From: John Conover <john@email.johncon.com>
Subject: Re: A year ago today, the NASDAQ was at an all time high
Date: 14 Mar 2001 00:50:31 -0000
And, what's all this mean? Complex systems are called that because
they are complex, right?
Wrong.
The tsdlogistic, (e.g., the discreet time logistic function,) program
works as follows:
x(t) = x(t - 1) * (a + b * x(t - 1))
which means that to get the next value, multiply the last value by b,
add a to it, and multiply that by the last value.
Not very complicated at all. 2 variables, (a and b,) and 3 operations
from the arithmetic, (one addition, two multiplies.)
If a = -b = 4, (or so,) it produces noise, (its a primitive
pseudo-random number generator; not a very good one, by today's
standards-although it was used as such for many years.)
For a = -b < 3.5, (or so,) it produces an oscillatory function, (that
just happens to model the indeterminacy of predator-prey ecology,
where it is used extensively.)
And, for a = -b = 3, (or so,) it produces an s-shaped curve that
models many saturated types of things, (like market saturation, etc.)
Its remarkable that such a simple equation/process could generate
such a rich assortment of characteristics.
John
BTW, if you make some time series from the tsdlogistic program, and
try to measure the two parameters, a and b, even knowing the equation,
you can't do it. You can use the tslsq program, (which will attempt to
curve fit a logistic function.) However, when you use the two
parameters derived from the tslsq program in the tsdlogistic program,
you end up with a different characteristic.
Its a fun thing to try, and observe that even the floating point unit
in the PC is not adequate for the job. These things are globally
stable, and everywhere, locally, unstable-driving the truncation
requirements of the machine to impractical limits. One can do short
term prediction, but not long. These systems are plagued with
numerical stability issues. The Rucker book explains why.
But its a fun thing to do.
John Conover writes:
>
> And, one last book.
>
> Fractal characteristics are pervasive in complex social and
> economic systems. Why? Section V, Chapter 12 of "Archimedes'
> Revenge", Paul Hoffman, Fawcett Crest, New York, New York, 1993,
> ISBN 0-449-21750-7, delves into the issue. Kenneth Arrow's
> so-called "Impossibility Theorem" was the first formalization of
> self-referential indeterminacy in complex systems, which is why
> they are called complex. The profession of economics is still
> doing damage control, since there is no answer to the optimal
> left-right, social welfare function, etc. A fractal solution is
> the only stable solution to such things, (if you can call a
> fractal, stable.) Self-referential indeterminacy is the engine of
> things fractal in complex systems, and leads back to the concepts
> in Rucker's book.
>
> John
>
> John Conover writes:
> >
> > While we are on the subject of good books about things fractal and
> > chaos, here are a few of my favorites:
> >
> > "Mind Tools", by the logician Rudy Rucker, (more noted as a
> > science fiction author, and a world renowned mathematician-he
> > personally worked with Kurt Goedel,) Houghton Mifflin Company,
> > Boston, Massachusetts, 1993, ISBN 0-395-46810-8, (paperback.)
> > Rucker is at the forefront of science in the 21'st century,
> > (looking after the formal issues,) and the book is about
> > complexity, and how information theory, math, logic, and
> > randomness are all tied together. It is non-technical, but that
> > doesn't mean the reader does not have to think. If I were king,
> > all high school diplomas would be issued only after a student had
> > demonstrated competency with the content of this book.
> >
> > The Stewart book, (mentioned previously,) "Does God Play Dice?:
> > The Mathematics of Chaos", Ian Stewart, Blackwell Publishers,
> > Cambridge, Massachusetts, 1992, ISBN 1-55786-106-4. It is very
> > heartening that a mathematician of the stature of Stewart would
> > set down and write a book about randomness for the lay
> > person. This book addresses predictability, randomness, and
> > entropy, and how they are related, (and how they are not.) Note
> > that we really don't know what the word random means. (Could you
> > explain it?)
> >
> > "The Jungles of Randomness: A Mathematical Safari", Ivars
> > Peterson, John Wiley & Sons, New York, New York, 1998, ISBN
> > 0-471-16449-6, a good introductory book on simple fractals, with
> > real-life examples, and how fractals, chance, and randomness work.
> > Perhaps available now in paperback.
> >
> > "What is Random?: Chance and Order in Mathematics and Life",
> > Edward J. Beltrami, Springer-Verlag, New York, New York, 1999,
> > ISBN 0-387-98737-1, (I don't know about paperback.) A bit more
> > technical and formal approach to randomness, and how it is tied
> > into information theory.
> >
> > John
> >
> > John Conover writes:
> > >
> > > 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/