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Test the ShannonEntropy procedure.

Don Pellegrino
May 7, 2008

Maple Equation

Examine the Shannon Entropy of a String

The Shannon Entropy of the string "aabc" should be 1.5.

The manual calculation should also be 1.5.

Maple Equation

Calculate the Shannon Entropy of a Matrix

A matrix is treated like a string. The matrix is a container of elements analogous to the way that a string is a container of characters. A matrix with the same contents as a string should therefore have the same entropy value.

We see that StringTools:-Entropy cannot operate on a matrix as the input must be a string.

Error, (in StringTools:-Entropy) argument must be a string

Maple Equation

Therefore, the custom ShannonEntropy procedure is used.

Maple Equation

Faster, Simpler

This section is an exploration of the faster, simpler version of the algorithm as presented by acer on MaplePrimes:

(http://www.mapleprimes.com/blog/optimusmapleprime/calculateshannonentropymatrix).

ImageTools:-Entropy

Joe Riel on MaplePrimes points out that the ImageTools:-Entropy procedure can be used on a matrix by way of the ImageTools:-Create procedure:

(http://www.mapleprimes.com/blog/optimusmapleprime/calculateshannonentropymatrix).

It seems that this procedure will not work when the matrix holds characters.

Error, (in ImageTools:-Create) unable to store '"a"' when datatype=float[8]

Maple Equation

A new matrix composed of floats is used to test the ImagesTools:-Entropy procedure.

It seems that the values in the matrix must be within 0..1.0 for the ImageTools package to work.

Normalization of the matrix might be sufficient to use ImageTools:-Entropy on numeric data however the LinearAlgebra:-Normalize procedure wants a vector rather than a matrix. The entropy calculation as implemented here treats the matrix as a generic container. Shannon's Entropy calculation treats the input as symmetric; the order of the elements does not effect the result. Therefore it should be possible to restructure the elements into a list or vector and then use the ImageTools procedures. ImageTools:-Create then wants a matrix rather than a vector so another conversion is needed. A possible algorithm would be:

1. Convert the matrix to a vector.

2. Normalize the vector with LinearAlgebra:-Normalize.

3. Convert the normalized vector to a matrix.

4. Run the ImageTools:-Create procedure on the matrix.

5. Run the ImageTools:-Entropy procedure on the image.

1. Convert the matrix to a vector.

2. Normalize the vector.

3. Convert the normalized vector to a matrix.

4. Run the ImageTools:-Create procedure on the matrix.

5. Run the ImageTools:-Entropy procedure on the image.

So this algorithm seems to work for processing a matrix of numeric data, however it will not work when the matrix contains character or string data. Could this algorithm be adapted so that ImageTools:-Entropy or StringTools:-Entropy could be used on a matrix containing those datatypes as well? How about a matrix with both numeric and string columns?

Maple Equation

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<center ><h1 >Test the ShannonEntropy procedure.</h1></center> <center >Don Pellegrino May 7, 2008</center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <h1 >Examine the Shannon Entropy of a String</h1> The Shannon Entropy of the string "aabc" should be 1.5. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> The manual calculation should also be 1.5. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <h1 >Calculate the Shannon Entropy of a Matrix</h1> A matrix is treated like a string. The matrix is a container of elements analogous to the way that a string is a container of characters. A matrix with the same contents as a string should therefore have the same entropy value. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> We see that StringTools:-Entropy cannot operate on a matrix as the input must be a string. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> Error, (in StringTools:-Entropy) argument must be a string <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> Therefore, the custom ShannonEntropy procedure is used. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <h1 >Faster, Simpler</h1> This section is an exploration of the faster, simpler version of the algorithm as presented by acer on MaplePrimes: (http://www.mapleprimes.com/blog/optimusmapleprime/calculateshannonentropymatrix). <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <h1 >ImageTools:-Entropy</h1> Joe Riel on MaplePrimes points out that the ImageTools:-Entropy procedure can be used on a matrix by way of the ImageTools:-Create procedure: (http://www.mapleprimes.com/blog/optimusmapleprime/calculateshannonentropymatrix). It seems that this procedure will not work when the matrix holds characters. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> Error, (in ImageTools:-Create) unable to store '"a"' when datatype=float[8] <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> A new matrix composed of floats is used to test the ImagesTools:-Entropy procedure. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> It seems that the values in the matrix must be within 0..1.0 for the ImageTools package to work. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> Normalization of the matrix might be sufficient to use ImageTools:-Entropy on numeric data however the LinearAlgebra:-Normalize procedure wants a vector rather than a matrix. The entropy calculation as implemented here treats the matrix as a generic container. Shannon's Entropy calculation treats the input as symmetric; the order of the elements does not effect the result. Therefore it should be possible to restructure the elements into a list or vector and then use the ImageTools procedures. ImageTools:-Create then wants a matrix rather than a vector so another conversion is needed. A possible algorithm would be: 1. Convert the matrix to a vector. 2. Normalize the vector with LinearAlgebra:-Normalize. 3. Convert the normalized vector to a matrix. 4. Run the ImageTools:-Create procedure on the matrix. 5. Run the ImageTools:-Entropy procedure on the image. 1. Convert the matrix to a vector. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> 2. Normalize the vector. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> 3. Convert the normalized vector to a matrix. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> 4. Run the ImageTools:-Create procedure on the matrix. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> 5. Run the ImageTools:-Entropy procedure on the image. <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> <center ><img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /></center> So this algorithm seems to work for processing a matrix of numeric data, however it will not work when the matrix contains character or string data. Could this algorithm be adapted so that ImageTools:-Entropy or StringTools:-Entropy could be used on a matrix containing those datatypes as well? How about a matrix with both numeric and string columns? <img src="http://mapleoracles.maplesoft.com" alt='Maple Equation' align='absMiddle' /> This post was generated using the MaplePrimes <a href="http://www.mapleprimes.com/filemanager">File Manager </a> <a href='http://mapleoracles.maplesoft.com:8080/maplenet/primes/worksheet/8408_ShannonEntropy-RegressionTest.mw'>View 8408_ShannonEntropy-RegressionTest.mw on MapleNet</a> or <a href='http://www.mapleprimes.com/files/8408_ShannonEntropy-RegressionTest.mw'>Download 8408_ShannonEntropy-RegressionTest.mw</a> <a href='http://www.mapleprimes.com/viewfile/2548'>View file details</a>

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