Deriving function equations from observed curves and particle paths

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cytochrome
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1) I want to learn how to look at a graph of any function and be able to derive the equation for the function given parameters that are observable. Is this parameterization? Are there any books on how to do this by hand?

2) If the function is too complex to observe and derive, I want to be able to enter parameters into a computer that has an algorithm to make a "best fit" function for the curve. What is this called? What kind of algorithm would a program such as this contain? More particularly, as an example: If a picture is taken of a fast particle in motion on a certain path, how could I derive the function for its position?
ThanksEDIT: To make it more specific about what I am trying to find out, I think it goes somewhere along the lines of mathematical modeling, parameterization, mutlivariate methods (maybe). Basically I just want to know how to either look at a function and derive it's equation, or make best fits equations for any curve that could represent the path of a particle. Is there a subject centered around deriving functions from observed paths of particles or observable curves in general?
 
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Hey cytochrome.

There is no "one" resource that will give you this crystal ball, but if I had to recommend some areas I would recommend data mining, statistics, signal processing, numerical analysis (including interpolation and approximation) and other forms of integral transforms not necessarily considered in the normal signal processing literature. Also (and this is important), read information theory.

The thing about what you are asking is that you are essentially asking is that you want some particular representation with the minimal Kolmogorov Complexity under a specific description.

This problem has no known public solution currently, but it is the answer to your question: to take something and find the kolmogorov complexity of that object.

When you start researching this yourself, you will soon see why this is a hard thing to calculate (there are again, no publically known general ways of calculating this).

Here is a wiki link to get you started:

http://en.wikipedia.org/wiki/Kolmogorov_complexity

and I recommend you take a serious look at data mining, since this is exactly the kind of thing they do:

http://en.wikipedia.org/wiki/Data_mining
 
Thanks a lot! That was very helpful
 
There is often no single equation that is 'best'

What, in any case, do you mean by best?

You will never have an infinite range of data and sometimes one equation fits one part better then another equation fits a different segment better.

Sometimes there is a question of accuracy for purpose. There is no point fitting a more accurate equation than is needed for the calculation in hand.

Examples might be whether to use the Ebers Moll equation in semiconducter electronics or some simpler model.

A good sequence of equations to study the development of are the gas equations.

The ideal gas laws
Van De Walls equation
The Virial Equation
Amagat and Andrews curves

More mathematical techniques, in additions to Chiro's long list to look at are

The method of undetermined coefficients.

Plotting non linear axes eg logarithmic or the ratio of two or more of the dataset variables or even more complicated expressions.
Normalisation