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As I try to process some physic experiment data that I don't have the closed form formula with unknown parameters, I have to use some regression models like polynomials or normal distributions . The problem here is that polynomials can't fit good enough when degree is low and have too many local extreme points when it is too high. And other closed forms have the similar or other problems.

Thus, it got me thinking whether there are methods or tools to just compute the most fitted curves or surface without getting a closed form, like some robust methods. And by only computable, it is used to predict values at certain points.

And by the most fitted, I mean that it has as less local extreme points as possible and do turn if the data has the trend and after all, differentiable or continuous at least.

I try to look up some materials in the google scholar and found nothing that I wanted. I wonder if there are on-going researches or already finished works on this. Thanks

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  • $\begingroup$ this is literally the hardest problem. $\endgroup$ – Zackkenyon Nov 11 '13 at 4:50
  • $\begingroup$ as just said by Zackkenyon, this is the most difficult problem when you do not have physics laws to suggest what to use. Did you try exponentionals or logarithms . I guess you did. Just for curiosity, could you post data here ? I shall look what I could find. Please let me know the inside of the problem. $\endgroup$ – Claude Leibovici Nov 11 '13 at 5:07
  • $\begingroup$ @ClaudeLeibovici :Thanks for offering the help. Actually, I do have the closed form with parameters that could be used to optimise the parameters. But the form itself is very hard to deduce and is not good enough for computation. And when I look at the data points that plotted on the graph, I "see" that there is, somehow, a very obvious curve there. So I wonder whether there is certain mathematical tool to help me just "draw" the line directly. And I guess it is really hard to theorise it. Thanks again ;) $\endgroup$ – Archimedes520 Nov 11 '13 at 8:34
  • $\begingroup$ @Archimedes520. If you have a closed form, the only problem is to obtain reasonable guesses for the parameters and, from there, start a non linear regression. Could you post at least your equation ? $\endgroup$ – Claude Leibovici Nov 11 '13 at 9:29
  • $\begingroup$ @ClaudeLeibovici: It is a RCL circuit in series to predict the frequency of the input that yields the maximum electric current, given a periodic voltage. But not component parameters are provided. And the equation goes like:$$ i = \frac{U}{\sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2}}$$ $\endgroup$ – Archimedes520 Nov 11 '13 at 12:08
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In general, it's a bad idea to try and find a general function that "fits" as best possible, since this function will clearly not have any physical meaning., and you will have no guarantee that the function found will "work" on a different set of observed data, due to different noise distributions. Another reason not to do so is that as you allow more parameters, the parameter values become highly susceptible to noise, and then again, these will have little or no physical meaning.

A better approach is to try and model your data, say, by assuming it is described by some known physical function $f(x)$, say a sum of Gaussians, with an added error term $g(x)$. In most cases$^®$, you should limit the noise to be a low-order polynomial, unless you know for a fact that there is a very good reason to assume otherwise.

If the problem you are having relates to having noisy individual outliers, you may want to try to use a different error function instead of the usual least squares regression. This "robust estimation" technique is known as "M-estimators", and the parameters can be calculated using a method known as "Iteratively Reweighted Least-squares, or IRLS.

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