Dependence of $F_{\rm P}(n)$ on the power $\lambda$

First we study how the rms fluctuation function $F_{\rm P}(n)$ for a power-law trend $u(i)=A_{\rm P} i^{\lambda }$ depends on the power $\lambda$. We find that

$$F_{\rm P}(n) \sim A_{\rm P}n^{\alpha_{\lambda}},$$ (19)

where $\alpha _{\lambda }$ is the effective exponent for the power-law trend. For positive $\lambda$ we observe no crossovers in $F_{\rm P}(n)$ (Fig. 10(a)). However, for negative $\lambda$ there is a crossover in $F_{\rm P}(n)$ at small scales $n$ (Fig. 10(b)), and we find that this crossover becomes even more pronounced with decreasing $\lambda$ or increasing the order $\ell$ of the DFA method, and is also shifted to larger scales [Fig. 11(a)].

Figure 11: Scaling behavior of rms fluctuation function $F_{\rm P}(n)$ for power-law trends, $u(i)\sim i^{\lambda }$, where $i=1,...,N_{max}$ and $N_{max}=2^{17}$ is the length of the signal. (a) For $\lambda <0$, $F_{\rm P}(n)$ exhibits crossover at small scales which is more pronounced with increasing the order $\ell$ of DFA-$\ell$ and decreasing the value of $\lambda$. Such crossover is not observed for $\lambda >0$ when $F_{\rm P}(n) \sim n^{\alpha_{\lambda}}$ for all scales $n$ [see Fig. 10(a)]. (b) Dependence of the effective exponent $\alpha _{\lambda }$ on the power $\lambda$ for different order $\ell =1,2,3$ of the DFA method. Three regions are observed depending on the order $\ell$ of the DFA: region I ( $\lambda >\ell -0.5)$, where $\alpha _{\lambda } \approx \ell +1$; region II ( $-1.5<\lambda <\ell -0.5$), where $\alpha _{\lambda }=\lambda +1.5$; region III ($\lambda <-1.5$), where $\alpha _{\lambda }\approx 0$. We note that for integer values of the power $\lambda =0,1,...,\ell -1$, where $\ell$ is the order of DFA we used, there is no scaling for $F_{\rm P}(n)$ and $\alpha _{\lambda }$ is not defined, as indicated by the arrows. (c) Asymptotic behavior near integer values of $\lambda$. $F_{\rm P}(n)$ is plotted for $\lambda \rightarrow 1$ when DFA-2 is used. Even for $\lambda -1 =10^{-6}$, we observe at large scales $n$ a region with an effective exponent $\alpha _{\lambda } \approx 2.5$, This region is shifted to infinitely large scales when $\lambda =1$.

Next, we study how the effective exponent $\alpha _{\lambda }$ for $F_{\rm P}(n)$ depends on the value of the power $\lambda$ for the power-law trend. We examine the scaling of $F_{\rm P}(n)$ and estimate $\alpha _{\lambda }$ for $-4<\lambda<4$. In the cases when $F_{\rm P}(n)$ exhibits a crossover, in order to obtain $\alpha _{\lambda }$ we fit the range of larger scales to the right of the crossover. We find that for any order $\ell$ of the DFA-$\ell$ method there are three regions with different relations between $\alpha _{\lambda }$ and $\lambda$ [Fig. 11(b)]:

(i) $\alpha _{\lambda } \approx \ell +1$ for $\lambda > \ell -0.5$ (region I);

(ii) $\alpha_{\lambda} \approx \lambda +1.5$ for $-1.5\le \lambda \le \ell-0.5$ (region II);

(iii) $\alpha _{\lambda }\approx 0$ for $\lambda <-1.5$ (region III).

Note, that for integer values of the power $\lambda$ ( $\lambda =0,1,...,m-1$), i.e. polynomial trends of order $m-1$, the DFA-$\ell$ method of order $\ell>m-1$ ($\ell$ is also an integer) leads to $F_{\rm P}(n) \approx 0$, since DFA-$\ell$ is designed to remove polynomial trends. Thus for a integer values of the power $\lambda$ there is no scaling and the effective exponent $\alpha _{\lambda }$ is not defined if a DFA-$\ell$ method of order $\ell > \lambda$ is used [Fig. 11]. However, it is of interest to examine the asymptotic behavior of the scaling of $F_{\rm P}(n)$ when the value of the power $\lambda$ is close to an integer. In particular , we consider how the scaling of $F_{\rm P}(n)$ obtained from DFA-2 method changes when $\lambda \rightarrow 1$ [Fig. 11(c)]. Surprisingly, we find that even though the values of $F_{\rm P}(n)$ are very small at large scales, there is a scaling for $F_{\rm P}(n)$ with a smooth convergence of the effective exponent $\alpha_{\lambda} \rightarrow 2.5$ when $\lambda \rightarrow 1$, according to the dependence $\alpha_{\lambda} \approx \lambda +1.5$ 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 $Y(i)$ (Eq. (3)) is smaller than the precision of the numerical simulation.