If the Curve Fits, Wear It

It is not uncommon in science and mathematics to have a collection of data points

(x₁, y₁),(x₂, y₂),...,(x_n, y_n)

for which we believe there is a functional relationship between the x_i and the y_i. Often, we want to get some idea of what might be happening at points other than those for which we have data. That is, we want to find some f so that f (x_i) = y_i for each of our points. Of course, in order to do this, we will need to have some knowledge (or make some guesses) about the function f, or sometimes about how precisely the points are known. In this chapter, we'll look at several different approaches to this problem of fitting a curve through (or near) some given data.