From: John Conover <john@email.johncon.com>
Subject: Re: A year ago today, the NASDAQ was at an all time high
Date: 13 Mar 2001 23:14:36 -0000
Yes, "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, pp. 694, is an outstanding
reference.
Its a text book, and fairly expensive, ($50 range,) and if I could buy
but one book, this would be it. I don't know if it is available in
paperback.
The book is the compendium of the modern study of things chaos and
fractal, and I found it interesting, and highly recommend it, (the
idea for the tsdeterministic program came from this book.)
Another favorite that covers Poincare stuff extensively, (for the lay,
and not so lay, reader,) is available in paperback, "Does God Play
Dice?: The Mathematics of Chaos", Ian Stewart, Blackwell Publishers,
Cambridge, Massachusetts, 1992, ISBN 1-55786-106-4. I highly recommend
this book, (and its fun to read.) Its SR is under $20 at Computer
Literacy, maybe cheaper at Amazon.
I was trying to avoid vernacular, but if you go to the library, (or
Computer Literacy,) search the indices for the phrases "Poincare map"
or "Poincare section". Those are the technical names for what we were
discussing.
John
BTW, also, "Fractals, Chaos, Power Laws", Manfred Schroeder,
W. H. Freeman and Company, New York, New York, 1991, ISBN
0-7167-2136-8, (another of my favorites,) pp. 243 and 324 looks into
the subject, too. It is available in paperback. This book is another
of my favorites-I have quoted this book more than any other.
Jeff Haferman writes:
>
> Thanks for the explanation, John.
> BTW, is the Peitgen et al reference a good place for
> background on the state-phase portrait? Can you
> recommend other literature?
> Jeff
>
>
> John Conover wrote:
> >
> >Hi Jeff. If you let your random number generator go long enough,
> >tsdeterministic will show that the pseudo-random sequence repeats.
> >
> >For a 32 bit machine, that will be a sequence of a minimum of 4
> >billion numbers before the sequence repeats for another 4 billion
> >numbers, (most random number generators use two integers, for a
> >sequence free interval of 1.8e19, which is near Heisenburg's
> >uncertainty of e22, which is practical enough for anything in the real
> >world.)
> >
> >The parametric geometric figure you get from tsdeterministic will not
> >be a parabola, (except for the logistic function, which is also known
> >as the discreet time parabolic function, because its parametric
> >geometrical map is a parabola-which you showed.) The logistic function
> >is but one of a family of discreet time functions-each of which has a
> >characteristic parametric geometric plot-including all pseudo-random
> >number generators.
> >
> >The reason you see a straight line when running tsdeterministic on the
> >the DJIA time series is that a parametric geometrical map of an
> >exponential is a straight line.
> >
> >The way you get from one time point, to the next, is to multiply the
> >point by a constant, (1.0003... in the case of the DJIA.)
> >
> >That's a formula for compound interest, which has an exponential
> >characteristic.
> >
> >Tsinvest does the same thing. If (v(n) - v(n - 1)) / v (n - 1) is
> >constant, then it is an exponential. The average, avg, and root mean
> >square, rms, of these can be used:
> >
> > P = (avg/rms + 1)/2
> > G = (1+rms)^P * (1-rms)^(1-P)
> >
> >to do the same thing. (If rms = avg, then P = 1, and G = 1 + rms.)
> >
> >The slope of the parametric geometric plot is G.
> >
> >So, you have just verified the validity of the methods used in
> >tsinvest, using non-linear dynamical system, (NLDS, e.g., chaos,)
> >methods, (a state-phase portrait, a la Poincare, to be exact; which is
> >kind of trick-the parametric geometric map, or state-phase portrait,
> >is not a function of time.)
> >
> > John
> >
--
John Conover, john@email.johncon.com, http://www.johncon.com/