Re: fitting surface with Mathematica 6.0.3 and range of fitted values

*To*: mathgroup at smc.vnet.net*Subject*: [mg99028] Re: fitting surface with Mathematica 6.0.3 and range of fitted values*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Fri, 24 Apr 2009 03:48:15 -0400 (EDT)

On 4/23/09 at 6:45 AM, simonematool at gmail.com wrote: >I'm trying to fit a surface with a Fit command and a 4th order >polynomial and I can get the fit and the model but then I try to >evaluate some data through the model and they are out of the range of >the original data, I have looked around but I could not find an option >to pass into Fit command to keep the fitted values in the range of the >possible cases. The fact you get a result using Fit is meaningless since Fit will always give you a result. But, there is no guarantee the result will be useful or meaningful. Nor is there any way to constrain the range of values returned by Fit for the fitted parameters. =46it only solves the linear regression problem. The solution to the linear regression problem can be expressed in closed form and requires no initial estimate of the solution. =46indFit accepts constraints on the fitted parameters. Alternatively, you can write you own function to compute the sum of squares or some other measure of goodness of fit and use NMimimize to find optimum values. NMimimize also accepts constaints on fitting parameters. But there is another issue here as well. While you did not specify what you are using as basis functions for Fit, I suspect you might be using simply powers of x, i.e., {1, x, x^2, x^3, x^4} as the set of functions to fit to your data. If so, this is a very bad idea. There are several issues here that will cause problems. First, the linear regression problem is increasingly numerically unstable when the degree of the polynomial increases when using powers of x as the set of basis functions. Since Fit does it thing by default with machine precision arithmetic, this means small changes in your data will result in large changes in the fitted values. Next, powers of x do not form an orthogonal set of basis functions. That means the fitted coefficients will not be independent of each other. Additionally, the variance in the fitted parameters scales as 1/(1-r) where r is the correlation coefficient between basis functions. If you absolutely need a polynomial fit, you should be using an orthogonal set of polynomials for your basis functions such as Chebyshev or Legendre polynomials. Chebyshev polynomials are particularly good.

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