Next: Conclusions
Up: Signals with different local
Previous: Dependence on the size
Scaling Expressions
To better understand the complexity in the scaling behavior of components
with correlated and anti-correlated segments at different scales, we employ
the superposition rule (see [61] and Appendix 7.1). For each
component we have
![\begin{displaymath}
F(n)/n=\sqrt{[F_{\rm corr}(n)/n]^2+[F_{\rm rand}(n)/n]^2},
\end{displaymath}](img131.png) |
(7) |
where
accounts for the contribution of the correlated or
anti-correlated non-zero segments, and
accounts for the
randomness due to ``jumps'' at the borders between non-zero and zero segments
in the component.
Components with correlated segments (
)
At small scales
, our findings presented in Fig. 6(b)
suggest that there is no substantial contribution from
. Thus
from Eq. (7),
![\begin{displaymath}
F(n)/n\approx F_{\rm corr}(n)/n\sim b_0\sqrt{p}n^{\alpha},
\end{displaymath}](img132.png) |
(8) |
where
is the r.m.s. fluctuation function for stationary
(
) correlated signals [Eq. (6) and [61]].
Similarly, at large scales
, we find that the contribution of
is negligible [see Fig. 7(a)], thus from
Eq. (7) we have
![\begin{displaymath}
F(n)/n\approx F_{\rm corr}(n)/n \sim b_0pn^{\alpha}.
\end{displaymath}](img134.png) |
(9) |
However, in the intermediate scale regime, the contribution of
to
is substantial. To confirm this we use the
superposition rule
[Eq. (7)] and our estimates for
at small
[Eq. (8)] and large [Eq. (9)] scales[65]. The result we
obtain from
![\begin{displaymath}
F_{\rm rand}(n)/n=\sqrt{[F(n)/n]^2-[b_0\sqrt{p}n^{\alpha}]^2-[b_0pn^{\alpha}]^2}
\end{displaymath}](img135.png) |
(10) |
overlaps with
in the intermediate scale regime, exhibiting a
slope of
:
[Fig. 9(a)]. Thus,
is indeed a contribution due to
the random jumps between the non-zero correlated segments and the zero
segments in the component [see Fig. 5(c)].
Figure 9:
(a) Scaling behavior of components containing correlated segments
(
).
exhibits two crossovers and three scaling regimes
at small, intermediate and large scales. From the superposition rule
[Eq. (7)] we find that the small and large scale regimes are
controlled by the correlations (
) in the segments
[
from Eqs. (8) and (9)] while the
intermediate regime [
from Eq. (10)]
is dominated by the random jumps at the borders between non-zero and zero
segments. (b) The ratio
in the
intermediate scale regime for fixed
and different values of
, and
the ratio
for fixed
and
.
is obtained from Eq. (10)
and the ratios are estimated for all scales
in the intermediate regime.
The two curves overlap for a broad range of values for the exponent
,
suggesting that
does not depend on
[see
Eqs. (11) and (16)].
![\begin{figure}\centerline{
\epsfysize=0.55\columnwidth{\epsfbox{wn09f33.eps}}
\epsfysize=0.55\columnwidth{\epsfbox{wn5r4.eps}}}\vspace*{0.25cm}
\end{figure}](img142.png) |
Further, our results in Fig. 8(b) suggest that in the intermediate
scale regime
for fixed fraction
[see
Sec. 5.2.2], where the
power-law exponent
may be a
function of the scaling exponent
characterizing the correlations in
the non-zero segments. Since at intermediate scales
dominates
the scaling [Eq. (10) and Fig. 9(a)], from
Eq. (7) we find
. We also find that at intermediate scales,
for fixed segment size
(see Appendix 7.2,
Fig. 10). Thus from Eq. (7) we find
. Hence we obtain the following general expression
![\begin{displaymath}
F_{\rm rand}(n)/n\sim h(\alpha)\sqrt{p(1-p)}W^{g_c(\alpha)}n^{0.5}.
\end{displaymath}](img148.png) |
(11) |
Here we assume that
also depends directly on the type of
correlations in the segments through some function
.
To determine the form of
in Eq. (11), we perform the
following steps:
(a) We fix the values of
and
, and from Eq. (10) we
calculate the value of
for two different values of the
segment size
, e.g., we choose
and
.
(b) From the expression in Eq. (11), at the same scale
in the
intermediate scale regime we determine the ratio:
![\begin{displaymath}
F_{\rm rand}(W_1)/F_{\rm rand}(W_2)=(W_1/W_2)^{g_c(\alpha)}.
\end{displaymath}](img151.png) |
(12) |
(c) We plot
vs.
on a linear-log scale
[Fig. 9(b)]. From the graph and Eq. (12) we obtain the
dependence
![\begin{displaymath}
g_c(\alpha)=\frac{\log[F_{\rm rand}(W_1)/F_{\rm rand}(W_2)]}...
... \leq 1$}\\
0.50, \mbox{ for $\alpha>1$},
\end{array} \right.
\end{displaymath}](img153.png) |
(13) |
where
. Note that
.
To determine if
depends on
in Eq. (11), we
perform the following steps:
(a) We fix the values of
and
and
calculate the value of
for two different values of the
scaling exponent
, e.g.,
and any other value of
from
Eq. (10).
(b) From the expression in Eq. (11), at the same scale
in
the intermediate scale regime we determine the ratio:
![\begin{displaymath}
\frac{F_{\rm rand}(\alpha)}{F_{\rm rand}(0.5)}=\frac{h(\alph...
...c(\alpha)-g_c(0.5)}= \frac{h(\alpha)}{h(0.5)} W^{g_c(\alpha)},
\end{displaymath}](img156.png) |
(14) |
since
from Eq. (13).
(c) We plot
vs.
on a linear-log
scale [Fig. 9(b)] and find that when
[in
Eqs. (12) and (14)] this curve
overlaps with
vs.
[Fig. 9(b)] for all values of the scaling exponent
. From this overlap and from Eqs. (12)
and (14), we obtain
![\begin{displaymath}
W^{g_c(\alpha)}=\frac{h(\alpha)}{h(0.5)}W^{g_c(\alpha)}
\end{displaymath}](img160.png) |
(15) |
for every value of
, suggesting that
and thus
can finally be expressed as:
![\begin{displaymath}
F_{\rm rand}(n)/n\sim \sqrt{p(1-p)}W^{g_c(\alpha)}n^{0.5}.
\end{displaymath}](img162.png) |
(16) |
Components with anti-correlated segments (
)
Our results in Fig. 6(a) suggest that at small
scales
there is no substantial contribution of
and
that:
![\begin{displaymath}
F(n)/n\approx F_{\rm corr}(n)/n\sim b_0\sqrt{p}n^{\alpha},
\end{displaymath}](img132.png) |
(17) |
a behavior similar to the one we find for components with correlated segments
[Eq. (8)].
In the intermediate and large scale regimes (
), from the plots in
Fig. 7(b) and Fig. 8(a) we find the scaling behavior of
is controlled by
and thus
![\begin{displaymath}
F(n)/n\approx F_{\rm rand}(n)/n\sim \sqrt{p(1-p)}W^{g_a(\alpha)}n^{0.5},
\end{displaymath}](img164.png) |
(18) |
where
for
[see Fig. 9(b)] and
the relation for
is obtained using the same procedure we
followed for Eq. (16).
Next: Conclusions
Up: Signals with different local
Previous: Dependence on the size
Zhi Chen
2002-08-28