Detrended fluctuation analysis

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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.

DFA was introduced by Peng et al. 1994 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.

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[edit] Calculation

Given a bounded time series xt, t \in \mathbb{N}, integration first converts this into an unbounded process Xt:

X_t=\sum_{i=1}^t (x_i-\langle x_i\rangle)

Xt is called cumulative sum or profile. This integration process converts, for example, an i.i.d. white noise process into a random walk.

Next, Xt is divided into time windows of length L samples, and a local least squares straight line fit (the local trend) is calculated by minimising the squared error E2 with respect to the slope and intercept parameters a,b:

E^2 = \sum_{i = 1}^L \left( X_i - ai - b \right)^2.

Next, the root-mean-square deviation from the trend, the fluctuation, is calculated over every window at every time scale:

F( L ) = \left[ \frac{1}{L}\sum_{i = 1}^L \left( X_i - ai - b \right)^2 \right]^{\frac{1}{2}}.

This detrending followed by fluctuation measurement process is repeated over the whole signal at a range of different window sizes L, and a log-log graph of L against F(L) is constructed.

A straight line on this graph indicates statistical self-affinity expressed as F(L) \propto L^{\alpha}. The scaling exponent α is calculated as the slope of a straight line fit to the log-log graph of L against F(L) using least-squares.

The fluctuation exponent can have different values:

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

There are different orders of DFA. In the described case, linear fits (n = 1) are applied to the profile, thus it is called DFA1. In general, DFAn, uses polynomial fits of order n. Due to the summation (integration) from xi to Xt, 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 xi. In general DFA of order n removes (polynomial) trends of order n − 1. For linear trends in the mean of xi at least DFA2 is needed. The Hurst R/S-Analysis removes constants trends in the original sequence and thus, in its detrending it is equivalent to DFA1.

Since in the fluctuation function F(L) the square(root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means α = α(2). The multifractal generalization (MF-DFA) uses a variable moment q and provides α(q). The Hurst exponent corresponds to the first moment: H = α(1)

[edit] Relations to other methods

In the case of power-law decaying auto-correlations, the correlation function decays with an exponent γ: C(L)˜L − γ. In addition the power spectrum decays with an P(f)˜f − β. The three exponent are related by:

  • γ = 2 − 2α
  • β = 2α − 1 and
  • γ = 1 − β.

The relations can be derived using the Wiener–Khinchin theorem.

Thus, α is tied to the slope of the power spectrum β used to describe the color of noise by this relationship: α = (β + 1) / 2.

For fractional Gaussian noise (FGN) β = [ − 1,1], and thus α = [0,1], and β = 2H − 1, where H is the Hurst exponent. α for FGN is equal to H.

For fractional Brownian motion (FBM) β = [1,3], and thus α = [1,2], and β = 2H + 1, where H is the Hurst exponent. α for FBM is equal to H − 1. In this context, FBM is the cumulative sum or the integral of FGN.


[edit] Pitfalls in interpretation

As with most methods that depend upon line fitting, it is always possible to find a number α by the DFA method, but this does not necessarily imply that the time series is self-similar. Self-similarity requires that the points on the log-log graph are sufficiently collinear across a very wide range of window sizes L.

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 α is not a fractal dimension sharing the all the desirable properties of the Hausdorff dimension, for example, although in certain special cases it can be shown to be related to the box-counting dimension for the graph of a time series.

[edit] References

  • Peng, C.K. et al. (1994) Mosaic organization of DNA nucleotides, Phys Rev E, 49 (2) 1685-1689.
  • Buldyrev et al. (1995) Phys Rev E, 51 (5) 5084-5091.
  • Bunde A. and Havlin S., Eds., (1996) Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York.
  • Heneghan et al. (2000) Phys Rev E, 62 (5) 6103-6110.
  • Bunde A. et al. (2000) Correlated and uncorrelated regions in heart-rate fluctuations during sleep, Phys Rev E, 85 (17) 3736-3739.
  • Kantelhardt J.W. et al. (2001) Detecting long-range correlations with detrended fluctuation analysis, Phys A, 295 (3-4) 441-454.
  • Hu, K. et al. (2001) Effect of trends on detrended fluctuation analysis, Phys Rev E, 64 (1) 011114.


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