From: John Conover <john@email.johncon.com>
Subject: Re: Chaos v. survival (was Re: General Equilibrium Model)
Date: 28 Dec 1998 21:36:06 GMT
Don Libby writes:
> conover@netcom.com wrote:
> >
> > jim blair writes:
> > > Goetz Kluge wrote:
> > >
> > > > Markets can be at equilibrium or close to it.
> > >
> > > I agree that it is theoritically possible for a market or a society to
> > > "be at or close to" equilibrium. Ancient Egypt might have been several
> > > thousand years ago, or maybe Europe from the fall or Rome until the late
> > > 1400's.
> > >
> > > But has any industrial nation in the 20th century ever been "at
> > > equilibrium"? I can't think of one.
> > >
> >
> > Or, perhaps the run lengths, (ie., duration,) of cultures assemble
> > their self into an erf (1 / sqrt (t)) distribution, which would be
> > expected if the durations could be modeled as a random walk, (and I
> > wouldn't attempt an interpretation of the meaning of that.)
> >
> > Although the data is sparse, and, at best, subjective, it does provide
> > a reasonably, (depending on one's POV, of course, of the historical
> > perspective,) accurate fit.
> >
> > If it really is, and if we select the time unit of the duration of
> > cultures to be centuries, the Egyptians, at about three millenia,
> > would have won the lottery, with a chance of one in 1 in 6-at least so
> > far, (the ancient cultures of India are not far behind.) And, the
> > "average" duration of a culture would be about four centuries-50%
> > would last less, 50% more, (and the historical perspective fits the
> > random walk model fairly accurately.)
> >
> > John
> >
>
> John Conover has some interesting ideas about "null" models for duration
> data, which might go some distance toward a formal definition for
> "sustainable societies". I am curious to know how his autopoeitic erf
> random-walk model fits in relation to other potential models for the
> statistical distribution of duration times (normal? gamma? Weibull?) and
> if this model has been formally tested?
>
> Survival analysis is the most widely applied statistical technique for
> the analysis of duration data, for everything from "failure times" in
> engineering research, to mortality in population biology, to "event
> histories" in sociology. Can John please provide a reference or two
> discussing (a) the origin of the "erf" distribution generated by random
> walks, (b) the fit of this distribution to data for the duration of
> civilizations, (c) formal equivalence of these "chaos theory"
> propositions to conventional survival analysis?
>
Hi Don. All I did was to make the assumption that such things are
fractal in nature, and picked the simplest fractal-a random walk-as a
first order approximation. Theoretically, if the assumptions are
correct, the distributions of run lengths should be the error function
(ie., erf,) of the reciprocal of the square root of time, eg., erf (1
/ sqrt (t)), which is 1 / sqrt (t) for t > 1. I then thumbed through
Michael Wood's "Legacy" and made a list of what
historians/anthropologists viewed the run lengths of various
cultures/societies to be, assembled them into a frequency
distribution, and compared this graph with erf (1 / sqrt (t)).
Probably not the best science, but intriguing since the two graphs
seem to have the same shape. (I wouldn't jump to conclusions, however,
since the data set size is pitifully small-and fractal analysis likes
a lot of data.)
The Legacy book is "Legacy: The Search for Ancient Cultures", Michael
Wood, Sterling Publishing Company, New York, New York, 1992, ISBN
0-8069-0863-7.
For a derivation/explanation of why run lengths assemble their self
into a erf (1 / sqrt (t)) frequency distribution, see "Fractals,
Chaos, Power Laws: Minutes from an Infinite Paradise", Manfred
Schroeder, W. H. Freeman and Company, New York, New York, 1991, ISBN
0-7167-2136-8, pp. 153.
Note that the general formula for the simple random walk fractal is
very similar to the formula for the logistic function, (AKA, the
iterated discreet time quadratic equation,) used in population studies
as suggested by Pierre Francois Verhulst in 1845, et al. The logistic
function is one of the simplest formulas that exhibits chaotic
phenomena, also. If you take the formula for a random walk, and add a
non-linear term, you have the logistic equation. For particulars, see
"Chaos and Fractals: New Frontiers of Science", Heinz-Otto Peitgen and
Hartmut Jurgens and Dietmar Saupe, Springer-Verlag, New York, New
York, 1992, ISBN 0-387-97903-4, 3-540-97903-4.
John
BTW, Don, the software I used is available at
http://www.johncon.com/ndustrix/archive/fractal.tar.gz, which contains the C
sources to about 50 programs useful for hacking fractal time series.
--
John Conover, john@email.johncon.com, http://www.johncon.com/