cs249 lecture notes, Fall 2001
Week 14, Monday

Quiz Thursday on matrix operations.


Measuring fractals
------------------

With non-fractals, length converges on a limit as the
resolution of the measurement increases.

As scale gets smaller, resolution gets bigger.

Define resolution = 1/scale.

If you plot log(length) vs. log(resolution) you get a
flat line. 


With (true, mathematical) fractals, the length increases
constantly as the scale gets smaller.

For example, in a Koch curve, the ruler with scale 1
yields length 3.

At scale=1/3, the length is 4.

Every time we cut the scale by 1/3, we discover a new
level of complexity, and the length increases by 4/3.

As a sequence, we could write

s = (1/3)^k    and    u = (4/3)^k

Better to write it in terms of resolution = 1/s

1/s = 3^k

Taking the log, base 3, of both equations yields

log 1/s = k    and    log u = k log (4/3)

Combining yields:

log u = log(4/3) log (1/s)

Which means that on a log-log scale the length as
a function of resolution is a straight line with slope

log (4/3), which is 0.2619


Measuring real curves
---------------------

Coastlines tend to look like fractals.  Can we CHARACTERIZE
their behavior using the notion of length and scale?

Empirically, we can measure maps with increasing resolution.

Plotting the results on a log-log scale provides a natural
statistical test for fractal behavior.

Eventually, of course, this behavior breaks down in the real
world.

But over a wide range of scales, real coastlines appear fractal.

Furthermore, we can estimate the parameter d, in the relationship

log u = d log (1/s) + b

The intercept, b, has no physical meaning.  It's the length
when measured with a compass that has unit length.  So it
depends on the units you choose.

But d doesn't depend on units, or even on the base of the logarithm.

It's the FRACTAL DIMENSION, and it describes how fast the
length grows as resolution increases.

Raising both sides of the equation to a power, we get

u = c (1/s)^d

In other words, d is a measure of cragginess!

1) a circle has zero cragginess

2) the Koch curve is comparable to real coastlines

3) Britain gets 0.36, Maine is higher, California is lower


Interesting comment in the reading about Spain and Portugal.



How is this related to SOC?
---------------------------

It's mostly not, except that we got some good info about
log-log curves.

In the one-dimensional sand model, the distribution of
topple-sizes is Gaussian.

In the two-dimensional model, the behavior is qualitatively
different.  There are many more small topples, and a small
number of very large topples.

There is a theory that suggests that the distribution of
topple sizes is Pareto, which means that the number of
topples of size t is proportional to (1/t)^a

N(t) ~= (1/t)^a

Taking the log of both sides yields

log n = a log t + c

So if we generate a histogram of topple-sizes, and plot
it on a log-log scale, we should get something that looks
like a straight line, and the slope estimates the coefficient
a.


Why is this an interesting model?
---------------------------------

Empirically, we have seen that earthquakes and some other
natural phenomena obey the same power law.

Cellular automaton models suggest two, somewhat suprising,
results

1) an explanatory model of power laws (good thing)

2) a dent in the notion that events have causes (bad thing?)

Like chaos, SOC undermines what seemed to be an obvious
truth about cause and effect.

We would like to be able to predict earthquakes.  How does
the SOC model affect our approach to this problem?