Polynomial fitting - how to fit and what is _polynomial fitting_

Question

I don't understand what is polynomial fitting.
Can anyone explain to me how to fit a curve to given points?

Answer 1

There are two concepts frequently conflated: interpolation and fitting. I'll discuss both since you don't seem to know what you really want.

Let's say you have a set of $n$ points

$$(x_i,y_i)\qquad i=1\dots n$$

Interpolation is the problem of finding the (unique) polynomial $p(x)$ that passes through all your given points (under the assumption that no two points have the same abscissa), i.e. $p(x_i)=y_i \qquad i=1\dots n$. Your resulting polynomial is of degree $n-1$. The usual techniques for finding the interpolating polynomial are the methods of Lagrange, Newton, and Neville-Aitken.

Fitting on the other hand assumes your data is contaminated with error, and you want the polynomial that is the "best approximation" to your data. Here polynomial interpolation does not make much sense since you do not want your function to be reproducing the inherent errors in your data as well. Least-squares is a common technique: it finds the polynomial $f(x)$ such that the quantity

$$\sum_{j=1}^{n}\left(f(x_i)-y_i\right)^2$$

which measures the departure of your polynomial from the ordinates is minimized (here the assumption is that your abscissas are error-free, and the error in your ordinates is normally distributed). The degree of $f(x)$ can be (and is often) less than $n$. A number of techniques for this are used as well: there's the normal equations, and then there are special matrix decompositions that can be used to efficiently solve this problem.