Problem 44508. Curve fitting (non-linear functions) & function handles

Created by David Verrelli

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>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 non-linear scalar function</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>y = mˣ + c</w:t></w:r><w:r><w:t>. Suppose</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 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>[3 2 1 4]</w:t></w:r><w:r><w:t>. The function would operate elementwise on</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>; for example, if</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=2</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=3</w:t></w:r><w:r><w:t> then</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 = [11 7 5 19]</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>. Every number will be provided in some form of an</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/matlab_prog/integers.html\"><w:r><w:t>integer</w:t></w:r></w:hyperlink><w:r><w:t> data type. The scalar 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 both whole numbers (although they're not provided directly).</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=\"2\"/></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 the relevant non-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 = mˣ + 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> (in any data type); and</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></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.base</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.translation</w:t></w:r><w:r><w:t> = 3.</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}='positive'</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}=uint8(3)</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} = 'positive'</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} = uint8(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>1</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>: the parameter</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> can be assumed to have values between -30 and +30 (inclusive); the parameter</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> can be assumed to have values between -10000 and +10000 (inclusive); the elements in</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> are positive.</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>2</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>: if the code runs too slowly, then it will also fail the Test Suite.</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>3</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>: there might not be only one unique pair of the m &amp; c parameter values that is correct — the individual values of m &amp; c are not tested here, only the values of y that they predict are examined.</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/44507\"><w:r><w:t>Problem 44507. Curve fitting (linear functions) &amp; function handles</w:t></w:r></w:hyperlink><w:r><w:t> — easier.</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 non-linear scalar function y = mˣ + c. Suppose <tt>x</tt> is the vector <tt>[3 2 1 4]</tt>. The function would operate elementwise on <tt>x</tt>; for example, if <tt>m=2</tt> and <tt>c=3</tt> then <tt>y = [11 7 5 19]</tt>. You are provided with both the vector <tt>x</tt> and the vector <tt>y</tt>. Every number will be provided in some form of an <a href = "https://au.mathworks.com/help/matlab/matlab_prog/integers.html">integer</a> data type. The scalar parameters <tt>m</tt> and <tt>c</tt> are both whole numbers (although they're not provided directly).So here you should output two things:<ol><li>the handle to a generic function that implements the relevant non-linear function (i.e. of the form y = mˣ + 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> (in any data type); and</li><li>a set of parameter values (m and c) wrapped into a single MATLAB variable (of any data type).</li></ol>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.base</tt> = 2 and <tt>param.translation</tt> = 3.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}='positive'</tt>, <tt>prms{2}=uint8(3)</tt>, <tt>prms{3} = 'positive'</tt> and <tt>prms{4} = uint8(2)</tt>.And so on.Note 1: the parameter <tt>m</tt> can be assumed to have values between -30 and +30 (inclusive); the parameter <tt>c</tt> can be assumed to have values between -10000 and +10000 (inclusive); the elements in <tt>x</tt> are positive.Note 2: if the code runs too slowly, then it will also fail the Test Suite.Note 3: there might not be only one unique pair of the m & c parameter values that is correct — the individual values of m & c are not tested here, only the values of y that they predict are examined.<ul><li>See also <a href = "https://www.mathworks.com/matlabcentral/cody/problems/44507">Problem 44507. Curve fitting (linear functions) & function handles</a> — easier.</li></ul>

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Last Solution submitted on Jun 12, 2020

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Tim on 1 Feb 2018

I think you may need to modify some of the tests to "...ensure both odd and even values of x" in all cases where m might be negative.

David Verrelli on 2 Feb 2018

Hi, Tim. Thanks for your feedback. Actually, from the Player's point of view, m 'could' be negative in any Test. I agree that in some situations there might not be one unique pair of the m & c parameter values that is correct. However, I never apply an assertion to the values of m & c in the Test Suite, rather I check what values of y the user-supplied parameters produce from the user-supplied function(handle). _Any_ valid combination of m & c should be able to REpredict the same values of y provided in the original input, from the same x values (or a subset thereof). There is only one test where I check for prediction of y using x values not included in the original input (extrapolation/interpolation), and for that one test I do have to be careful to ensure there is one unique pair of m & c values that is correct. So I would say this is intentional (I'll add a short note to the Problem Statement). But please let me know if there's a flaw in my logic. —DIV

Tim on 3 Feb 2018

Aha--I see now. To quote Emily Litella: "Never mind."

David Verrelli on 7 Feb 2018

I should clarify one detail for other Players. Given y = mˣ + c (elementwise), suppose x is the vector [7 8 5 6] and y is the vector [123 123 123 123]. Then there are two valid combinations of m & c, namely m=0 & c=123 and m=1 & c=122. So actually there is not just one unique set of m & c values in this case. However, given x>0 (in the Problem Statement), either of these alternatives can be correctly extrapolated or interpolated from.

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