Sunday, March 29, 2009

A math primer

Not a very light topic for a Sunday, perhaps, but I like math so this post seems like fun to me. I'll try to keep it brief, but I think I need to pull together some of the myths that drive thinking about mathematical topics. I've written about one or more these before, but it's important to keep them in mind.

1) "Exponential" growth

One reads all the time about something that's experiencing exponential growth. Cell phone penetration, chip utilization, and so forth: many things are seen as following an inexorable growth of growth tendency.

Of course, that's not true, not possible, and I wrote about it about a year ago. But it bears repeating. Growth in any real situation more likely follows the logistic curve, the elongated S that features slow growth at the beginning, apparent exponential growth in its maturity, then a flattening as some kind of natural limit presents itself.

There is an upper limit on the number of cell phones we can possibly have; we can differ on that limit (I'd say 30 billion is higher than we will ever see), but it exists. Therefore, any business model or op-ed that depends on "exponential growth" is bound to be wrong. More importantly, at the middle of the S, the rate of growth begins to slow down. Finding this inflection point is vital to understanding the profile of the market in question.

2) Curve fitting

Much of the current discussion about what kind of recession we're having depends on historical precedent. Economists take what happened in the past and extrapolate it to the present, despite a lack of relevant data points (is our current situation like the Great Depression? Who knows, we've only had one?).

This attitude is typified by a recent post from Kevin Drum, where he argues:
If you want to know what's going to happen in the future, you should pay attention to what's happened before...."This time it's different" is probably the most dangerous phrase in the world. It's especially dangerous because every once in a while it's true. But not often.
Drum is arguing for a kind of determinism that is actually more dangerous than what he's saying. I'm not contending that one should accept convoluted arguments to ignore what is patently obvious, just that most real-world models are complex, made up of many components. Ignoring changes to assumptions or conditions has its own perils.

Some simple math. If you tell a student to fit a curve to data points (1, 2) and (2, 4), most will pretty easily come up with y = 2x. But here's what I can do, I can fit a slightly more complicated curve to these points: y = 2x + k(x - 1)(x - 2), where k is any number I want. This is a family of parabolas that pass through the two given points, but I can make one pass through any other point I want, simply by choosing the right k. (And there are literally an infinite number of other curves I could fit, including circles and squares and triangles.)

My point is, the fewer data points we have, the less likely we are to get a model that predicts anything. If I give you those two points, and ask you what y will be when x = 3, you may say 6, but I can make a model that will give any answer I desire. We need to be quite wary about using the past to predict the future.

3) Nonlinearity

[I'm going to be brief and imprecise here, give just enough detail to make my point.]

Nonlinear systems are, roughly, any real system of any complexity. They feature components that depend on one another, that are changed by the changes they themselves generate in other components. Weather is one such system. I suspect the global financial system is the same.

The main result of the study of such systems is that small changes in initial conditions can bring about huge changes down the road. One person decides to go out for a drive in Iowa, and the extra heat generated by burning the gas brings about a monsoon in Bangladesh. (That's an extreme example, and there is evidence that some systems, including weather, feature dampening effects that keep such extremes from happening, but the math is consistent with that.)

Therefore, it's virtually impossible to get a precise forecast of what any real nonlinear system will do. I would wager that we'll never have absolutely exact weather forecasts ("that corner will be 74ยบ with 48% humidity and a NNE wind at 7.6 mph a week from Thursday").

By the same token we can't really predict what, for example, a given stimulus package will do. It is not impossible that an $800 billion package will work perfectly, creating jobs and improving GDP, while $801 billion will force the economy into a chaotic condition that will be worse than what we have now (and, oddly, $802 billion might lead to a perfect result again). But we don't know that ahead of time, and we can't.

So we need to take all the certainties and forget them, and just hope that we will move in the right direction. But we also need to be realistic and understand one really knows.


Anonymous said...


Androcass said...

Thank you, Anon. The thing I'm proudest of is that I kept it relatively brief, because I thought of a whole bunch of stuff I wanted to throw in. I think it was wise to resist (besides, I had to leave time to watch basketball).

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