

Data Types: char | string

Distribution — Normalization method

"scale" (default) | "global"

Normalization method to use to obtain the empirical probability distribution for the wavelet transform coefficients, specified as "scale" or "global".

• "global" — The function normalizes the squared magnitudes of the coefficients by the total sum of squared magnitudes of all coefficients. Each scale in the wavelet transform yields a scalar and the vector of these values forms a probability vector. The function performs entropy calculations on this vector and the overall entropy is a scalar.

• "scale" — The function normalizes the wavelet coefficients at each scale separately and calculates the entropy by scale.

  • If the input is time series data, the output ent is of size (Ns+1)-by-1, where Ns is the number of scales.

  • If the input is a cell array or matrix, ent is of size M-by-1, where M is the length of the cell array or number of rows in the matrix.

Scaled — Scale wavelet entropy

true or 1 (default) | false or 0

Scale wavelet entropy logical, specified as a numeric or logical 1 (true) or 0 (false). If specified as true, the wentropy function scales the wavelet entropy by the factor corresponding to a uniform distribution for the specified entropy.

• For the Shannon and Renyi entropies, the factor is 1/log(Nj), where Nj is the length of the data in samples by scale if Distribution is "scale", or the number of scales if Distribution is "global".

• For the Tsallis entropy, the factor is (Exponent-1)/(1-Nj^(1-Exponent)).

Setting Scaled=false does not scale the wavelet entropy.

Data Types: logical

EnergyThreshold — Energy threshold

1e-8 (default) | nonnegative scalar

Energy threshold, specified as a nonnegative scalar. The function replaces all coefficients with energy by scale below EnergyThreshold with 0. A positive EnergyThreshold prevents the function from treating wavelet or wavelet packet coefficients with nonsignificant energy as a sequence with high entropy.

Data Types: single | double

Output Arguments

ent — Entropy

scalar | vector

Entropy of X, returned as a scalar or vector.

• If X is time data, ent is a real-valued (Ns+1)-by-1 vector of entropy estimates by scale, where Ns is the number of scales.

• If X is a wavelet or wavelet packet transform input, ent is a real-valued column vector with length equal to the length of X if X is a cell array or the row dimension of X if X is a matrix.

See Distribution to obtain global estimates of the wavelet entropy. The wentropy function uses the natural logarithm to compute the entropy.

Data Types: single | double

re — Relative wavelet energy

vector | matrix

Relative wavelet energy, returned as a vector or matrix.

• If Distribution="scale", the function returns the relative wavelet energies by coefficient and scale.

• If Distribution="global", the function returns the relative wavelet energies by scale.

Scales where the coefficient energy is below the value of EnergyThreshold are equal to 0.

Data Types: single | double

More About

Wavelet Entropy

Wavelet entropy (WE) is often used to analyze nonstationary signals. WE combines a wavelet or wavelet decomposition with a measure of order within the wavelet coefficients by scale. These measures of order are referred to as entropy measures. WE treats the normalized wavelet coefficients as an empirical probability distribution and calculates its entropy.

You can normalize the wavelet coefficients wt in one of two ways.

• The function normalizes all the coefficients by the total sum of their squared magnitudes: $$E={\displaystyle \sum _i{\displaystyle \sum _j|}}w{t}_{ij}{|}^2,$$where j corresponds to time, and i corresponds to scale. The probability mass function is: $$\mathbb{P}\left(w{t}_{ij}\right)=|w{t}_{ij}{|}^2/E.$$

• The function normalizes the coefficients at each scale separately by the sum of their squared magnitudes: $$E_i={\displaystyle \sum _j|}w{t}_{ij}{|}^2.$$ The probability mass function is: $$\mathbb{P}\left(w{t}_{ij}\right)=|w{t}_{ij}{|}^2/E_i.$$

The wentropy function supports three entropy measures.

• Shannon Entropy

  For a discrete random variable X, the Shannon entropy is defined as:

  $$H(X)=-{\displaystyle \sum _i\mathbb{P}}(X=x_i)\ln (\mathbb{P}(X=x_i)),$$

  where the sum is taken over all values that the random variable can take. By convention, 0 ln(0) = 0.

