Algorithm edit

Given: a time series ${\displaystyle x_{1},x_{2},...,x_{N}}$ .

Compute its average value ${\displaystyle \langle x\rangle ={\frac {1}{N}}\sum _{t=1}^{N}x_{t}}$ .

Sum it into a process ${\displaystyle X_{t}=\sum _{i=1}^{t}(x_{i}-\langle x\rangle )}$ . This is the cumulative sum, or profile, of the original time series. For example, the profile of an i.i.d. white noise is a standard random walk.

Select a set ${\displaystyle T=\{n_{1},...,n_{k}\}}$  of integers, such that ${\displaystyle n_{1}<n_{2}<\cdots <n_{k}}$ , the smallest ${\displaystyle n_{1}\approx 4}$ , the largest ${\displaystyle n_{k}\approx N}$ , and the sequence is roughly distributed evenly in log-scale: ${\displaystyle \log(n_{2})-\log(n_{1})\approx \log(n_{3})-\log(n_{2})\approx \cdots }$ . In other words, it is approximately a geometric progression.^[2]

For each ${\displaystyle n\in T}$ , divide the sequence ${\displaystyle X_{t}}$  into consecutive segments of length ${\displaystyle n}$ . Within each segment, compute the least squares straight-line fit (the local trend). Let ${\displaystyle Y_{1,n},Y_{2,n},...,Y_{N,n}}$  be the resulting piecewise-linear fit.

Compute the root-mean-square deviation from the local trend (local fluctuation):

${\displaystyle F(n)={\sqrt {{\frac {1}{N/n}}\sum _{i=1}^{N/n}F(n,i)^{2}}}.}$ 

(If ${\displaystyle N}$  is not divisible by ${\displaystyle n}$ , then one can either discard the remainder of the sequence, or repeat the procedure on the reversed sequence, then take their root-mean-square.^[3])

Make the log-log plot ${\displaystyle \log n-\log F(n)}$ .^[4][5]

Interpretation edit

A straight line of slope ${\displaystyle \alpha }$  on the log-log plot indicates a statistical self-affinity of form ${\displaystyle F(n)\propto n^{\alpha }}$ . Since ${\displaystyle F(n)}$  monotonically increases with ${\displaystyle n}$ , we always have ${\displaystyle \alpha >0}$ .

The scaling exponent ${\displaystyle \alpha }$  is a generalization of the Hurst exponent, with the precise value giving information about the series self-correlations:

• ${\displaystyle \alpha <1/2}$ : anti-correlated
• ${\displaystyle \alpha \simeq 1/2}$ : uncorrelated, white noise
• ${\displaystyle \alpha >1/2}$ : correlated
• ${\displaystyle \alpha \simeq 1}$ : 1/f-noise, pink noise
• ${\displaystyle \alpha >1}$ : non-stationary, unbounded
• ${\displaystyle \alpha \simeq 3/2}$ : Brownian noise

Because the expected displacement in an uncorrelated random walk of length N grows like ${\displaystyle {\sqrt {N}}}$ , an exponent of ${\displaystyle {\tfrac {1}{2}}}$  would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is fractional Gaussian noise.

Pitfalls in interpretation edit

Though the DFA algorithm always produces a positive number ${\displaystyle \alpha }$  for any time series, it does not necessarily imply that the time series is self-similar. Self-similarity requires the log-log graph to be sufficiently linear over a wide range of ${\displaystyle n}$ . Furthermore, a combination of techniques including maximum likelihood estimation (MLE), rather than least-squares has been shown to better approximate the scaling, or power-law, exponent.^[6]

Also, there are many scaling exponent-like quantities that can be measured for a self-similar time series, including the divider dimension and Hurst exponent. Therefore, the DFA scaling exponent ${\displaystyle \alpha }$  is not a fractal dimension, and does not have certain desirable properties that the Hausdorff dimension has, though in certain special cases it is related to the box-counting dimension for the graph of a time series.

Generalization to polynomial trends (higher order DFA) edit

The standard DFA algorithm given above removes a linear trend in each segment. If we remove a degree-n polynomial trend in each segment, it is called DFAn, or higher order DFA.^[7]

Since ${\displaystyle X_{t}}$  is a cumulative sum of ${\displaystyle x_{t}-\langle x\rangle }$ , a linear trend in ${\displaystyle X_{t}}$  is a constant trend in ${\displaystyle x_{t}-\langle x\rangle }$ , which is a constant trend in ${\displaystyle x_{t}}$  (visible as short sections of "flat plateaus"). In this regard, DFA1 removes the mean from segments of the time series ${\displaystyle x_{t}}$  before quantifying the fluctuation.

Similarly, a degree n trend in ${\displaystyle X_{t}}$  is a degree (n-1) trend in ${\displaystyle x_{t}}$ . For example, DFA1 removes linear trends from segments of the time series ${\displaystyle x_{t}}$  before quantifying the fluctuation, DFA1 removes parabolic trends from ${\displaystyle x_{t}}$ , and so on.

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

Generalization to different moments (multifractal DFA) edit

DFA can be generalized by computing

Multifractal systems scale as a function ${\displaystyle F_{q}(n)\propto n^{\alpha (q)}}$ . Essentially, the scaling exponents need not be independent of the scale of the system. In particular, DFA measures the scaling-behavior of the second moment-fluctuations.

Kantelhardt et al. intended this scaling exponent as a generalization of the classical Hurst exponent. The classical Hurst exponent corresponds to ${\displaystyle H=\alpha (2)}$  for stationary cases, and ${\displaystyle H=\alpha (2)-1}$  for nonstationary cases.^[8][9][10]

The DFA method has been applied to many systems, e.g. DNA sequences,^[11][12] neuronal oscillations,^[10] speech pathology detection,^[13] heartbeat fluctuation in different sleep stages,^[14] and animal behavior pattern analysis.^[15]

The effect of trends on DFA has been studied.^[16]

For signals with power-law-decaying autocorrelation edit

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:^[11]

• ${\displaystyle \gamma =2-2\alpha }$ 
• ${\displaystyle \beta =2\alpha -1}$  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.^[17]

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 edit

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 ${\displaystyle H}$ .^[18]

For fractional Brownian motion edit

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 ${\displaystyle H+1}$ .^[9] In this context, FBM is the cumulative sum or the integral of FGN, thus, the exponents of their power spectra differ by 2.

• Multifractal system
• Self-organized criticality
• Self-affinity
• Time series analysis
• Hurst exponent

• Tutorial on how to calculate detrended fluctuation analysis Archived 2019-02-03 at the Wayback Machine in Matlab using the Neurophysiological Biomarker Toolbox.
• FastDFA MATLAB code for rapidly calculating the DFA scaling exponent on very large datasets.
• Physionet A good overview of DFA and C code to calculate it.
• MFDFA Python implementation of (Multifractal) Detrended Fluctuation Analysis.