Dependence of $F_{\rm P}(n)$ on the signal length $N_{max}$

Here, we study how the rms fluctuation function $F_{\rm P}(n)$ depends on the length $N_{max}$ of the power-law signal $u(i)=A_{\rm P} i^{\lambda }$ ( $i=1,...,N_{max}$). We find that there is a vertical shift in $F_{\rm P}(n)$ with increasing $N_{max}$ [Fig. 13(a)]. We observe that when doubling the length $N_{max}$ of the signal the vertical shift in $F_{\rm P}(n)$, which we define as $F^{2N_{max}}_{\rm P}/F^{N_{max}}_{\rm P}$, remains the same, independent of the value of $N_{max}$. This suggests a power-law dependence of $F_{\rm P}(n)$ on the length of the signal:

$$F_{\rm P}(n) \sim \left(N_{max}\right )^{\gamma},$$ (21)

where $\gamma$ is an effective scaling exponent.

Next, we ask if the vertical shift depends on the power $\lambda$ of the power-law trend. When doubling the length $N_{max}$ of the signal, we find that for $\lambda <\ell -0.5$, where $\ell$ is the order of the DFA method, the vertical shift is a constant independent of $\lambda$ [Fig. 13(b)]. Since the value of the vertical shift when doubling the length $N_{max}$ is $2^{\gamma}$ (from Eq. (21)), the results in Fig. 13(b) show that $\gamma$ is independent of $\lambda$ when $\lambda <\ell -0.5$, and that $-\log 2^{\gamma} \approx -0.15$, i.e. the effective exponent $\gamma \approx -0.5$.

For $\lambda > \ell -0.5$, when doubling the length $N_{max}$ of the signal, we find that the vertical shift $2^{\gamma}$ exhibits the following dependence on $\lambda$: $-\log _{10} 2^{\gamma} =\log _{10} 2^{\lambda-\ell}$, and thus the effective exponent $\gamma$ depends on $\lambda$ -- $\gamma =\lambda -\ell$. For positive integer values of $\lambda$ ($\lambda =\ell$), we find that $\gamma=0$, and there is no shift in $F_{\rm P}(n)$, suggesting that $F_{\rm P}(n)$ does not depend on the length $N_{max}$ of the signal, when DFA of order $\ell$ is used [Fig. 13]. Finally, we note that depending on the effective exponent $\gamma$, i.e. on the order $\ell$ of the DFA method and the value of the power $\lambda$, the vertical shift in the rms fluctuation function $F_{\rm P}(n)$ for power-law trend can be positive ( $\lambda >\ell$), negative ($\lambda <\ell$), or zero ($\lambda =\ell$).

Figure 13: Dependence of the rms fluctuation function $F_{\rm P}(n)$ for power-law trend $u(i)=A_{\rm P} i^{\lambda}$, where $i=1,...,N_{max}$, on the length of the trend $N_{max}$. (a) A vertical shift is observed in $F_{\rm P}(n)$ for different values of $N_{max}$ -- $N_{1max}$ and $N_{2max}$. The figure shows that the vertical shift , defined as $F^{N_{1max}}_{\rm P}(n)/F^{N_{2max}}_{\rm P}(n)$, does not depend on $N_{max}$ but only on the ratio $N_{1max}/N_{2max}$, suggesting that $F_{\rm P}(n) \sim \left(N_{max}\right )^{\gamma}$. (b) Dependence of the vertical shift on the power $\lambda$. For $\lambda <\ell -0.5$ ($\ell$ is the order of DFA), we find a flat (constant) region characterized with effective exponent $\gamma =-0.5$ and negative vertical shift. For $\lambda > \ell -0.5$, we find an exponential dependence of the vertical shift on $\lambda$. In this region, $\gamma =\lambda -\ell$, and the vertical shift can be negative (if $\lambda <\ell$) or positive (if $\lambda >\ell$). the slope of $-\log_{10}\left(F^{2N_{max}}_{\rm P}(n)/F^{N_{max}}_{\rm P}(n)\right )$ vs. $\lambda$ is $-\log_{10}2$ due to doubling the length of the signal $N_{max}$. This slope changes to $-\log_{10}m$ when $N_{max}$ is increased $m$ times while $\gamma$ remains independent of $N_{max}$. For $\lambda =\ell$ there is no vertical shift, as marked with $\times$. Arrows indicate integer values of $\lambda <\ell$, for which values the DFA-$\ell$ method filters out completely the power-law trend and $F_{\rm P}=0$.