Thread: Least Squares Solution for a Parabola

Hi there,

I have a set of data points

{(X1,Y1), (X2,Y2),........(Xn,Yn)}

I would like to approximate the relation between X and Y as a parabola. Assuming the parabola has form y = Ax^2+Bx+C, can somebody please provide me with the least squares regression formulas for A, B and C.

Any help is greatly appreciated.

Thanks,

AP

This is very simple. Use matrices. I'll use your model as is... .
We don't need the epsilon since we are only fitting a curve.
Using your model the design matrix consists of three columns.

The first column is through , the second column is through and the last column is all 1's.

The vector Y consists of through .

The solution is the vector

where

MathEagle,

Thanks for your reply. Unfortunately, matices to me, are like chinese to an englishman........

The general matrix solution for the regression of a polynomial of any degree is all over the web. But I dont get it.

I was looking for a specific solution to the parabola regression. Something that has a heap of "sum of XiYi" and "sum of Xi squared"..........

Any chance you could help with that??

Thanks again,

AP

Well, that makes our work harder.
I bet I can find it on the web.
One can solve it by differentiating and obtaining the normal equations.


with respect to a,b,c and setting each equal to zero.

But now you'll tell me you never had calculus.

I figured I could find it on the web, matrices is better and they didn't finish it either.
They have just one more step after what I told you to do.
http://www.efunda.com/math/leastsqua...sqr2dcurve.cfm
I googled parabolic least squares, maybe you cna find other useful links.
I'm sure others have done this.
But between matrices and JMP on my pc I don't need to look at the normal equations.

Here's more with a parabolic fit, but I see matrices and a weird looking robot
http://www.bsu.edu/web/jkshim/mathan...eastsquare.htm

MathEagle,

I do have calculus, and I should understand matrices (a long, long time ago I did a degree in mathematics). I have been trying to solved the equations for the past 2 days, and I just keep stuffing it up. Thus, I was hoping someone had it in a nice neat form for me.

I have previously found the pages you suggest. The robot was a mystery to me also.

Thanks for you time and effort.

AP.

try the following page for the normal equations to fit a parabola in summation form
mixture: normal equation to fit a parabola


for a straight line
http://keral2008.blogspot.com/2009/0...of-y-on-x.html

As I stated yesterday, the normal equations can be found in the box in Least-Squares Parabola

Eagle and QP,

Thanks very much for your help, I am all sorted now.

To be honest, I am more than a little embarrassed I couldn't sort this out for myself.

Cheers,

AP