Sunday, September 03, 2006

The religion of numbers

I had a chance to read a paper on interpolation of interest rates not long ago. The idea is, given a few data points, to generate a "smooth" curve that goes through them. Now, natural cubic splines are the ideal tool for this because it is them who minimize the integral of the absolute value of their second derivative --- in other words, they are the function that oscillates less around the sampling points.

To my surprise, this paper criticized natural cubic splines because, when sampled outside the range of sample points, they behave like a line and lose their curviness. Well, of course they will, because they are designed to minimize their waviness.

But what was the solution? Just to add a sample point further out in the future according to what is desired, right? No. To interpolate these values with quartic splines instead to keep curvy behavior when extrapolating.

Now, quartic splines are considerably more complicated to manage, they are not the smoothest interpolating function, and yet the decision is made to use them? Exactly why is this? Because a particular behavior is wanted when extrapolating.

But if you already know the kind of values you want, why come up with a more complicated curve instead of saying what the desired values are? Why not add a point further out in time so that cubic splines can be curvy too? Besides, it is what you want, right?

Perhaps there's some interesting argument I am missing. From what I saw in this paper, though... I can't help feeling that this is just a matter of a function being trusted more than a data point. Why put faith on a function to give you something you find close enough to what you want, instead of specifying what you already knew you want for an interpolation mechanism to use it? I don't get it.

2 comments:

Anonymous said...

Hehe. I remember how surprised I was to learn in the nuclear industry that 13th order polynomial fits were the standard used, before micronodal simulations became feasible with enough memory.

Travis

Andres said...

Nuclear plants riding Runge's phenomenon!!! Aaaaiiieeee!!!