Next: Dependence of on the
Up: Noise with Power-law trends
Previous: Noise with Power-law trends
Dependence of
on the power
First we study how the rms fluctuation function
for a
power-law trend
depends on the power
. We find that
![\begin{displaymath}
F_{\rm P}(n) \sim A_{\rm P}n^{\alpha_{\lambda}},
\end{displaymath}](img293.png) |
(19) |
where
is the effective exponent for the power-law trend. For positive
we observe no crossovers in
(Fig. 10(a)). However, for negative
there is a crossover in
at small scales
(Fig. 10(b)), and we find that this crossover becomes even more pronounced with decreasing
or increasing the order
of the DFA method, and is also shifted to larger scales [Fig. 11(a)].
Figure 11:
Scaling behavior of rms fluctuation function
for power-law trends,
, where
and
is the length of the signal. (a) For
,
exhibits crossover at small scales which is more pronounced with increasing the order
of DFA-
and decreasing the value of
. Such crossover is not observed for
when
for all scales
[see Fig. 10(a)]. (b) Dependence of the effective exponent
on the power
for different order
of the DFA method. Three regions are observed depending on the order
of the DFA: region I (
, where
; region II (
), where
; region III (
), where
. We note that for integer values of the power
, where
is the order of DFA we used, there is no scaling for
and
is not defined, as indicated by the arrows. (c) Asymptotic behavior near integer values of
.
is plotted for
when DFA-2 is used. Even for
, we observe at large scales
a region with an effective exponent
, This region is shifted to infinitely large scales when
.
![\begin{figure}
\centerline{
\epsfysize=0.55\textwidth{\epsfbox{mdev.eps}}
\ep...
...\epsfysize=0.55\textwidth{\epsfbox{near1.eps}}}
\vspace*{0.5cm}
\end{figure}](img295.png) |
Next, we study how the effective exponent
for
depends on the value of the power
for the power-law trend. We examine the scaling of
and estimate
for
. In the cases when
exhibits a crossover, in order to obtain
we fit the range of larger scales to the right of the crossover. We find that for any order
of the DFA-
method there are three regions with different relations between
and
[Fig. 11(b)]:
- (i)
for
(region I);
- (ii)
for
(region II);
- (iii)
for
(region III).
Note, that for integer values of the power
(
), i.e. polynomial trends of order
, the DFA-
method of order
(
is also an integer) leads to
, since DFA-
is designed to remove polynomial trends. Thus for a integer values of the power
there is no scaling and the effective exponent
is not defined if a DFA-
method of order
is used [Fig. 11]. However, it is of interest to examine the asymptotic behavior of the scaling of
when the value of the power
is close to an integer. In particular , we consider how the scaling of
obtained from DFA-2 method changes when
[Fig. 11(c)]. Surprisingly, we find that even though the values of
are very small at large scales, there is a scaling for
with a smooth convergence of the effective exponent
when
, according to the dependence
established for region II [Fig. 11(b)]. At smaller scales there is a flat region which is due to the fact that the fluctuation function
(Eq. (3)) is smaller than the precision of the numerical simulation.
Next: Dependence of on the
Up: Noise with Power-law trends
Previous: Noise with Power-law trends
Zhi Chen
2002-08-28