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
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
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
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.