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Excellent job, but if in the end result the system is so chaotic, I'm not
sure that it could be proven. We really need a direct and simply elegant
proof.
Brent
-----Original Message-----
From: Alex Matulich [mailto:alex@xxxxxxxxxxxxxx]
Sent: Friday, January 10, 2003 12:45 PM
To: omega-list@xxxxxxxxxx
Subject: Re: FW: Random walk
Brent wrote:
> I know that randomness can be proven. Can non-randomness be proven?
Yes. I've done graduate study in Chaos Theory. Many seemingly random
phenomena have deterministic causes, and sometimes that determinism can be
derived from observing the 'random' data. There are even universal
constants associated with chaos theory which can be derived from dynamical
systems that are truly deterministic yet chaotic. Fractals play a big
role, and the fractal nature of markets was demonstrated over two decades
ago (see an early book by Edgar J. Peters for example) as an alternative
to the Efficient Market Hypothesis (the 'random walk theory').
I can give you a simple real-world example of random events for which one
can derive a deterministic relationship. Set up a faucet to drip from a
small orifice; preferably the drops should come from a small tube, but not
so small the the flow rate causes the water to spray out rather than drip.
You start out with drip, drip, drip, drip. Increase the flow rate slowly
and you eventually get a bifurcation: dripdrip, dripdrip, dripdrip.
Increase some more, and if your drops are small enough to resolve, you get
another bifurcation: dripdripdripdrip, dripdripdripdrip, dripdripdripdrip.
Increase the flow rate some more, and at some point the drips become
completely random; you can make a histogram of the time interval between
drips and find a reasonably uniform distribution of random numbers. For
all intents and purposes you could use them as random numbers.
But wait! Is it really random? If you generate a list of time intervals
between successive drops, t1, t2, t3, t4, etc., it looks like a list of
random numbers, but if you PLOT the points (t1,t2), (t2,t3), (t3,t4), etc.
on a graph, you get a bunch of dots that eventually fill into a smooth
curve. This means that the any given time interval is determined by the
one preceding it! Voila! You have derived the deterministic nature of
these chaotic "random" events. Therefore they're non-random.
You asked "can non-randomness be proven?" The answer is yes.
> If you doubt it, just tell me what the exact price of the Dow will be 5
> days from today at 9:00est.
Chaotic systems don't work that way. There's this aspect called SDIC, or
sensitive dependence on initial conditions. You can't predict a time
interval between two drops at 11:43:02 tomorrow just from knowing a time
interval today and iterating the nonlinear function you derived from
observations. A time interval of 0.15224 seconds might imply a time
interval of 0.02386 seconds at that time tomorrow, but any uncertainty in
that last decimal place can mean huge differences in tomorrow's
prediction, so 0.15223 seconds would predict 0.19000 seconds tomorrow.
My example with the drops is a really simple "map" between one event and
the next. Theoreticians have tried to derive a similar "map" for markets.
It's not one-dimensional like the drip map. And it's not necessarily
discrete either. And it's polluted by noise in the data. So it's a tough
problem.
> Could the truth be that the markets are both random and non-random? I
> guess that you didn't think of that.
Insofar as noise is random, markets have a random nature, yes.
> On a cloudy day you can see shapes in the clouds, like ships, faces,
What I see is a number of levels of fractal similarity.
-Alex
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