Deriving The Quad. Eq. Using least squares.

[juice34]

Could someone show me exactly how to derive the quadratic equation from the least squares method? I have no idea where to start. I will appreciate it very much. Thankyou.

 

[Doc Al]

The method is called "completing the square" (not least squares). Google it!

 

[juice34]

Really, my professor said that its the least squares method. Is it the same thing i assume then. I appreciate it very much doc.

 

[Redbelly98]

No, they are different. "Least squares" is a method of fitting a best line or other equation to data.

Are you looking for a quadratic equation that fits some data, or are you trying to solve for x in

a x² + b x + c = 0

 

[juice34]

Im tryin to derive the least squares method for a quadratic equation. I also want to find expressions for the three coefficients in terms of sums Sx, Sy, Sxx, etc.

 

[Redbelly98]

Okay, so you are fitting a quadratic equation to some data.

You must make some attempt at a solution before we can help you (PF rules). You might look at how fitting a linear equation using least squares works, and go from there.

 

[juice34]

Here is what i have Y=ax^2+bx+c. Then the residual is d(i)=y(xi) so di=yi-(ax^2i+bxi+c). Then i take the sum of the square of the residuals. &(a,b,c)=E di^2=E((ax^2i+bxi+c))^2 and then I am lost after that!

 

[Redbelly98]

So far so good.

Next step is to take partial derivatives of & with respect to a, b, and c.

 

[Doc Al]

Really, my professor said that its the least squares method. Is it the same thing i assume then. I appreciate it very much doc.

Oops. I thought you were looking for a derivation of the quadratic formula itself (which happens to be done via "completing the square"). But you are looking to derive the parameters of a least squares fit to a quadratic function. My bad!

Redbelly98's got you covered. (I'll move this back to Calc & Beyond.)

 

[Redbelly98]

Here is what i have Y=ax^2+bx+c. Then the residual is d(i)=y(xi) so di=yi-(ax^2i+bxi+c). Then i take the sum of the square of the residuals. &(a,b,c)=E di^2=E((ax^2i+bxi+c))^2 and then I am lost after that!

Wait, I just spotted an error here. Might be just a typo on your part, but since

di = yi-(axi^2+bxi+c)

then

di^2 = ( yi - (axi^2+bxi+c))^2

not ((axi^2+bxi+c))^2

 

[juice34]

Yes you are correct Redbelly98. Ok so i differentiated with respect to a, b, and c then rearranged to get 3 equations.
1.aEx^4+bEx^3+cEx^2=Eyx^2
2.aEx^3+bEx^2+cEx=Exy
3.aEx^2+bEx+cn=Ey Where E is a summation. Now my problem is how do i define these into expressions for the three coefficients in terms of sums Sx, Sy, Sxx, etc. And how do i know how many of these S terms i will need and how do i define them. Defining them being where do i get the x,y, xx, xy, yy, etc from? Thank you all for who contributed.

 

[Redbelly98]

You have 3 equations in the 3 unknowns (a, b, and c are the unknowns), so use standard techniques for solving linear systems of equations.

Sx is Ex, Sxx is Ex^2, and Sxy is Exy, aren't they?

 

[juice34]

Redbelly 98, Thank you for your help your amazing!