Problem 45851. Least Absolute Deviations (L1-norm) line fit - degree 1

Created by jmac

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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:t>Matlab's polyfit function is very handy to find least-squares regression. It minimizes the (L2-norm) of the estimation errors, by solving a linear system.</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/ref/polyfit.html\"><w:r><w:t>&lt;https://www.mathworks.com/help/matlab/ref/polyfit.html</w:t></w:r></w:hyperlink><w:r><w:t>&gt;</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>An often overlooked way to deal with these situations is to use Least Absolute Deviations (LAD) line fitting. It minimizes the L1-norm of the residuals, thus it is less sensitive to outliers that fall far away from the underlying model</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Least_absolute_deviations\"><w:r><w:t>https://en.wikipedia.org/wiki/Least_absolute_deviations</w:t></w:r></w:hyperlink></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>- - -</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>You are given two vectors X and Y (coordinates of observations on a plane). Return a row vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>P = [a, b]</w:t></w:r><w:r><w:t> with the coefficients of the best-fit line, in the L1-norm sense. I.e., find</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>b</w:t></w:r><w:r><w:t> that minimize</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>sum( abs( Y - a*X - b ) )</w:t></w:r><w:r><w:t> .</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>(compare your results with polyfit on the test suite!)</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>"}]}

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Last Solution submitted on Sep 18, 2020

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Alfonso Nieto-Castanon on 23 Aug 2020

please fix testsuite (e.g. change "polyval(P_lad,X)" to "polyval(P,X)")

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fit optimization regression

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Problem 45851. Least Absolute Deviations (L1-norm) line fit - degree 1

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Problem 45851. Least Absolute Deviations (L1-norm) line fit - degree 1

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