Thursday, September 26, 2013

Chaotic Systems

Corrections has yet to meet anyone who is good at forecasting much of anything.  Why might this be?  One reason is bad statistical models.  Another reason (that we are not overly sympathetic towards!) might be chaotic systems.  The present may perfectly and completely determine the future, but the near present may not have any power at predicting the future.

One simple example of this is the sequence x(t+1)=4x(t)*(1-x(t)).  The sequence will bounce around for a while between 0 and 1 (given we avoid a few bad starting states like {0, 0.25, 0.5, 0.75, 1}) and be completely deterministic.  Surely it wouldn't be hard to forecast, right?  

Wrong.  If your starting point (initial information used for forecasting) deviates the slightest amount, your sequence soon becomes completely different than if you had used the true starting point.  Below, Corrections depicts two such starting values:  X(0)=0.1 and X(0)=0.24, and plot the series for 100 periods (click to enlarge).
What if we were really, really, really close?  If we start out with a percent error of merely 0.0001%, then shouldn't our forecasts match up?  They do, for a while, but diverge rather quickly for having a one-part-in-ten-million difference (click to enlarge).
Does one series provide any forecast of the other, or have a recognizable pattern?  Below, Corrections depicts the two series against one another after the 20th period:  they no longer have a discernible relationship (click to enlarge).
This is one possible reason why the vast majority of sophisticated forecasts Corrections has heard (that don't suffer from selection) have been wrong.   We don't put much stock in it, however.

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