Problem 258. linear least squares fitting

Created by Tomasz

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:t>Inputs:</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:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>f</w:t></w:r><w:r><w:t>: cell-array of function handles</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:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>x</w:t></w:r><w:r><w:t>: column vector of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>x</w:t></w:r><w:r><w:t> values</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:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>y</w:t></w:r><w:r><w:t>: column vector of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>y</w:t></w:r><w:r><w:t> values, same length as</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>x</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Output:</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:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t>: column vector of coefficients, same length as</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>f</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>In a correct answer the coefficients</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> take values such that the function</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[   fit = @(x) a(1)*f{1}(x) + a(2)*f{2}(x) + a(3)*f{3}(x) +...+ a(end)*f{end}(x)]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>minimizes the sum of the squared deviations between</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>fit(x)</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:rFonts w:cs=\"monospace\"/></w:rPr><w:t>y</w:t></w:r><w:r><w:t>, i.e. sum((fit(x)-y).^2) is minimal.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Remarks:</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>The functions will all be vectorized, so e.g.</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>f{1}(x)</w:t></w:r><w:r><w:t> will return results for the whole vector x</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>The absolute errors of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> must be smaller than 1e-6 to pass the tests</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>"}]}

Solve

Solution Stats

174 Solutions
46 Solvers

Last Solution submitted on Feb 28, 2021

Last 200 Solutions

Problem Comments

6 Comments

6 Comments

Show 3 older comments

Alfonso Nieto-Castanon on 9 Jan 2013

your functions are not all truly 'vectorized'; I would change @(x)1 to @(x)ones(size(x)) for a more consistent 'vectorized' behavior...

Tomasz on 10 Jan 2013

Yup, you're right. I made the adjustment...

Alfonso Nieto-Castanon on 10 Jan 2013

not meaning to be picky, but in the third test you might want to use aref.' instead of aref' (if x take negative values, which occurs at random for this test, then log(x) takes complex values, and the aref' portion takes the conjugate of the complex valued results); perhaps it is simpler to make sure x takes only positive values...

Alfonso Nieto-Castanon on 10 Jan 2013

and nice problem, by the way!

Tomasz on 10 Jan 2013

Thanks! Nice catch, I fixed that second issue too (those randomized tests have their drawbacks)

Tikay on 26 Oct 2020

Sorry, but I can not understand the problem to solve.
Can anyone explain to me, please?

Problem Recent Solvers46

Problem Tags

Community Treasure Hunt

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

Start Hunting!

Problem 258. linear least squares fitting

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers46

Suggested Problems

More from this Author7

Problem Tags

Community Treasure Hunt

Problem 258. linear least squares fitting

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers46

Suggested Problems

More from this Author7

Problem Tags

Community Treasure Hunt

Players