Thread: Least squares to matrix to poly coefficients

I am writing programming code to develop a polynomial equation from x,y data. I am using the least squares procedure to develop various sums to fill a matrix. Once the matrix is filled, computing poly coefficients is straight forward matrix algebra.


I have 29 reps of data that look like:


Rep x₁ x₂ x₃ y
——————————————————————————————————————————
1
2
…
29
——————————————————————————————————————————




x₂ and x₃ have the form:


x₂ = x₁^2
x₃ = x₁^3




My problem is figuring out how to fill the 3x3 matrix cells that take the sum of products (SP).


The diagonal cells take the sum of squares (SS) which is:


SS = ∑X² - (∑x)²/n This is for cells (1,1), (2,2) and (3,3), using x₁, x₂ and x₃ respectively.


The non-diagonal cells take the sum of products (SP) which is:


SP = ∑(x₁*x₂) - ((∑x₁)(∑x₂))/n This is for cell (1,2)


The mirror cells are:
(2,1) = (1,2)
(3,1) = (1,3)
(3,2) = (2,3)


So the matrix I now have is:

∑X12 - (∑x1)2/n	∑(x1*x2)-((∑x1)(∑x2))/n	???
mirror of cell (1,2)	∑X22 - (∑x2)2/n	???
mirror of cell (1,3)	mirror of cell (2,3)	∑X32 - (∑x3)2/n

The question marks are where I’m stumped. This procedure works great for a 2x2 matrix and a 2nd degree polynomial. But now I’m expanding to a 3x3 matrix for a 3rd degree polynomial. I’ve tried every combo of SP’s for cells (1,3) and (2,3), but to no avail.


I suspect this is simple, and aesthetically pleasing. I have not found the answer in the literature.


Any help is appreciated.


TJ

Murphy’s Law

I should have guessed. As soon as I join Math Forum, I figure out the problem. I had an error in the way I was transposing x data, which was showing up as a matrix error. The sum of products (SP) that I tried and failed, was really the right solution.

Here is the final least squares matrix for solving a 3rd degree polynomial:

∑X12 - (∑x1)2/n ∑(x1*x2)-((∑x1)(∑x2))/n ∑(x1*x3)-((∑x1)(∑x3))/n
mirror of cell (1,2) ∑X22 - (∑x2)2/n ∑(x2*x3)-((∑x2)(∑x3))/n
mirror of cell (1,3) mirror of cell (2,3) ∑X32 - (∑x3)2/n


Now just invert and multiply by the y matrix to get your equation coefficients.

TJ

PS. These posts eliminate subscript and superscript, so I hope you can tell which is which. ∑X22 is the sum of x2 squared, etc. Email me if you want a readable pdf copy.