Problem 44933. Vogel-Dobbener entropy

Created by Christian Schröder

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Vogel-Dobbener entropy</w:t></w:r><w:r><w:t> is a measure of dispersion for ordinal variables.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Given an ordered list of distinct observations u_(1), ..., u_(k) with observed relative frequencies f_1, ..., f_k, the Vogel-Dobbener entropy is defined as</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ VD = -(sum_{i=1}^{k-1} (F_i log2(F_i) + (1 - F_i) log2(1 - F_i))]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>where F_1, ... F_k are the cumulative relative frequences, i.e.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ F_i = sum_{j=1}^i f_i]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The Vogel-Dobbener entropy of a sample satisfies 0 &lt;= VD &lt;= (k - 1); the normalized Vogel-Dobbener entropy is thus defined as</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ VD* = VD / (k - 1)]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>For example</w:t></w:r><w:r><w:t>, suppose that your sample is [2.7 3.3 2.0 3.3 1.7 3.7]. Then:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>k = 5 (there are five distinct observations);</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>u_(1), ..., u_(k) = [1.7 2.0 2.7 3.3 3.7]; (note that 3.3 only appears once);</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>f_1, ..., f_k = [1/6 1/6 1/6 2/6 1/6];</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>F_1, ..., F_k = [1/6 2/6 3/6 5/6 6/6];</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>VD = 3.2183; (approx.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>VD* = 0.8046. (approx.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Your task</w:t></w:r><w:r><w:t> is to write a function that, given a list of observations (unordered and possibly containing duplicates) computes the normalized Vogel-Dobbener entropy VD* of the sample.</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Round to four decimal digits.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Hint</w:t></w:r><w:r><w:t>: if all observations in the sample are the same, then k = 1, the sum in the definition of VD is empty, and VD equals zero.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

Vogel-Dobbener entropy is a measure of dispersion for ordinal variables.Given an ordered list of distinct observations u_(1), ..., u_(k) with observed relative frequencies f_1, ..., f_k, the Vogel-Dobbener entropy is defined as<pre> VD = -(sum_{i=1}^{k-1} (F_i log2(F_i) + (1 - F_i) log2(1 - F_i))</pre>where F_1, ... F_k are the cumulative relative frequences, i.e.<pre> F_i = sum_{j=1}^i f_i</pre>The Vogel-Dobbener entropy of a sample satisfies 0 &lt;= VD &lt;= (k - 1); the normalized Vogel-Dobbener entropy is thus defined as<pre> VD* = VD / (k - 1)</pre>For example, suppose that your sample is [2.7 3.3 2.0 3.3 1.7 3.7]. Then:<ul><li>k = 5 (there are five distinct observations);</li><li>u_(1), ..., u_(k) = [1.7 2.0 2.7 3.3 3.7]; (note that 3.3 only appears once);</li><li>f_1, ..., f_k = [1/6 1/6 1/6 2/6 1/6];</li><li>F_1, ..., F_k = [1/6 2/6 3/6 5/6 6/6];</li><li>VD = 3.2183; (approx.)</li><li>VD* = 0.8046. (approx.)</li></ul>Your task is to write a function that, given a list of observations (unordered and possibly containing duplicates) computes the normalized Vogel-Dobbener entropy VD* of the sample. Round to four decimal digits.Hint: if all observations in the sample are the same, then k = 1, the sum in the definition of VD is empty, and VD equals zero.

*Vogel-Dobbener entropy* is a measure of dispersion for ordinal variables.

Given an ordered list of distinct observations u_(1), ..., u_(k) with observed relative frequencies f_1, ..., f_k, the Vogel-Dobbener entropy is defined as

VD = -(sum_{i=1}^{k-1} (F_i log2(F_i) + (1 - F_i) log2(1 - F_i))

where F_1, ... F_k are the cumulative relative frequences, i.e.

F_i = sum_{j=1}^i f_i

The Vogel-Dobbener entropy of a sample satisfies 0 <= VD <= (k - 1); the normalized Vogel-Dobbener entropy is thus defined as

VD* = VD / (k - 1)

*For example*, suppose that your sample is [2.7 3.3 2.0 3.3 1.7 3.7]. Then:

* k = 5 (there are five distinct observations);
* u_(1), ..., u_(k) = [1.7 2.0 2.7 3.3 3.7]; (note that 3.3 only appears once);
* f_1, ..., f_k = [1/6 1/6 1/6 2/6 1/6];
* F_1, ..., F_k = [1/6 2/6 3/6 5/6 6/6];
* VD = 3.2183; (approx.)
* VD* = 0.8046. (approx.)

*Your task* is to write a function that, given a list of observations (unordered and possibly containing duplicates) computes the normalized Vogel-Dobbener entropy VD* of the sample. *Round to four decimal digits.*

*Hint*: if all observations in the sample are the same, then k = 1, the sum in the definition of VD is empty, and VD equals zero.

Solve

Solution Stats

24 Solutions
8 Solvers

Last Solution submitted on Sep 06, 2021

Last 200 Solutions

Problem Comments

Problem Recent Solvers8

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!