Information Technology Reference
In-Depth Information
The autocorrelation function is defined by
(τ)
=
Q
(
t
+
τ)
Q
(
t
)
C
Q
2
)
(2.42)
(
t
)
=
for a zero-centered process, that is, a process with
It is worthwhile to point
out that the two scaling relations imply that the underlying process is stationary in time,
meaning that it depends only on the difference between the two times, not on the times
separately, which can be expressed formally as
Q
(
t
0
.
.
Finally, the spectral density for this time series is given by the Fourier transform of
the autocorrelation function,
Q
(
t
1
)
Q
(
t
2
)
=
Q
(
t
1
−
t
2
)
Q
(
0
)
∞
e
i
ω
t
C
S
(ω)
=
FT
[
C
(
t
)
;
ω
]≡
(
t
)
dt
.
(2.43)
−∞
Introducing a scaling parameter into (
2.43
), inserting the scaling relation (
2.41
) and
changing variables,
z
=
bt
,
yields
e
i
ω
z
C
z
b
dz
b
=
∞
∞
1
b
2
μ
−
1
e
i
ω
z
C
S
(
b
ω)
=
(
z
)
dz
−∞
−∞
and subsequently the scaling behavior
b
1
−
2
μ
S
S
(
b
ω)
=
(ω).
(2.44)
The solutions to each of these three scaling equations, the second moment (
2.40
), the
autocorrelation function (
2.41
) and the spectrum (
2.44
), are of precisely the algebraic
form (
2.33
) of the solution to the renormalization-group relation, with the modulation
amplitude fixed at a constant value.
The scaling properties given here are usually assumed to be the result of long-time
memory in the underlying statistical process. Beran [
9
] discusses these power-law
properties of the spectrum and autocorrelation function, as well as a number of
other properties involving long-time memory for discrete time series. Here the result
corresponds to a second moment of the form
Q
2
t
2
μ
,
(
t
)
(2.45)
∝
which for
2this
is the second moment for an
anomalous
diffusive process. We discuss both classical
and anomalous diffusion subsequently in terms of both dynamics and statistical distri-
butions. But for the time being we are content with the realization that such processes
exist and are observed in nature.
μ
=
1
/
2 corresponds to ordinary or classical diffusion and for
μ
=
1
/
2.2.3
Autocorrelation functions and fractal dimensions
Before ending this section let us make a few observations about the generality of the
autocorrelation functions, the interpretation of the scaling indices and the properties of
