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October 1st, 2012, 06:57 AM   #1
Joined: Oct 2012

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Polynomial equation for surface fit

Hi, newcomer to the forum here!

I am an engineer and need to estimate a surface using an equation. I use the solver in Excel to solve the equation and it has been working quite well. I have a problem with the latest surface, though. I attach an image of the surface. I tried using the following equations to fit the surface:
z = a + bx + cy + dx^2 + ey^2 + fx^3 + gy^3 + hxy + ix^2y + jxy^2
z = Ax^2 + By^2 + Cxy + Dx + Ey + F
z = (ax^2 + Bx + c) * (dy^2 + ey + f)

The second on converges quite well and I get an estimate of the surface, but the accuracy is not very good. The first and third equations converge to surfaces that don't look anything like the one I want.

My questions:
- How do I know what equation to use for a specific surface?
- Do you have some tips on determining start values for the variables?

Any help will be highly appreciated!

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File Type: png Surface.png (47.2 KB, 398 views)
Beiteltjie is offline  
October 5th, 2012, 12:18 AM   #2
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Re: Polynomial equation for surface fit

Hey Beiteltjie.

In terms of finding general forms of a surface, one approach you could take is to fit to a very general model and then keep reducing the model until the fit differences go over some pre-defined limit: this is in fact what is done with linear models in statistics and its known as back elimination.

But with regards to your problem, I'd recommend looking at fitting data to what's known as a Spline if you can particularly the non-uniform kinds if you have access to a routine that does this (it's going to be really involved to tell you up front).

With regards to the first suggestion, basically you choose some large number n and generate a multi-variate polynomial orthogonal basis that can be projected on to and to do this you need to look at the inner product that is used which uses the integral definiton in fourier analysis. Fourier analysis discusses this inner product and also how to construct an orthonormal basis across an interval (which will be the domain of your signal).

Once a program constructs the different bases for the various fitting bases, you look at the fit difference between each reduction and do exactly what the back-elimination method does in regression modelling, but instead of the ratio of variances you look instead as the ratio of the integrated residuals between the true signal and the fitted model (basically integrate a function that looks at the square of the distance between the fitted and the original data).

You can build on this idea if you want, but it's one of the simplest when it comes to finding the simplest general model for any kind of signal data without having to "guess" the fitted function and as I said above, this idea is already used in statistical regression modelling.
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