Obtaining the functional form of a curve

Question

The following is the plot of a curve f(r), where r is the radial coordinate, and plotted for different values of a parameter as shown:

However, I don't know the functional form of the curve and I am interested to find the same. Are there any numerical methods which can be used to find the functional form of f(r) in terms of the radial coordinate and the parameter?

Answer 1

I had found a solution of the problem based on the suggestion by ja72 to use the Eureqa software which churns through the data to create accurate predictive models using evolutionary search algorithm.

In the question, the different curves corresponds to different values of . So, initially I obtained the best fit equation for different values of and found that the following model equation is suitable for my purpose:

Then, I repeated the process for a large number of values of and calculated the values of the four functions for different values of and then individually fitted these four functions. The following are the results that I obtained:

N.B.: Eureqa gave several other better fitting formulas than those mentioned in the answer. But the formulas that I mentioned are sufficiently accurate for my purpose and have minimum complexity.

Answer 2

A blind curve fit without an underlying model is a dangerous thing.

You need to have an understanding of the physical model behind the data to create a successful fit. The reason is that if r is distance and the best fit curve uses r^0.4072 for example, that dimension raised to a decimal power bears no meaning and it hides any underlying assumptions.Like some other dimension l not included in the model, whereas only the dimensionless quantity (r/l) would make sense to raise to the decimal power.

From a function analysis standpoint

These curves are not the result of any standard math function. Well I am not that familiar with bessel functions, gamma functions and legendre polynomials. But none of the standard functions you find in a scientific calculator jumps out here.

If r is assumed to be dimensionless, then you try to match the asymptotic behavior when r -> 0 and when r -> ∞. The would be the baseline curve. To me it does not look hyperbolic, but rather close to 1/LN(1+r).

So change the variables make g=1/LN(1+r) and plot f(r) against g(r) and see what that looks like. Then try another round of curve fitting in the new curves ... and so on.

Nobody can answer this question

Nobody else could effectively answer this question but you, because a) you have the data, and b) you need to make assumptions about what region is important or not, and what is acceptable deviation.