Detrended fluctuation analysis

Template:Short description In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory processes (diverging correlation time, e.g. power-law decaying autocorrelation function) or 1/f noise.

The obtained exponent is similar to the Hurst exponent, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are non-stationary (changing with time). It is related to measures based upon spectral techniques such as autocorrelation and Fourier transform.

Peng et al. introduced DFA in 1994 in a paper that has been cited over 3,000 times as of 2022 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.

Systematic studies of the advantages and limitations of the DFA method were performed by PCh Ivanov et al. in a series of papers focusing on the effects of different types of nonstationarities in real-world signals: (1) types of trends; (2) random outliers/spikes, noisy segments, signals composed of parts with different correlation; (3) nonlinear filters; (4) missing data; (5) signal coarse-graining procedures and comparing DFA performance with moving average techniques (cumulative citations > 4,000).  Datasets generated to test DFA are available on PhysioNet.

Definition

DFA on a Brownian motion process, with increasing values of $n$.

Algorithm

Given: a time series $x_1,x_2,...,x_N$.

Compute its average value $⟨x⟩=\frac{1}{N}\sum _{t=1}^N{x}_t$.

Sum it into a process $X_t=\sum _{i=1}^t(x_i-⟨x⟩)$. 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 $T=\{n_1,...,n_k\}$ of integers, such that $n_1<n_2<\cdots <n_k$, the smallest $n_1\approx 4$, the largest $n_k\approx N$, and the sequence is roughly distributed evenly in log-scale: $\log (n_2)-\log (n_1)\approx \log (n_3)-\log (n_2)\approx \cdots$. In other words, it is approximately a geometric progression.

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

Compute the root-mean-square deviation from the local trend (local fluctuation):$$F(n,i)=\sqrt{\frac{1}{n}\sum _{t=in+1}^{in+n}{(X_t-Y_{t,n})}^2}.$$And their root-mean-square is the total fluctuation:

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

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

Make the log-log plot $\log n-\log F(n)$.

Interpretation

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

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

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

Because the expected displacement in an uncorrelated random walk of length N grows like $\sqrt{N}$, an exponent of $\frac{1}{2}$ 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 $\alpha$ 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 $n$. 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.

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 $\alpha$ 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.

Generalizations

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.

Since $X_t$ is a cumulative sum of $x_t-⟨x⟩$, a linear trend in $X_t$ is a constant trend in $x_t-⟨x⟩$, which is a constant trend in $x_t$ (visible as short sections of "flat plateaus"). In this regard, DFA1 removes the mean from segments of the time series $x_t$ before quantifying the fluctuation.

Similarly, a degree n trend in $X_t$ is a degree (n-1) trend in $x_t$. For example, DFA1 removes linear trends from segments of the time series $x_t$ before quantifying the fluctuation, DFA1 removes parabolic trends from $x_t$, 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$$F_q(n)={(\frac{1}{N/n}\sum _{i=1}^{N/n}F(n,i{)}^q)}^{1/q}.$$then making the log-log plot of $\log n-\log F_q(n)$, If there is a strong linearity in the plot of $\log n-\log F_q(n)$, then that slope is $\alpha (q)$. DFA is the special case where $q=2$.

Multifractal systems scale as a function $F_q(n)\propto n^{\alpha (q)}$. 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 $H=\alpha (2)$ for stationary cases, and $H=\alpha (2)-1$ for nonstationary cases.

Applications

The DFA method has been applied to many systems, e.g. DNA sequences; heartbeat dynamics in sleep and wake,  sleep stages, rest and exercise, and across circadian phases; locomotor gate and wrist dynamics, neuronal oscillations, speech pathology detection, and animal behavior pattern analysis.

Relations to other methods, for specific types of signal

For signals with power-law-decaying autocorrelation

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

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

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

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

For fractional Gaussian noise

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

For fractional Brownian motion

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

See also

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External links

• Tutorial on how to calculate detrended fluctuation analysis Template:Webarchive 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.