anonymous
  • anonymous
How can I use the method of least squares to fit a line to the points { (0,0), (1,2), (2,1), (3,4) }?
OCW Scholar - Multivariable Calculus
katieb
  • katieb
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anonymous
  • anonymous
the method of least squares gives rise to the entire concept of karl pearsons coefficient.Try doing some research on that subject.And then retry the question.
anonymous
  • anonymous
let the best fit line be y = ax + b then u using least square method "a" and "b" satisfies \[(\sum_{1}^{4} x _{i} ^{2}).a + (\sum_{1}^{4} x _{i}).b = \sum_{1}^{4} x _{i} y _{i}\] \[(\sum_{1}^{4} x _{i}).a + 4b = \sum_{1}^{4} y _{i}\] 14a + 7b = 16 7a + 4b = 7 on solving a= 15/7 , b = -2 => best fit y=(15/7)x -2
anonymous
  • anonymous
Write down the equations of the line (for all the points) in a form \[Mx_{i}+N=y_{i}\] Then assemble the matrix equation: Ax = b, Where x is the vector of unknown line coeffiients x= [M,N] In general this equation cannot be solved unless vector b is in the column space of matrix A. Solve equation Ax = p instead, where p is projection of b on the column space of A. The error you make by taking the projection and not the real vector is e = b - p e is perpendicular to p and all vectors in column space of A so: \[A ^{T}e=0\]\[A ^{T}(b-p) = 0\] \[A ^{T}(b-Ax)=0\]Solving the equation for unknow x gives: \[x=(A ^{T}A)^{-1}A ^{T}b\]which contains unknown line coefficients

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