Combined effect on $F_{\rm P}(n)$ of $\lambda$, $\ell$ and $N_{max}$

We have seen that, taking into account the effects of the power $\lambda$ (Eq. (19)), the order $\ell$ of DFA-$\ell$ (Eq. (20)) and the effect of the length of the signal $N_{max}$ (Eq. (21)), we reach the following expression for the rms fluctuation function $F_{\rm P}(n)$ for a power-law trend $u(i)=A_{\rm P} i^{\lambda }$:

$$F_{\rm P}(n) \sim A_{\rm P} \cdot n^{\alpha_{\lambda}}\cdot... ...au(\lambda)} \cdot \left(N_{max}\right )^{\gamma(\lambda)},$$ (22)

For correlated noise, the rms fluctuation function $F_{\rm\eta }(n)$ depends on the box size $n$ (Eq. (6)) and on the order $\ell$ of DFA-$\ell$ (Sec. 5.2 and Fig. 12(a), (d)), and does not depend on the length of the signal $N_{max}$. Thus we have the following expression for $F_{\rm\eta }(n)$

$$F_{\rm\eta}(n) \sim n^{\alpha}\ell^{\tau(\alpha)},$$ (23)

To estimate the crossover scale $n_{\times }$ observed in the apparent scaling of $F_{\rm\eta P}(n)$ 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 $n_{\times }$ as the intercept between $F_{\rm P}(n)$ and $F_{\rm\eta }(n)$:

$$n_{\times} \sim \left[A l^{\tau(\lambda)-\tau(\alpha)} \lef... ...max}\right )^{\gamma}\right]^{1/(\alpha-\alpha_{\lambda})}.$$ (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 ($\lambda =1$) when DFA-1 ($\ell=1$) is applied, we have $\alpha_{\lambda}=2$ (see Appendix 7.3 and Sec. 5.1, Fig. 11(b)). Since in this case $\lambda = \ell =1 > \ell -0.5$ we have $\gamma =\lambda-\ell =0$ (see Sec.5.3 Fig. 13(b)), and from Eq. (24) we recover Eq. (9).