Method

To illustrate the DFA method, we consider a noisy time series, $u(i)$ ( $i=1,..,N_{max}$ ). We integrate the time series $u(i)$,

$$y(j) = \sum\limits_{i=1}^{j} (u(i) - <u>),$$ (1)

where

$$<u>=\frac{1}{N_{max}} \sum\limits_{j=1}^{N_{max}} u(i),$$ (2)

and is divided into boxes of equal size, $n$. In each box, we fit the integrated time series by using a polynomial function, $y_{fit}(i)$, which is called the local trend. For order-$\ell$ DFA (DFA-1 if $\ell=1$, DFA-2 if $\ell=2$ etc.), $\ell$ order polynomial function should be applied for the fitting. We detrend The integrated time series, $y(i)$ by subtracting the local trend $y_{fit}(i)$ in each box, and we calculate the detrended fluctuation function

$$Y(i) = y(i)-y_{fit}(i).$$ (3)

For a given box size $n$, we calculate the root mean square (rms) fluctuation

$$F(n) =\sqrt{\frac{1}{N_{max}}\sum\limits_{i=1}^{N_{max}}\left [Y(i)\right ]^2}$$ (4)

The above computation is repeated for box sizes $n$ (different scales) to provide a relationship between $F(n)$ and $n$. A power-law relation between $F(n)$ and the box size $n$ indicates the presence of scaling: $F(n) \sim n^{\alpha}$. The parameter $\alpha$, called the scaling exponent or correlation exponent, represents the correlation properties of the signal: if $\alpha=0.5$, there is no correlation and the signal is an uncorrelated signal (white noise); if $\alpha < 0.5$, the signal is anticorrelated; if $\alpha > 0.5$, there are positive correlations in the signal.