Best Fit Quadratic CurveDate: 02/21/2002 at 01:57:27 From: Avin Sinanan Subject: Best fit Quadratic Curve Hello, I would please like to know how, given a scatter plot of X and Y cordinates, one finds the best-fitting quadratic curve. Let's say that Y = A*X^2 + B*X + C is the best fit curve. How does one find A, B, and C for the best quadratic fit curve? I looked on many Web sites and they mention that it can be done, but they never show how to do it. Thanks, Yours respectfully, Avin Sinanan Date: 02/21/2002 at 07:52:20 From: Doctor Rob Subject: Re: Best fit Quadratic Curve Thanks for writing to Ask Dr. Math, Avin. Suppose the points you are given are {(X[i],Y[i]): 1 <= i <= N}. Then you want the values of A, B, and C that minimize the sum of squares of the deviations of Y[i] from A*X[i]^2 + B*X[i] + C. They will give you the best-fitting quadratic equation. Let the sum of the squares of the deviations be N F(A,B,C) = SUM (A*X[i]^2+B*X[i]+C-Y[i])^2. i=1 To minimize this, take partial derivatives of F with respect to the three variables A, B, and C, set them all equal to zero, and solve simultaneously: dF/dA = SUM 2*(A*X[i]^2+B*X[i]+C-Y[i])*X[i]^2 = 0, dF/dB = SUM 2*(A*X[i]^2+B*X[i]+C-Y[i])*X[i] = 0, dF/dC = SUM 2*(A*X[i]^2+B*X[i]+C-Y[i]) = 0. (Here all sums range over i = 1, 2, ..., N.) Dividing by 2 and rearranging, you can see that these are three simultaneous linear equations in the three unknowns A, B, and C: (SUM X[i]^4)*A + (SUM X[i]^3)*B + (SUM X[i]^2)*C = SUM X[i]^2*Y[i], (SUM X[i]^3)*A + (SUM X[i]^2)*B + (SUM X[i])*C = SUM X[i]*Y[i], (SUM X[i]^2)*A + (SUM X[i])*B + (SUM 1)*C = SUM Y[i]. I leave it to you to solve these three equations. Feel free to write again if I can help further. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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