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Dependence of
on the signal length
Here, we study how the rms fluctuation function
depends on the length
of the power-law signal
(
). We find that there is a vertical shift in
with increasing
[Fig. 13(a)]. We observe that when doubling the length
of the signal the vertical shift in
, which we define as
, remains the same, independent of the value of
. This suggests a power-law dependence of
on the length of the signal:
![\begin{displaymath}
F_{\rm P}(n) \sim \left(N_{max}\right )^{\gamma},
\end{displaymath}](img323.png) |
(21) |
where
is an effective scaling exponent.
Next, we ask if the vertical shift depends on the power
of the power-law trend. When doubling the length
of the signal, we find that for
, where
is the order of the DFA method, the vertical shift is a constant independent of
[Fig. 13(b)]. Since the value of the vertical shift when doubling the length
is
(from Eq. (21)), the results in Fig. 13(b) show that
is independent of
when
, and that
, i.e. the effective exponent
.
For
, when doubling the length
of the signal, we find that the vertical shift
exhibits the following dependence on
:
, and thus the effective exponent
depends on
--
. For positive integer values of
(
), we find that
, and there is no shift in
, suggesting that
does not depend on the length
of the signal, when DFA of order
is used [Fig. 13]. Finally, we note that depending on the effective exponent
, i.e. on the order
of the DFA method and the value of the power
, the vertical shift in the rms fluctuation function
for power-law trend can be positive (
), negative (
), or zero (
).
Figure 13:
Dependence of the rms fluctuation function
for power-law trend
, where
, on the length of the trend
. (a) A vertical shift is observed in
for different values of
--
and
. The figure shows that the vertical shift , defined as
, does not depend on
but only on the ratio
, suggesting that
. (b) Dependence of the vertical shift on the power
. For
(
is the order of DFA), we find a flat (constant) region characterized with effective exponent
and negative vertical shift. For
, we find an exponential dependence of the vertical shift on
. In this region,
, and the vertical shift can be negative (if
) or positive (if
). the slope of
vs.
is
due to doubling the length of the signal
. This slope changes to
when
is increased
times while
remains independent of
. For
there is no vertical shift, as marked with
. Arrows indicate integer values of
, for which values the DFA-
method filters out completely the power-law trend and
.
![\begin{figure}
\centerline{
\epsfysize=0.55\textwidth{\epsfbox{length.eps}}
\epsfysize=0.55\textwidth{\epsfbox{vertical_shift.eps}}}
\end{figure}](img335.png) |
Next: Combined effect on of
Up: Noise with Power-law trends
Previous: Dependence of on the
Zhi Chen
2002-08-28