How do you generate a mathematical model?

[brushman]

For example, suppose you have a random set of points (x, y). How do you generate a mathematical model/function for these points?

Is it possible to have a set of information that cannot be described by a mathematical model?

How accurate are mathematical models? What determines how accurate a mathematical model is?

 

[VeeEight]

If you have just a few points, take a look at Lagrange Interpolation. It is a simple method for creating a function satisfying some set of ordered pairs and it is very intuitive.

Take a look at this article: http://en.wikipedia.org/wiki/Polynomial_interpolation
It is very general but describes a lot of things.

As for modeling, properties will depend on the object that you are studying and the research that has been done there (for example, fluid, elementary particles, biological systems, etc). I don't know much about this field but Numerical Analysis, and Differential Equations (which is essentially the language for studying these things) are of great use.
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[hotvette]

Pretty broad set of questions. Here are some comments to ponder and perhaps stimulate responses from others.

Not sure if you really meant random points or not. If so, by definition, there isn't any pattern to the points to fit a function.

Assumming you mean the points resemble some sort of pattern, you are likely wondering how one might derive a function in the form of y = f(x) to represent the collection of points.

As mentioned by VeeEight, a polynomial of degree n can be fit exactly to a set of n+1 points using Lagrange Interolation.

Least squares is a numerical method to estimate the coefficients of a function that approximates the points, but you have the know the specific function (i.e. polynomial, exponential, logarithmic, rational, periodic, parametric) to try. The fit will rarely be exact. The quality of fit is the sum of the squares of the vertical distances between the function and the points.

If the points are generated by some experimental process, the nature of the function that describes the points is often known or surmised by theoretical considerations.

If there is no information at all about the nature of the function, hints of reasonable ones to try can possibly be obtained by just looking at the data. For example, if the points appear to approach a horizontal asymptote, an exponential function might be a good candidate. If the points form a periodic pattern, that's a good clue that trig functions might help. A pattern that is periodic with decreasing/increasing in amplitude could be a trig function multiplied by an exponential. Plotting the log of x, y, or both can sometimes be revealing.

Rather than find a single function in the form of y = f(x), another approach is to use a series of connected polynomial splines (e.g. cubic splines) to fit the data.

I know of no way to systematically and comprehensively analyze data to determine the best fit function. Some software programs will have a standard set of 10 or 20 function types that it will try and find the one with the best fit, but only within it's limited menu of functions. Since there are an infinite number of possible functions, an argument can be made that finding the function that best fits a set of points is impossible.