Algorithm

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):

$${\displaystyle F(n,i)={\sqrt {{\frac {1}{n}}\sum _{t=in+1}^{in+n}\left(X_{t}-Y_{t,n}\right)^{2}}}.}$$

And their root-mean-square is the total 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

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

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 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)

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)

DFA can be generalized by computing

$${\displaystyle F_{q}(n)=\left({\frac {1}{N/n}}\sum _{i=1}^{N/n}F(n,i)^{q}\right)^{1/q}.}$$

then making the log-log plot of ${\displaystyle \log n-\log F_{q}(n)}$, If there is a strong linearity in the plot of ${\displaystyle \log n-\log F_{q}(n)}$, then that slope is ${\displaystyle \alpha (q)}$.^[8] DFA is the special case where ${\displaystyle q=2}$.

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]

For signals with power-law-decaying autocorrelation

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

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

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.