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We have seen that, taking into account the effects of the power
(Eq. (19)), the order
of DFA-
(Eq. (20)) and the effect of the length of the signal
(Eq. (21)), we reach the following expression for the rms fluctuation function
for a power-law trend
:
![\begin{displaymath}
F_{\rm P}(n) \sim A_{\rm P} \cdot n^{\alpha_{\lambda}}\cdot...
...au(\lambda)}
\cdot \left(N_{max}\right )^{\gamma(\lambda)},
\end{displaymath}](img336.png) |
(22) |
For correlated noise, the rms fluctuation function
depends on the box size
(Eq. (6)) and on the order
of DFA-
(Sec. 5.2 and Fig. 12(a), (d)), and does not depend on the length of the signal
. Thus we have the following expression for
![\begin{displaymath}
F_{\rm\eta}(n) \sim n^{\alpha}\ell^{\tau(\alpha)},
\end{displaymath}](img337.png) |
(23) |
To estimate the crossover scale
observed in the apparent scaling of
for a correlated noise superposed with a power-law trend [Fig. 10(a), (b) and Fig. 12(a)], we employ the superposition rule (Eq. (18)). From Eq. (22) and Eq. (23), we obtain
as the intercept between
and
:
![\begin{displaymath}
n_{\times} \sim \left[A l^{\tau(\lambda)-\tau(\alpha)} \lef...
...max}\right )^{\gamma}\right]^{1/(\alpha-\alpha_{\lambda})}.
\end{displaymath}](img338.png) |
(24) |
To test the validity of this result, we consider the case of correlated noise with a linear trend. For the case of a linear trend (
) when DFA-1 (
) is applied, we have
(see Appendix 7.3 and Sec. 5.1, Fig. 11(b)). Since in this case
we have
(see Sec.5.3 Fig. 13(b)), and from Eq. (24) we recover Eq. (9).
Next: Conclusion and Summary
Up: Noise with Power-law trends
Previous: Dependence of on the
Zhi Chen
2002-08-28