From: John Conover <john@email.johncon.com>
Subject: Re: SFI
Date: Sat, 26 Oct 1996 17:40:59 -0700
I have a few minutes while the compiler runs out. In 1/f to the n'th
stuff, ie., 1/f^n, measuring n can be done with a spectrum
analyzer. Obviously, if n = 2, then it is a Gaussian noise source that
has been passed through an integrator. (Unfortunately, conventional
statistics won't work any more-you have to differentiate it to work on
it with those kind of statistics.) Many, (most?,) times in nature, n
is not 2, but can be fractional, (1.5, etc.,) thus the name fractional
integration, or fractal process. If n = 2, then it is called
fractional Brownian motion. And here is the corker. Two fractional
Brownian voltages do not add root mean square-they add linearly. (So,
trying to play games by restricting the bandwidth because the noise
voltage goes down faster than the information/data rate voltage
doesn't work any more.) In point of fact, n is related to the root
mean square in v0^2 = v1^2 + v2^2, and in fractional Brownian process'
it is v0^1 = v1^1 + v2^1.
Note that there are other implications. If n = 2, then the range of
variation in a graph will be proportional to the square root of time,
ie., if we measure the range of a stock's price over one year, then
over a two year period, we would expect the range to increase by 1.414
times. The root means square of the stock's price would remain
constant. (And, within reason, this is the case.) We would also expect
manufacturing variances to do the same, (irregardless of what MBO and
Harvard say-which are based on the wrong kind of statistics.)
Bottom line is that the conventional statistics are good only if n =
2, and are simply a single subset of a whole family, (actually
infinite,) of statistical systems.
If n > 3, then the system is termed "chaotic." These systems exhibit
cyclic, (not to be confused with periodic,) phenomena. Sun spots are
an example. (They are not periodic, because the "period" seems to
wonder around at random.) Spice is a good program, (using electronic
chaos generators,) to experiment with such things-see "IEEE Circuits
and Systems." Someone is always coming up with a new chaos generator.
A good way of thinking about such things is a coin toss game. The
graph of the gambler's capital is a fixed increment fractal, (Brownian
in nature, but not fractional.) Note that the capital will have large
swings, going away from zero, (ie., the original starting point,) for
many tosses, ie., it will look like, (and is,) a 1/f type of process.
Note that it is a trend-reinforcing process, (once it goes in one
direction, chances are, it will continue for a long time-ie., it is no
longer a 50-50 proposition where conventional statistics are
applicable.) Note that if you play two games concurrently, the two
capitals add linearly, and not root mean square.
John
--
John Conover, john@email.johncon.com, http://www.johncon.com/