Blogtrek

Predictable/Unpredictable

With situations of the kind that hurricanes posed, it seems that things are either in fine agreement or they are haywire all over the place. There is no middle ground. With Isabel, the first few runs were all over the place: sea, Florida, Baltimore, New England, Nova Scotia. But then after a few days they consistently pointed a finger at North Carolina and Virginia. There was no mildly unpredictable phase. It is either precise enough to predict accurately or you can say absolutely nothing about it.

Similar situations happen in mathematical systems. For example, in estimating a quantity, such as the length of the tropical year in solar days, 365.2422, the first three decimals are just about absolutely accurate. The fourth one is about right, the 2, but it is fuzzy and I have heard estimates that begin with 365.2421. After that the digits are totally unpredictable and depend on the irregular rotation of the earth. Another example would be to observe that if x is the square root of 2, then x = 1 + 1/(1+x) so if one guesses a value for x, substitute it into 1 + 1/(1+x) and make that your revised value of x, one may hope to obtain sqrt(2). If you set x = sqrt(2) you get sqrt(2) back, but if you are off by a little bit, say x = 1.414213562373, then the next few approximations are about this value but they lose a decimal place of accuracy each iteration. Still, it is good enough for practical purposes. When this gets back to the units digits, the whole thing becomes unpredictable. Try it out on a spreadsheet and you see what I mean.

Why does this happen? Both the hurricane and the iteration formula are unstable equilibria. If you are right on the money, you remain so, but if you are not, you eventually get thrown off the track. And that is what happened to the first few predictions for Isabel, 12 days ago.