Calculus, How can you find an equation from a series of numbers?

Question

I'm analyzing financial data and would like to find the inflection points of a line. I know I can do this using derivatives, but first I need an equation. Is there a way to generate an equation based off of a series of numbers. I would need to do this programmaticly.

Answer 1

Spline interpolation is probably more useful for you than polynomial interpolation: if you fit a polynomial, it must inevitably head off to +/- infinity outside your data range.

You will also want a method which allows a slightly loose fit: financial data is often a bit noisy which can result in very weird curves if you try to fit it exactly.

Answer 2

There are established procedures for turning a set of existing data points into a polynomial; this is called Polynomial Interpolation. This article in Wikipedia: http://en.wikipedia.org/wiki/Polynomial_interpolation explains it mathematically. You can probably Google for algorithms easily enough.

Given enough points, your polynomial tracks the original, unknown function reasonably well, so the polynomial's inflection points should roughly coincide with the peaks and troughs of your data.

On the other hand, we all know there's not really a function behind financial data. So if I were you I'd scan along those points and find every point that has a smaller value to either side of it, and declare that a high; and vice versa for lows. Force-fitting this data into a fictitious function isn't going to make it any more useful.

Update: Tom Smith advises that spline interpolation is to be preferred to polynomial interpolation for this kind of thing, and Wikipedia bears him out. Or rather, it's bullish on his answer.

Answer 3

What you are thinking is analytical calculus ... when having discrete data (e.g. points), you have to do it numerically. Now, a line usually doesn't have inflection points, so I guess you're thinking of a curve. You can either interpolate some kind of it through the points, then calculate the first derivative (also numerically, but for a larger number of points), or you can just calculate the first derivation from the points you have (which will be better depends on how many points you actually have).

But really, this is just theory since we don't know the nature of data, or the language or anything.

For more on the subject search: numerical analysis on wiki, and go from there.

Answer 4

I think curve fitting might help you in this case. Here is a discussion which might be handy.

cheers