Problem 44933. Vogel-Dobbener entropy

Created by Christian Schröder

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{"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.

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