Question

How can I use the method of least squares to fit a line to the points { (0,0), (1,2), (2,1), (3,4) }?

Answer 1

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.

Answer 2

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

Answer 3

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