• Renyi Entropy

  The Renyi entropy is defined as:

  $$H_r(X)=\frac{1}{1-\alpha }\text{ln}\left({\displaystyle \sum _i{\left(\mathbb{P}(X=x_i)\right)}^{\alpha }}\right),\alpha \ge \text{ 0}\text{.}$$

  In the limit, the Renyi entropy becomes the Shannon entropy: $$\underset{\alpha \to 1}{\lim }{H}_r(X)=H(X).$$

• Tsallis Entropy

  The Tsallis entropy is defined as:

  $$H_t(X)=\frac{1}{q-1}\left(1-{\displaystyle \sum _i{\left(\mathbb{P}(X=x_i)\right)}^q}\right),q\in ℝ,q\ne 1.$$

  Similar to the Renyi entropy, in the limit, the Tsallis entropy becomes the Shannon entropy: $$\underset{q\to 1}{\lim }{H}_t(X)=H(X).$$

References

[1] Zunino, L., D.G. Pérez, M. Garavaglia, and O.A. Rosso. “Wavelet Entropy of Stochastic Processes.” Physica A: Statistical Mechanics and Its Applications 379, no. 2 (June 2007): 503–12. https://doi.org/10.1016/j.physa.2006.12.057.

[2] Rosso, Osvaldo A., Susana Blanco, Juliana Yordanova, Vasil Kolev, Alejandra Figliola, Martin Schürmann, and Erol Başar. “Wavelet Entropy: A New Tool for Analysis of Short Duration Brain Electrical Signals.” Journal of Neuroscience Methods 105, no. 1 (January 2001): 65–75. https://doi.org/10.1016/S0165-0270(00)00356-3.

[3] Alcaraz, Raúl, ed. "Wavelet Entropy: Computation and Applications." Special issue, Entropy 17 (2015). https://www.mdpi.com/journal/entropy/special_issues/wavelet-entropy.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The values of the Wavelet, Transform, and Distribution name-value arguments must be constant at compile time. Use coder.Constant (MATLAB Coder).

• Vector inputs must have one dimension fixed at 1 at compile time. For example, to allow for row vector input with unbounded size, specify the first input argument at compile time as {coder.typeof(0,[1 Inf],[0 1]])}. For more information, see coder.typeof (MATLAB Coder).

• When you compile with variable-size dimensions for both row and column input, the generated code expects matrix input. For example, if you specify the first input argument at compile time as {coder.typeof(0,[1 Inf],[1 1])}, the generated code errors for row vector input.

• The syntax used in the old version of the wentropy function is not supported. For more information, see Version History.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• The "dwpt" transform is not supported.

• The syntax used in the old version of the wentropy function is not supported. For more information, see Version History

Version History

Introduced before R2006a

R2023a: wentropy supports C/C++ code generation and gpuArray objects

The wentropy function supports:

• C/C++ code generation. You must have MATLAB^® Coder™ to generate C/C++ code.

• gpuArray object inputs. You must have Parallel Computing Toolbox™ to use gpuArray objects.

R2022b: wentropy input syntax has changed

The syntax used in the old version of wentropy continues to work, but is no longer recommended. The old version provides you minimal control over how to estimate the entropy. The wentropy function automatically determines from the input syntax which version to use.

You can specify the Shannon entropy in both versions of wentropy. However, because the old version makes no assumptions about the input data, reproducing the same results as the new version can require extensive effort.

Old Version	New Version
load wecg n = numel(wecg); lev = 3; wt = modwt(wecg,lev); energy = sum(abs(wt).^2,2); wt2 = abs(wt)./sqrt(energy); ent = zeros(lev+1,1); for k=1:lev+1 ent(k) = wentropy(wt2(k,:),'shannon')/log(n); end ent ent = 0.3925 0.6512 0.6985 0.9329	load wecg ent = wentropy(wecg,Level=3) ent = 0.3925 0.6512 0.6985 0.9329