DFA-1 on noise with a linear trend

Using the algorithm of Makse[63], we generate correlated noise with standard deviation one, with a given correlation property characterized by a given scaling exponent $\alpha$. We apply DFA-1 to quantify the correlation properties of the noise and find that only in certain good fit region the rms fluctuation function $F_{\rm\eta }(n)$ can be approximated by a power-law function [see Appendix 7.1]

$$F_{\rm\eta}(n) = b_0 n^{\alpha}$$ (6)

where $b_0$ is a parameter independent of the scale $n$. We find that the good fit region depends on the correlation exponent $\alpha$ [see Appendix 7.1]. We also derive analytically the rms fluctuation function for linear trend only for DFA-1 and find that [see Appendix 7.3]

$$F_{\rm L}(n) = k_0 A_{\rm L} n^{\alpha_{L}}$$ (7)

where $k_0$ is a constant independent of the length of trend $N_{max}$, of the box size $n$ and of the slope of the trend $A_{\rm L}$. We obtain $\alpha_{L} = 2$.

Figure 1: Crossover behavior of the root mean square fluctuation function $F_{\rm \eta L}(n)$ for noise (of length $N_{max}=2^{17}$ and correlation exponent $\alpha =0.1$) with superposed linear trends of slope $A_{\rm L}=2^{-16}, 2^{-12}, 2^{-8}$. For comparison, we show $F_{\rm \eta}(n)$ for the noise (thick solid line) and $F_{\rm L}(n)$ for the linear trends (dot-dashed line) (Eq.(7)). The results show that a crossover at a scale $n_{\times }$ for $F_{\rm \eta L}(n)$. For $n < n_{\times }$, the noise dominates and $F_{\rm \eta L}(n) \approx F_{\rm \eta}(n)$. For $n > n_{\times }$, the linear trend dominates and $F_{\rm \eta L}(n) \approx F_{\rm L}(n)$. Note that the crossover scale $n_{\times }$ increases when the slope $A_{\rm L}$ of the trend decreases.

Next we apply the DFA-1 method to the superposition of a linear trend with correlated noise and we compare the rms fluctuation function $F_{\rm\eta L}(n)$ with $F_{\rm\eta }(n)$ [see Fig.1]. We observe a crossover in $F_{\rm\eta L}(n)$ at scale $n = n_{\times}$. For $n < n_{\times }$, the behavior of $F_{\rm\eta L}(n)$ is very close to the behavior of $F_{\rm\eta }(n)$, while for $n > n_{\times }$, the behavior of $F_{\rm\eta L}(n)$ is very close to the behavior of $F_{\rm L}(n)$. A similar crossover behavior is also observed in the scaling of the well-studied biased random walk [61,62]. It is known that the crossover in the biased random walk is due to the competition of the unbiased random walk and the bias [see Fig.5.3 of [62]]. We illustrate this observation in Fig. 2, where the detrended fluctuation functions (Eq. (3)) of the correlated noise, $Y_{\rm\eta }(i)$, and of the noise with a linear trend, $Y_{\rm\eta L}(i)$ are shown. For the box size $n < n_{\times }$ as shown in Fig. 2(a) and (b), $Y_{\rm\eta L}(i) \approx Y_{\rm\eta}(i)$. For $n > n_{\times }$ as shown in Fig. 2(c) and (d), $Y_{\rm\eta L}(i)$ has distinguishable quadratic background significantly different from $Y_{\rm\eta }(i)$. This quadratic background is due to the integration of the linear trend within the DFA procedure and represents the detrended fluctuation function $Y_{L}$ of the linear trend. These relations between the detrended fluctuation functions $Y(i)$ at different time scales $n$ explain the crossover in the scaling behavior of $F_{\rm\eta L}(n)$: from very close to $F_{\rm\eta }(n)$ to very close to $F_{\rm L}(n)$ (observed in Fig.1).

Figure 2: Comparison of the detrended fluctuation function for noise $Y_{\rm \eta}(i)$ and for noise with linear trend $Y_{\rm \eta L}(i)$ at different scales. (a) and (c) are $Y_{\rm \eta}$ for noise with $\alpha =0.1$; (b) and (d) are $Y_{\rm \eta L}$ for the same noise with a linear trend with slope $A_{\rm L}=2^{-12}$ (the crossover scale $n_{\times } = 320$ see Fig. 1). (a) (b) for scales $n < n_{\times }$ the effect of the trend is not pronounced and $Y_{\rm \eta} \approx Y_{\rm \eta L}$ (i.e. $Y_{\rm \eta} \gg Y_{\rm L}$); (c)(d) for scales $n > n_{\times }$, the linear trend is dominant and $Y_{\rm \eta} \ll Y_{\rm \eta L}$.

