Problem 44507. Curve fitting (linear functions) & function handles

Created by David Verrelli

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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>In this problem you are provided some raw data. You need to find a way of summarising the data with just a few parameters, so that it can be reconstructed. You then need to provide a handle to your own (generic) custom function that will indeed reproduce the raw data when provided with the parameters that you determine.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Consider a linear function</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>y = m x + c</w:t></w:r><w:r><w:t>. Let's say</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> is a vector of uniformly spaced numbers, such 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>1:100</w:t></w:r><w:r><w:t>. The function operates on the data to produce, say,</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 = 2:2:200</w:t></w:r><w:r><w:t>. You are provided with both the vector</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> and the vector</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>. The parameters</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>m</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>c</w:t></w:r><w:r><w:t> are scalars; in this example</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>m</w:t></w:r><w:r><w:t> is 2 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>c</w:t></w:r><w:r><w:t> is zero.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>So here you should output two things:</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 handle to a</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>generic</w:t></w:r><w:r><w:t> function that implements a linear function (i.e. of the form</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>y = m x + c</w:t></w:r><w:r><w:t>) taking two inputs, namely</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>1</w:t></w:r><w:r><w:t> a set of parameters and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>2</w:t></w:r><w:r><w:t> the vector</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>, and outputting the corresponding vector</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>; and</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>a set of parameter values (m and c) wrapped into a single MATLAB variable (of any data type).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>As the parameters will be used in your own function, the data type will be set by you.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>So, for the above example, you</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>could</w:t></w:r><w:r><w:t> return a</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/function-handles.html\"><w:r><w:t>function handle</w:t></w:r></w:hyperlink><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>@myFunc</w:t></w:r><w:r><w:t> that you have defined, along with the variable</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>param</w:t></w:r><w:r><w:t> that has two</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/structures.html\"><w:r><w:t>fields</w:t></w:r></w:hyperlink><w:r><w:t> such that</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>param.amplification</w:t></w:r><w:r><w:t> = 2 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>param.verticalShift</w:t></w:r><w:r><w:t> = NaN.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Or, if you have defined your function differently, then you</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>could</w:t></w:r><w:r><w:t> return the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/function-handles.html\"><w:r><w:t>function handle</w:t></w:r></w:hyperlink><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>@myFn</w:t></w:r><w:r><w:t> along with a</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/cell-arrays.html\"><w:r><w:t>cell array</w:t></w:r></w:hyperlink><w:r><w:t> variable</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>prms</w:t></w:r><w:r><w:t> that has four elements such that</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>prms{1}='no'</w:t></w:r><w:r><w:t>,</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>prms{2}=NaN</w:t></w:r><w:r><w:t>,</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>prms{3} = 'yes'</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>prms{4} = 2</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>And so on.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Note: </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:b/><w:i/></w:rPr><w:t>all</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> of the relevant numbers (</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>m</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>,</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> and the elements of</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>x</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> and</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>y</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>) are</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:b/><w:i/></w:rPr><w:t>integers</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> (i.e. whole numbers, not decimals or fractions), even though they have been implicitly specified as being of type</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/double.html\"><w:r><w:rPr><w:i/></w:rPr><w:t>double</w:t></w:r></w:hyperlink><w:r><w:rPr><w:i/></w:rPr><w:t>.</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>See also</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44508\"><w:r><w:t>Problem 44508. Curve fitting (non-linear functions) &amp; function handles</w:t></w:r></w:hyperlink><w:r><w:t> — more difficult.</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>"}]}

In this problem you are provided some raw data. You need to find a way of summarising the data with just a few parameters, so that it can be reconstructed. You then need to provide a handle to your own (generic) custom function that will indeed reproduce the raw data when provided with the parameters that you determine.Consider a linear function y = m x + c. Let's say <tt>x</tt> is a vector of uniformly spaced numbers, such as <tt>1:100</tt>. The function operates on the data to produce, say, <tt>y = 2:2:200</tt>. You are provided with both the vector <tt>x</tt> and the vector <tt>y</tt>. The parameters <tt>m</tt> and <tt>c</tt> are scalars; in this example <tt>m</tt> is 2 and <tt>c</tt> is zero.So here you should output two things:<ul><li>the handle to a generic function that implements a linear function (i.e. of the form y = m x + c) taking two inputs, namely 1 a set of parameters and 2 the vector <tt>x</tt>, and outputting the corresponding vector <tt>y</tt>; and</li><li>a set of parameter values (m and c) wrapped into a single MATLAB variable (of any data type).</li></ul>As the parameters will be used in your own function, the data type will be set by you.So, for the above example, you could return a <a href = "https://au.mathworks.com/help/matlab/function-handles.html">function handle</a> <tt>@myFunc</tt> that you have defined, along with the variable <tt>param</tt> that has two <a href = "https://au.mathworks.com/help/matlab/structures.html">fields</a> such that <tt>param.amplification</tt> = 2 and <tt>param.verticalShift</tt> = NaN.Or, if you have defined your function differently, then you could return the <a href = "https://au.mathworks.com/help/matlab/function-handles.html">function handle</a> <tt>@myFn</tt> along with a <a href = "https://au.mathworks.com/help/matlab/cell-arrays.html">cell array</a> variable <tt>prms</tt> that has four elements such that <tt>prms{1}='no'</tt>, <tt>prms{2}=NaN</tt>, <tt>prms{3} = 'yes'</tt> and <tt>prms{4} = 2</tt>.And so on.Note: all of the relevant numbers ( <tt>m</tt>, <tt>c</tt> and the elements of <tt>x</tt> and <tt>y</tt>) are integers (i.e. whole numbers, not decimals or fractions), even though they have been implicitly specified as being of type <a href = "https://au.mathworks.com/help/matlab/ref/double.html">double</a>.<ul><li>See also <a href = "https://www.mathworks.com/matlabcentral/cody/problems/44508">Problem 44508. Curve fitting (non-linear functions) & function handles</a> — more difficult.</li></ul>

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Tim on 29 Jan 2018

It might be nice to replace assert(isequal(y,y_correct)) by something along the lines of assert(max(abs(y-y_correct))<1e-9) to allow for roundoff errors.

David Verrelli on 30 Jan 2018

Hello, Tim. Thank-you for your suggestion. Although it hadn't been mentioned, _all_ of the relevant numbers (x(i), y(i), m and c) are integers, so I didn't expect rounding would be a problem. [Certainly it wasn't in my reference code.] In any case, the secondary application I had in mind was lossless compression, so I am not keen to change the assertions in this particular problem (although I agree that often, elsewhere, it is indeed good to implement assertions as in your snippet). I will instead add a note to the Problem Statement. —David

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Problem 44507. Curve fitting (linear functions) & function handles

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