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Chaos Theory Cleared Up

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More than a little interested in statistical analysis

Chaos Cleared Up (almost)

Many people errantly confuse chaos with randomness. But:

  • You have a random sequence of events when there's no correlation between each event.
  • In chaos theory, a seemingly random event causes another event which causes another event. The outcome appears to have happened randomly, but in fact it was caused by a sequence of events.

The most famous illustration of this is the butterfly that flaps its wings in some far off place. This causes a minor air disturbance, which causes the angle of airflow to change ever so slightly. But angles over a distance can mean enormous displacements. Thus, a butterfly flapping its wings leads to a hurricane some weeks later. This particular illustration is why chaos theory is often called the "butterfly effect."

For more modern audiences, this video illustrates chaos with a new twist on the butterfly effect:

The events weren't random. There were causally related, and they occurred in a definite sequence. Take any one of them out of sequence, and the man's roof would still be intact.

Some people try to explain chaos theory in terms of probability. But have you ever noticed that the examples used are quite improbable? While chaos theory did arise from statistical analysis, it's not really a statistical tool. Randomness, however, is. In fact, you need randomness (as a control) for statistical validity. Randomness is not haphazard, and it can be measured. Since randomness isn't chaos, let's now leave it behind.

One factor in chaos theory is recurrence. This means the return of a system to its starting point (or initial conditions), how it was before an event triggered a change. This concept is key in electrical theory, but we don't need to go into the particulars here. The amount, speed, direction, and other characteristics of recurrence depend on how sensitive the system is to the initial condition, a characteristic we call dependence. This term means other things in other disciplines, so don't read too much into it. Recurrence and dependence are the two main drivers of the subsequent events. We call that flow of events chaotic motion.

Unfortunately for analysts trying to predict outcomes such as the weather or stock market prices, recurrence and dependence alter each other and can "oscillate" out of the existing pattern plus other variables can change these two and each other, really making a mash of things. This means there's a natural time limit to the accuracy of predictions. This isn't linear, either. Think more in terms of a J curve.

Another factor in chaos theory is the correlation of the events. Events with a low correlation (they aren't relevant to each other) can be excluded from a study of chaos. Why? Because, as noted, in chaos theory you're looking at a sequence of related events. Typically, the relationship is causal (one event causes another), dependent (the event would not have happened without the preceding event(s)), or both.

As you can probably see, chaos theory is complex. Which brings up another point. It's related to complexity theory. This theory is typically stated as "Critically interacting components self-organize to form potentially evolving structures exhibiting a hierarchy of emergent system properties."

We typically think of "chaos" as a state of disorder (thus the confusion with "random"). But as you can see now, it's not disorder. It's just an unexpected order. The challenge is to predict that order. This challenge has given rise to an entire field of mathematics, called chaos theory. It's a very useful tool for predicting changes in dynamic systems when there's a minor change to initial conditions. The problems, though are:

  • The math makes your head spin.
  • Chaos theory models can almost never include all data and all variables. Thus, accuracy is time-limited.
  • Recurrence and dependency interact, changing each other. Unknown variables change these and each other, further limiting the time window of accuracy.
  • Confidence in results takes a nosedive after a time; knowing exactly when that time is would require a complete analysis in itself, subject to confidence degradation itself.

So, this should help you get a better idea of what chaos theory is about. And what it's not about.

 

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