Higher order DFA on pure sinusoidal trend

In the previous Sec. 4.2, we discussed how sinusoidal trends affect the scaling behavior of correlated noise when the DFA-1 method is applied. Since DFA-1 removes only constant trends in data, it is natural to ask how the observed scaling results will change when we apply DFA of order $\ell$ designed to remove polynomial trends of order lower than $\ell$. In this section, we first consider the rms fluctuation $F_{\rm S}$ for a sinusoidal signal and then we study the scaling and crossover properties of $F_{\rm\eta S}$ for correlated noise with superimposed sinusoidal signal when higher order DFA is used.

We find that the rms fluctuation function $F_{\rm S}$ does not depend on the length of the signal $N_{max}$, and preserves a similar shape when different order-$\ell$ DFA method is used [Fig. 9]. In particular, $F_{\rm S}$ exhibits a crossover at a scale $n_{2\times }$ proportional to the period $T$ of the sinusoidal: $n_{2\times } \sim T^{\theta _{\rm T2}}$ with $\theta _{\rm T2}\approx 1$. The crossover scale shifts to larger values for higher order $\ell$ [Fig. 5 and Fig. 9]. For the scale $n < n_{2\times }$, $F_{\rm S}$ exhibits an apparent scaling: $F_{\rm S}\sim n^{\alpha_{\rm S}}$ with an effective exponent $\alpha_{\rm S}~=~\ell +~1~$. For DFA-1, we have $\ell=1$ and recover $\alpha_{\rm S}~=~2~$ as shown in Eq. (12). For $n > n_{2\times }$, $F_{\rm S}(n)$ is a constant independent of the scale $n$, and of the order $\ell$ of the DFA method in agreement with Eq. (13).

Next, we consider $F_{\rm\eta S}(n)$ when DFA-$\ell$ with a higher order $\ell$ is used. We find that for all orders $\ell$, $F_{\rm\eta S}(n)$ does not depend on the length of the signal $N_{max}$ and exhibits three crossovers -- at small, intermediate and large scales -- similar behavior is reported for DFA-1 in Fig. 6. Since the crossover at small scales, $n_{1\times }$, and the crossover at large scale, $n_{3\times }$, result from the ``competition'' between the scaling of the correlated noise and the effect of the sinusoidal trend (Figs. 6 and 7), using the superposition rule [Eq. (10)] we can estimate $n_{1\times }$ and $n_{3\times }$ as the intercepts of $F_{\rm\eta }(n)$ and $F_{\rm S}(n)$ for the general case of DFA-$\ell$.

For $n_{1\times }$ we find the following dependence on the period $T$, amplitude $A_{\rm S}$, the correlation exponent $\alpha$ of the noise, and the order $\ell$ of the DFA-$\ell$ method:

$$n_{1\times} \sim \left(T/A_{\rm S}\right )^{1/(\ell+1-\alpha)}$$ (16)

For DFA-1, we have $\ell=1$ and we recover Eq. (14). In addition, $n_{1\times }$ is shifted to larger scales when higher order DFA-$\ell$ is applied, due to the fact that the value of $F_{\rm S}(n)$ decreases when $\ell$ increases ( $\alpha_{\rm S}=\ell+1$, see Fig. 9).

For the third crossover observed in $F_{\rm\eta S}(n)$ at large scale $n_{3\times }$ we find for all orders $\ell$ of the DFA-$\ell$ the following scaling relation:

$$n_{3\times} \sim (T A_{\rm S})^{1/\alpha} .$$ (17)

Since the scaling function $F_{\rm\eta }(n)$ for correlated noise shifts vertically to lower values when higher order DFA-$\ell$ is used [see the discussion in Appendix 7.1 and Sec. 5.2], $n_{3\times }$ exhibits a slight shift to larger scales.

For the crossover $n_{2\times }$ in $F_{\rm\eta S}(n)$ at $F_{\rm\eta S}(n)$ at intermediate scales, we find: $n_{2\times}~\sim~T$. This relation is independent of the order $\ell$ of the DFA and is identical to the relation found for $F_{\rm S}(n)$ [Eq. (11)]. $n_{2\times }$ also exhibits a shift to larger scales when higher order DFA is used [see Fig. 9].

The reported here features of the crossovers in $F_{\rm\eta S}(n)$ can be used to identify low-frequency sinusoidal trends in noisy data, and to recognize their effects on the scaling properties of the data. This information may be useful when quantifying correlation properties in data by means of scaling analysis.

Figure 9: Comparison of the results of different order DFA on a sinusoidal trend. The sinusoidal trend is given by the function $64\sin({2\pi i}/2^{11})$ and the length of the signal is $N_{max}=2^{17}$. The spurious singularities (spikes) arise from the discrete data we use for the sinusoidal function.