Many dynamical systems create solution paths, or trajectories, that look strange and complex. These solution plots are called "strange attractors".
Some strange attractors have a fractal structure. For example, we saw in Part 3 that it was easy to create a fractal called a Sierpinski Carpet by using a
Here is a dynamical system that I ran across while doing computer art. I labeled it "Grow Brain" because of its structure. To see Grow Brain in action, make sure Java is enabled on your browser (you can turn it off afterward) and click here. (The truly paranoid can, of course, compile their own applet, since I provide the source code, as usual.)
The trajectory of Grow Brain is amazingly complex. But is it a fractal? That is, at some larger or smaller scale, will similar structures repeat themselves? Unlike the case of the Sierpinski Carpet, the answer to this question is not obvious for Grow Brain.
Some dynamical systems create fractal structures in
And some systems are all wet. Or maybe not, as the case may be.
For centuries, perhaps millennia, the yearly flooding of the Nile was the basis of agriculture which supported much of known civilization. The annual overflowing of the river deposited rich top soil from the Ethiopian Highland along the river banks. The water and silt were distributed by irrigation, and the staple crops of wheat, barley, and flax were planted. The grain was harvested and stored in silos and granaries, where it was protected from rodents by guard cats, whom the Egyptians worshipped and turned into a cult (of the goddess Bast) because of their importance for survival of the grain, and hence for human survival.
The amount of Nile flooding was critical. A good flood meant a good harvest, while a low-water flood meant a poor harvest and possible food shortage. The flooding came (and still comes) from tropical rains in the Upper Nile Basin in Ethiopia (the Blue Nile) and in the East African Plateau (the White Nile). The river flooding would begin in the Sudan in April, and reach Aswan in Egypt by July. (This would occur about the time of the heliacal rising of the Dog-Star Sirus, or Sothis, around July 19 in the Julian calendar.) The waters would then continue to rise, peaking in mid-September in Aswan. Further down the river at Cairo, the peak wouldn’t occur until October. The waters would then fall rapidly in November and December, and continue to fall afterward, reaching their low point in the March to May period. Ancient Egypt had three seasons, all determined in reference to the river:
A British government bureaucrat named Hurst made a study of records of the Nile’s flooding and noticed something interesting. Harold Edwin Hurst was a poor Leicester boy who made good, eventually working his way into Oxford, and later became a British "civil servant" in Cairo in 1906. He got interested in the Nile. He looked at 800 years of records and noticed that there was a tendency for a good flood year to be followed by another good flood year, and for a bad (low) flood year to be followed by another bad flood year.
That is, there appeared to be non-random runs of good or bad years. Later Mandelbrot and Wallis [1] used the term
Of course, even if the yearly flows were independent, there still could be runs of good or bad years. So to pin this down, Hurst calculated a variable which is now called a
Let me give a specific example of Hurst exponent calculation which will illustrate the general rule. Suppose there are 99 yearly observations of the height
Calculate a location
The first thing is to remove any trend, any tendency over the century for
The set of
Next we form partial sums of these random variables, each partial sum
Since the
{
will have a maximum and a minimum:
If we adjust
rescaled range =
Now, the probability theorist William Feller [2] had proven that if a series of random variables like the
where k is a constant (in particular, k = (p
/2)
In particular, for
Now, the latter equation implies log(R/c) = log k + ½ log 99. So if you ran a regression of log(R/c) against log(n) [for a number of rescaled ranges (R/c) and their associated number of years n] so as to estimate an intercept
log(R/c) =
you should find that
But that wasn’t what Hurst found. Instead, he found that in general the rescaled range was governed by a power law
where the Hurst exponent
This implied that succeeding
That this would be true in general for H > ½ , of course, needs to be proven. Nevertheless, to summarize, for reference, for the Hurst exponent H: H = ½ : the flood level deviations from the mean are independent, random; the
½ < H <=1: the flood level deviations are
0<=H< 1/2: the flood level deviations are
Recall that Bachelier had noted that the probability range of the log of a stock price would increase with the square root of time T. The probability range, starting at log S, would grow with T according to:
(log S – k T
where
(log S – k T
Hurst similarly said the rescaled range of the flood level varied according to (setting n = T):
R/c = k T
So it is tempting to equate the Hurst exponent H with the reciprocal of the Hausdorff dimension D, to equate H with 1/D = 1/a . But we must be careful.
Recall that symmetric stable distributions, with a
< 2, have infinite variance (for them, variance is a blob measure that is not meaningful). However, here in discussing the Hurst exponent we are assuming that the variance, and standard deviation (the scale
Nevertheless, the formal equation H = 1/D or D = 1/H yields the correct exponent for T in the case ½<= H <=1. Even though a
=2, the calculation of the Hausdorf dimension D yields D<2 if the increments are not independent. Hence D can take a minimum value of 1, D = 1/H = 1/1 = 1 when H=1, so that the process accumulates variation (rescaled range) much like a Cauchy sequence (T = T^{H}^{1/2}), or ordinary Brownian motion. [4]
Mandelbrot called these types of processes where a
=2, but where H ¹
½,
We are, of course, used to the idea of persistent phenomena in the stock market and foreign exchange markets. The NASD rises relentlessly for a period of time. Then it falls just as persistently. There are bull and bear markets, implying the price rise or decline is a
The US dollar rises relentless for a period of years, then (as it is doing now) begins a relentless decline for another period of years. In the case of the Nile, the patterns of rising and falling are partly governed by the weather patterns in the green rain forest of the Ethiopian highlands. In the case of the US dollar, the patterns of rising and falling are partly governed by the span of Green in the Washington D.C. lowlands.
[1] B.B. Mandelbrot & J. R. Wallis, "Noah, Joseph, and Operational Hydrology."
[2] W. Feller, "The asymptotic distribution of the range of sums of independent random variables."
[3] H. E. Hurst, "Long-term storage capacity of reservoirs."
[4] See also the discussion on pages 251-2 in Benoit B. Mandelbrot, J. Orlin Grabbe is the author of International Financial Markets, and is an internationally recognized derivatives expert. He has recently branched out into cryptology, banking security, and digital cash. His home page is located at http://orlingrabbe.org/ .
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from The Laissez Faire City Times, Vol 5, No 3, January 15, 2001 |