The experimental results presented in Figs.1 and 2 suggest that the rms fluctuation function for a signal which is a superposition of a correlated noise and a linear trend can be expressed as:

$$\left [F_{\rm\eta L}(n)\right ]^2 = \left [F_{\rm L}(n)\right ]^2 + \left [F_{\rm\eta}(n)\right ]^2$$ (8)

We provide an analytic derivation of this relation in Appendix 7.2, where we show that Eq.(8) holds for the superposition of any two independent signals -- in this particular case noise and a linear trend. We call this relation the ``superposition rule''. This rule helps us understand how the competition between the contribution of the noise and the trend to the rms fluctuation function $F_{\rm\eta L}(n)$ at different scales $n$ leads to appearance of crossovers [61].

Next, we ask how the crossover scale $n_{\times }$ depends on: (i) the slope of the linear trend $A_{\rm L}$, (ii) the scaling exponent $\alpha$ of the noise, and (iii) the length of the signal $N_{max}$. Surprisingly, we find that for noise with any given correlation exponent $\alpha$ the crossover scale $n_{\times }$ itself follows a power-law scaling relation over several decades: $n_{\times } \sim \left (A_{\rm L}\right )^{\theta }$ (see Fig. 3). We find that in this scaling relation, the crossover exponent $\theta$ is negative and its value depends on the correlation exponent $\alpha$ of the noise -- the magnitude of $\theta$ decreases when $\alpha$ increases. We present the values of the ``crossover exponent'' $\theta$ for different correlation exponents $\alpha$ in Table I.

Figure 3: The crossover $n_{\times }$ of $F_{\rm \eta L}(n)$ for noise with a linear trend. We determine the crossover scale $n_{\times }$ based on the difference $\Delta$ between $\log F_{\rm \eta}$ (noise) and $\log F_{\rm \eta L}$ (noise with a linear trend). The scale for which $\Delta =0.05$ is the estimated crossover scale $n_{\times }$. For any given correlation exponent $\alpha$ of the noise, the crossover scale $n_{\times }$ exhibits a long-range power-law behavior $n_{\times} \sim \left(A_{\rm L}\right)^{\theta}$, where the crossover exponent $\theta$ is a function of $\alpha$ [see Eq.(9) and Table I.

Table I: The crossover exponent $\theta$ from the power-law relation between the crossover scale $n_{\times }$ and the slope of the linear trend $A_{\rm L}$ -- $n_{\times } \sim \left (A_{\rm L}\right )^{\theta }$ --for different values of the correlation exponents $\alpha$ of the noise [Fig. 3]. The values of $\theta$ obtained from our simulations are in good agreement with the analytical prediction $-1/(2-\alpha)$ [Eq. (9)]. Note that $-1/(2-\alpha)$ are not always exactly equal to $\theta$ because $F_{\rm\eta }(n)$ in simulations is not a perfect simple power-law function and the way we determine numerically $n_{\times }$ is just approximated.

$\alpha$	$\theta$	$-1/(2-\alpha)$
0.1	-0.54	-0.53
0.3	-0.58	-0.59
0.5	-0.65	-0.67
0.7	-0.74	-0.77
0.9	-0.89	-0.91

To understand how the crossover scale depends on the correlation exponent $\alpha$ of the noise we employ the superposition rule [Eq.(8)] and estimate $n_{\times }$ as the intercept between $F_{\rm\eta }(n)$ and $F_{\rm L}(n)$. From the Eqs. (6) and (7), we obtain the following dependence of $n_{\times }$ on $\alpha$:

$$n_{\times} = \left(A_{\rm L}\frac{k_0}{b_0}\right)^{1/(\alp... ...L})} = \left(A_{\rm L}\frac{k_0}{b_0}\right)^{1/(\alpha-2)}$$ (9)

This analytical calculation for the crossover exponent $-1/(\alpha_{L}-\alpha)$ is in a good agreement with the observed values of $\theta$ obtained from our simulations [see Fig.3 and Table I].

Finally, since the $F_{\rm L}(n)$ does not depend on $N_{max}$ as we show in Eq.(7) and in Appendix 7.3, we find that $n_{\times }$ does not depend on $N_{max}$. This is a special case for linear trends and does not always hold for higher order polynomial trends [see Appendix 7.4].