Information Technology Reference
In-Depth Information
|
q
=
λ
(
H
+
1
)
q
λ
−
q
α
α
|
q
Q
(λ
t
)
|
Q
(
t
)
|
ξ
q
ξ
λ
ζ(
q
)
,
=|
Q
(
t
)
|
(5.85)
yielding the
q
th-order correlation coefficient
ζ(
q
)
. We assume the statistics of the
fractional index
α
to be Gaussian such that
λ
−
q
α
e
−
σ
q
2
ln
λ
.
α
=
(5.86)
Thus, we obtain the correlation coefficient
q
2
ζ(
q
)
=
(
q
+
1
)
H
−
σ
.
(5.87)
The solution to the fractional Langevin equation is monofractal when the correlation
coefficient is linear in
q
, that is, when
σ
=
0; otherwise the process is multifractal.
Properties of multifractals
Partition functions [
4
] have been the dominant measure used to determine the multifrac-
tal behavior of time series covered by a uniform mesh of cell size
δ
. The typical scaling
behavior of the partition function
Z
q
(δ)
δ
→
in the limit of vanishing grid scale [
4
,
5
]
0,
Z
q
(δ)
≈
δ
τ(
q
)
,
(5.88)
determines the mass exponent
. However, there is no unique way to determine
the partition function and consequently a variety of data-processing methods has been
used for its calculation. Wavelet transforms have been used in studying the prop-
erties of cerebral blood flow [
32
], as well as in the investigation of stride-interval
time-series data [
25
], but the latter time series have also been analyzed using a random-
walk approach to determining the partition function. However, reviewing these data-
processing techniques would take us too far from our path, so we restrict our discussion
to the results.
The mass exponent is related to the generalized dimension
D
τ(
q
)
(
q
)
by the relation
τ(
q
)
=
(
1
−
q
)
D
(
q
),
(5.89)
where
D
(
0
)
is the fractal or box-counting dimension,
D
(
1
)
is the information dimension
obtained by applying l'Hôpital's rule to the ratio
q
→
1
τ(
)
q
)
=−
τ
(
(
)
=
)
D
1
lim
q
(5.90)
(
−
1
q
and
D
is the correlation dimension [
5
]. The moment
q
therefore accentuates differ-
ent aspects of the underlying dynamical process. For
q
(
2
)
0, the partition function
Z
q
(δ)
emphasizes large fluctuations and strong singularities through the generalized dimen-
sions, whereas for
q
>
0, the partition function stresses the small fluctuations and the
weak singularities. This property of the partition function deserves a cautionary note
because the negative moments can easily become unstable, introducing artifacts into
the calculation, and it was to handle various aspects of the instability that the different
ways of calculating the partition function were devised.
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