Information Technology Reference
In-Depth Information
generalized Weierstrass function (GWF)
W
satisfies the scaling relation (
2.33
) and
consequently shares the scaling properties of the renormalization solution (
2.36
). The
non-differentiability of the GWF arises because each new term in the series inserts an
order of magnitude more wiggles (in base
b
) on a previous wiggle, but each new wig-
gle is smaller in amplitude by an order of magnitude (in base
a
). Recall Bernoulli's
game of chance in which the scale of the wager increases by a factor
b
for an outcome
whose likelihood decreases by a factor of 1
(
t
)
/
a
. This Bernoulli scaling appears in the
Weierstrass function as well.
Weierstrass was the teacher of Cantor. Later, as a colleague, Cantor was interested in
questions of singularities and the various kinds of infinities; the apprentice challenged
the master to construct a function that was continuous everywhere but was nowhere
differentiable. The resulting Weierstrass function became the first non-analytic function
having subsequent utility in the physical sciences. The scaling of the GWF does not in
itself guarantee that it is a fractal function even though it is non-analytic, but Mauldin
and Williams [
47
] examined its formal properties and concluded that for
b
>
1 and
b
>
a
>
0 the dimension is essentially given by
D
=
2
−
μ,
(2.39)
a result established numerically somewhat earlier by Berry and Lewis [
10
].
Now we restrict the discussion to a special class of complex phenomena having the
scaling properties found in a large number of data sets. This focus has the advantage of
providing a tractable analytic treatment for a wide class of webs where all or almost all
the stated properties are present. Consider the homogeneous function
Z
introduced
above. We use the renormalization-group argument to define the scaling observed in the
averages of an experimental time series with long-time
memory
.
1
Suppose the second
moment of a stochastic process
Q
(
t
)
(
t
)
having long-time memory is given by
Q
2
b
2
μ
2
(
bt
)
=
Q
(
t
)
,
(2.40)
where the angle brackets denote an average over an ensemble of realizations of the
fluctuations in the time series. Although it is probably not clear why such a function
should exist, or why it is appropriate to discuss such functions here, let us assure you
that this scaling is closely connected to the scaling we have observed in the data of the
last two chapters.
For the same process whose second moment is given by (
2.40
) a different but related
scaling is given by the autocorrelation function
b
2
μ
−
2
C
C
(
bt
)
=
(
t
).
(2.41)
1
In this chapter we use the intuitive term “memory” to introduce the mathematical concept of time correla-
tion. A more formal definition of memory is provided subsequently by introducing memory functions as
kernels in integro-differential equations for the fractional calculus.
