Detrended fluctuation analysis

Trends of higher order can be removed by higher order DFA, in which a linear fit is replaced by a polynomial fit.[4] In the described case, linear fits (${\displaystyle i=1}$) are applied to the profile, thus it is called DFA1. To remove trends of higher order, DFA${\displaystyle i}$, uses polynomial fits of order ${\displaystyle i}$.

Owing to the summation (integration) from ${\displaystyle x_{i}}$ to ${\displaystyle X_{t}}$, linear trends in the mean of the profile represent constant trends in the initial sequence, and DFA1 only removes such constant trends (steps) in the ${\displaystyle x_{i}}$. In general, DFA of order ${\displaystyle i}$ removes (polynomial) trends of order ${\displaystyle i-1}$. For linear trends in the mean of ${\displaystyle x_{i}}$ at least DFA2 is needed.

The Hurst R/S analysis removes constant trends in the original sequence and thus, in its detrending it is equivalent to DFA1.

Since in the fluctuation function ${\displaystyle F(n)}$ the square (root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means ${\displaystyle \alpha =\alpha (2)}$. The multifractal generalization (MF-DFA)[5] uses a variable moment ${\displaystyle q}$ and provides ${\displaystyle \alpha (q)}$. Kantelhardt et al. intended this scaling exponent as a generalization of the classical Hurst exponent. The classical Hurst exponent corresponds to the second moment for stationary cases ${\displaystyle H=\alpha (2)}$ and to the second moment minus 1 for nonstationary cases ${\displaystyle H=\alpha (2)-1}$.[6][7][5]

Essentially, the scaling exponents need not be independent of the scale of the system. In the case ${\displaystyle \alpha }$ depends on the power ${\displaystyle q}$ extracted from

${\displaystyle F_{q}(n)=\left({\frac {1}{N}}\sum _{t=1}^{N}\left(X_{t}-Y_{t}\right)^{q}\right)^{1/q},}$

where the previous DFA is ${\displaystyle q=2}$. Multifractal systems scale as a function ${\displaystyle F_{q}(n)\propto n^{\alpha (q)}}$. To uncover multifractality, Multifractal Detrended Fluctuation Analysis is one possible method.[8]

In the case of power-law decaying auto-correlations, the correlation function decays with an exponent ${\displaystyle \gamma }$: ${\displaystyle C(L)\sim L^{-\gamma }\!\ }$. In addition the power spectrum decays as ${\displaystyle P(f)\sim f^{-\beta }\!\ }$. The three exponents are related by:[9]

• ${\displaystyle \gamma =2-2\alpha }$
• ${\displaystyle \beta =2\alpha -1}$ and
• ${\displaystyle \gamma =1-\beta }$.

The relations can be derived using the Wiener–Khinchin theorem. The relation of DFA to the power spectrum method has been well studied.[15]

Thus, ${\displaystyle \alpha }$ is tied to the slope of the power spectrum ${\displaystyle \beta }$ and is used to describe the color of noise by this relationship: ${\displaystyle \alpha =(\beta +1)/2}$.

For fractional Gaussian noise (FGN), we have ${\displaystyle \beta \in [-1,1]}$, and thus ${\displaystyle \alpha \in [0,1]}$, and ${\displaystyle \beta =2H-1}$, where ${\displaystyle H}$ is the Hurst exponent. ${\displaystyle \alpha }$ for FGN is equal to ${\displaystyle H}$.[16]

For fractional Brownian motion (FBM), we have ${\displaystyle \beta \in [1,3]}$, and thus ${\displaystyle \alpha \in [1,2]}$, and ${\displaystyle \beta =2H+1}$, where ${\displaystyle H}$ is the Hurst exponent. ${\displaystyle \alpha }$ for FBM is equal to ${\displaystyle H+1}$.[6] In this context, FBM is the cumulative sum or the integral of FGN, thus, the exponents of their power spectra differ by 2.