Curve Fitting AlgorithmDate: 6/20/96 at 15:36:45 From: Ramon Handel Subject: Curve Fitting Algorithm Dear Doctor Math: I have seen a method called "Least squares approximation," but I don't understand how it works. Could you please explain it to me? Yours truly, Ramon van Handel RVHANDEL@DDS.NL Date: 6/20/96 at 17:39:10 From: Doctor Anthony Subject: Re: Curve Fitting Algorithm I will just indicate the method for 'least squares' curve fitting. In the case of a straight line of best fit, you assume its equation is of the form y = ax+b. If you want the line of regression of y on x, then x is the independent variable, and the theoretical y values will be given by this equation. The method of least squares is used to find a and b. If (x1,y1), (x2,y2), (x3,y3) .... (xn,yn) are the experimentally determined points, then the RESIDUALS are (y1-ax1-b), (y2-ax2-b), ...(yn-axn-b), and we must now minimize the SUMS OF SQUARES OF THE RESIDUALS. Let z = (y1-ax1-b)^2 + (y2-ax2-b)^2 + etc, etc We now find partial(dz/db) and partial(dz/da) and equate each of these to zero. This process leads to the 'normal' equations: SUM(y) = aSUM(x) + nb SUM(xy) = aSUM(x^2) + bSUM(x) After some more algebra we obtain the equation of the line of best fit: y-mean(y) = {cov(xy)/var(x)}[x-mean(x)] The method of 'least squares' is not confined to straight lines. You can also find parabolas or cubics etc of best fit, but clearly the algebra involved becomes progressively more difficult. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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