Mathematical functions from data sets?

[MathWarrior]

I feel like I have gone pretty far in math now, but I keep finding myself asking the same question.

Say I had a series of data points from like a randomly collected survey or stock stock price graph over time etc.

Is there a way to take this seemingly random and scattered data and turn it into a mathematical function which I can then use calculus on to find things like optimal points of selling stock, maximum price a customer might pay based on survey data etc? What is this process called?

I was thinking perhaps you could use sigma notation which directly correlates with the data set or something but I am not positive this would be the correct way?

 

[micromass]

It seems like you want the concept of "random variable"? See http://en.wikipedia.org/wiki/Random_variable

Basically, a random variable takes a data set and associates with the data set a certain number. For example, a random variable could be the minimum value of the data set, or the maximum value of the data set.

Or, we can also take X_n to be the n'th value of the data set. Then it's possible to form things like $$X_1+X_2$$...

Is this what you're looking for?

 

[MathWarrior]

I was thinking more the concept of taking a set of data and converting it into a mathematical function. Or approximating it with a function I guess? I am not sure what it would be I've always wondered how you would go about getting a function from the data.

 

[micromass]

Ah, then maybe regression analysis/Least squares approximation is the thing you're looking for. It creates a function that lies very close to the data set. And you can use calculus on the function to get to know things about it...

 

[coelho]

The process of finding a function that fits some given set of points is known as curve fitting.

There are infinitely many functions that can be fit to the same set of points. The person fitting it must choose what are the desired properties of the function it wants. One can fit a straight line, a parable, a cubic, etc, to the same set of points, depending on the "fitter" 's choice.

Anyway, you can find more information about this here:

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

 

[Studiot]

When you deduce the 'next week's lottery numbers function' from your dataset, don't post it here, send it to me by PM.

 

[HallsofIvy]

The difficulty is that given any finite number of data points, there exist an infinite number of different functions that will give those data points. You need to assume some other properties to reduce the possible functions. It is true that if, for example, you have n data points there exist a unique n-1 degree polynomial that fits those points. It is more common, recently, to use "splines", piecewise defined lower degree polynomials (cubic is most common) that can be fitted to any number of data points.