Least Squares Parabola

**rfhrth** · June 8th 2012, 11:21 AM

Hi all

I have been trying to find the values of the coefficients for the least squares parabola:

y = a + bx + cx^2

I have taken partial derivatives wrt a, b, c and now have three equations, but I don't know what to do next...any suggestions? Have you solved this before?

thank you very much!!

**BobP** · June 8th 2012, 01:20 PM

$S=\sum_{i=1}^{n}(y_{i}-a-bx_{i}-cx_{i}^{2})^{2}.$

Differentiate $S$ partially wrt $a,b$ and $c,$ equate each derivative to zero and solve the resulting equations.

**rfhrth** · June 10th 2012, 01:21 PM

Hi

Thank you for response. I am try to show that a,b,c are minimization. I took derivatives of proper equation wrt a,b,c set to zero but now am unsure of how to solve for a,b,c explicit

Thank you very much

**BobP** · June 11th 2012, 12:45 AM

The equation that you should be differentiating is the one for $S.$

Differentiating wrt $a$ gives you

$\frac{\partial S}{\partial a}=-2\sum_{i=1}^{n}(y_{i}-a-bx_{i}-cx^{2}_{i})$

and on putting this equal to zero this can be rewritten as

$\sum_{i=1}^{n} y_{i}=an+b\sum_{i=1}^{n} x_{i}+c\sum_{i=1}^{n} x_{i}^{2}$

The summations are taken over the $n$ data points and will therefore be numbers.

Differentiate to find the other two equations and then solve them simultaneously for $a,b$ and $c.$

BTW, do not double post. The response that you are getting from your other posting is an alternative method, they produce the same end result.

**Prove It** · June 11th 2012, 01:10 AM

Originally Posted by BobP

The equation that you should be differentiating is the one for $S.$

Differentiating wrt $a$ gives you

$\frac{\partial S}{\partial a}=-2\sum_{i=1}^{n}(y_{i}-a-bx_{i}-cx^{2}_{i})$

and on putting this equal to zero this can be rewritten as

$\sum_{i=1}^{n} y_{i}=an+b\sum_{i=1}^{n} x_{i}+c\sum_{i=1}^{n} x_{i}^{2}$

The summations are taken over the $n$ data points and will therefore be numbers.

Differentiate to find the other two equations and then solve them simultaneously for $a,b$ and $c.$

BTW, do not double post. The response that you are getting from your other posting is an alternative method, they produce the same end result.

Much quicker than this method by the way...

Mods, would you please merge this thread with the other identical thread?

**rfhrth** · June 11th 2012, 10:59 AM

Hello
thank you for reply
My onlly question was how to solve for a,b,c simultaneously. I already done the derivatives wrt a,b,c from proper equation (S)

thank you very much

**BobP** · June 11th 2012, 02:15 PM

If you are working 'on paper', then a routine elimination is best.
Subtract multiples of the first equation from the second and third equations to eliminate $a,$ say, from both of them, then take a multiple of one of the new equations from the other to eliminate $b.$
That allows you to calculate $c$ and you can back substitute to calculate $b$ and $a.$

Thread: Least Squares Parabola

LinkBack

Thread Tools

Display

Least Squares Parabola

Re: Least Squares Parabola

Re: Least Squares Parabola

Re: Least Squares Parabola

Re: Least Squares Parabola

Re: Least Squares Parabola

Re: Least Squares Parabola

Similar Math Help Forum Discussions

Magic squares of Squares

Sum of THREE squares

sum of squares - coefficients of least squares parabola.

Least Squares Solution for a Parabola

Sum of squares?

Search Tags