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>

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 |x| is a vector of uniformly spaced numbers, such as |1:100|.  The function operates on the data to produce, say, |y = 2:2:200|.  You are provided with both the vector |x| and the vector |y|.  The parameters |m| and |c| are scalars;  in this example |m| is 2 and |c| is zero.

So here you should output two things:

* 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 |x|, and outputting the corresponding vector |y|;  and
* a set of parameter values (m and c) wrapped into a single MATLAB variable (of any data type).

As the parameters will be used in your own function, the data type will be set by you.

Or, if you have defined your function differently, then you _could_ return the <https://au.mathworks.com/help/matlab/function-handles.html function handle> |@myFn| along with a <https://au.mathworks.com/help/matlab/cell-arrays.html cell array> variable |prms| that has four elements such that |prms{1}='no'|, |prms{2}=NaN|, |prms{3} = 'yes'| and |prms{4} = 2|.

And so on.

_Note: *all* of the relevant numbers ( |m|, |c| and the elements of |x| and |y|) are *integers* (i.e. whole numbers, not decimals or fractions), even though they have been implicitly specified as being of type <https://au.mathworks.com/help/matlab/ref/double.html double>._

* See also <https://www.mathworks.com/matlabcentral/cody/problems/44508 Problem 44508. Curve fitting (non-linear functions) & function handles> — more difficult.

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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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