One particularly clear example is the
fern,
in which each frond is composed
of many sub-fronds, each a miniature copy of it.
This scaling goes further; each sub-frond is composed of sub-sub-fronds, and
would extend to arbitrarily small scales for a fully fractal mathematical
object.
The Cantor set
is an example of a one-dimensional fractal structure which
extends to arbitrarily small scales.
One particular mathematical construction of this set begins with the unit
interval, then the middle third is removed.
Next the middle thirds of both remaining parts, are removed, then the middle
thirds of the four smaller pieces, and so on without limit.
The resulting set consists of two exact copies of itself, each of which is
one third the size of the whole.
Frequently, objects are self-scaling in a statistical sense, and, when
scaled, parts of the whole fit the whole in distribution, rather than being
exact copies.
Again, nature provides many examples, one broad class of random fractals
being branching processes.
For many objects, such as river networks, blood supply systems, and
bronchial trees, the parts resemble the whole but do not match it
exactly. One example is the
Grand Canyon.
Indeed, the Cantor set
can be similarly randomized to produce a
one-dimensional random fractal, or fractal stochastic process.
Scaling leads mathematically to power-law dependencies in the scaled quantities. Consider a real, continuous function for which scaling the argument results in scaling the function value by some other factor, related to the scale but independent of the argument:
The only non-trivial solution of this scaling equation for arbitrary a and x is
for some constants c and d. Thus fractals and power-law functions are closely related.
One-dimensional stochastic processes are simply random functions of time, of which there are two main types. A continuous-time random process has a well-defined value at all times, and often takes on a wide range of values at different times. Moments of the instantaneous value of the process, autocorrelation function, and higher-order moments are useful statistics for these processes. In contrast, an (unmarked) point processes (or dust) is defined as a random collection of points in time, the points usually taken to be indistinguishable from each other. Relevant statistics include the probabilities of obtaining a certain number of points in a specified time, the probability of adjacent points being separated by a certain time interval, and more complicated statistics such as descriptions of how the points cluster together.
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