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How to use LeastSquares[] or QRDecomposition[] to Fit y = mx + b

Mitchell Sandlin

Posted 20 hours ago

I am trying to calculate the m and b values as a best fit to the equation of the line of y=mx + b - see attached notebook. In the first section of the notebook, I performed the detail calculations manually to obtain the variables of m = 1.1 and b = 1 to create the equation of the line of y = 1.1x + 1. Then I tried to use two Mathematica functions (QRDecomposition[] and LeastSquares[]) in an attempt to streamline the calculations and ran into problems. I tried to use examples in both Mathematica as well as examples in the WolframU course on Linear Algebra in which I managed to thoroughly confuse myself because none of the new calculations even remotely resembles the calculations or the end results that I performed manually. If there is one, I would certainly appreciate someone showing me the recommended method of using Mathematica functions to perform the calculations that I performed manually in the first section of the attached notebook. Thank you, Mitch Sandlin Attachments: QR_Fact.nb

POSTED BY: Mitchell Sandlin

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 hours ago

You can do it with `Minimize` or `FindFit`: y = {2, 3, 5, 5}; x = {{1, 1}, {2, 1}, {3, 1}, {4, 1}}; \[Beta] = {m, b}; Minimize[(y - x . \[Beta]) . (y - x . \[Beta]), {m, b}] data = Transpose[{x[[All, 1]], y}]; Block[{x}, FindFit[data, m*x + b, {m, b}, x]]

You can do it with Minimize or FindFit:

y = {2, 3, 5, 5};
x = {{1, 1}, {2, 1}, {3, 1}, {4, 1}};
\[Beta] = {m, b};
Minimize[(y - x . \[Beta]) . (y - x . \[Beta]), {m, b}]
data = Transpose[{x[[All, 1]], y}];
Block[{x}, FindFit[data, m*x + b, {m, b}, x]]

POSTED BY: Gianluca Gorni

Michael Rogers

Michael Rogers, Emory University

Posted 15 hours ago

Here's how I would do it. (I made up my own data vecs `xx`, `yy`.) xx = Subdivide[0., 5., 10]; yy = 1 + 1.1xx + RandomVariate[NormalDistribution[0., 0.2], Length@xx]; design = Transpose@{ConstantArray[1., Length@xx], xx}; bmvec = LeastSquares[design, yy] ( {0.936894, 1.11642} ) {qq, rr} = QRDecomposition[design]; LinearSolve[rr, qq . yy] ( {0.936894, 1.11642} *) ListPlot[Transpose[{xx, yy}], Epilog -> {Red, Line[{{0, Last[bmvec]}, {5, bmvec . {1, 5}}}]}]

Here's how I would do it. (I made up my own data vecs xx, yy.)

xx = Subdivide[0., 5., 10];
yy = 1 + 1.1*xx + 
   RandomVariate[NormalDistribution[0., 0.2], Length@xx];
design = Transpose@{ConstantArray[1., Length@xx], xx};
bmvec = LeastSquares[design, yy]
(*  {0.936894, 1.11642}  *)

{qq, rr} = QRDecomposition[design];
LinearSolve[rr, qq . yy]
(*  {0.936894, 1.11642}  *)

ListPlot[Transpose[{xx, yy}], 
 Epilog -> {Red, Line[{{0, Last[bmvec]}, {5, bmvec . {1, 5}}}]}]

data and best-fit line

POSTED BY: Michael Rogers

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