 Detrended fluctuation analysis

In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical selfaffinity of a signal. It is useful for analysing time series that appear to be longmemory processes (diverging correlation time, e.g. powerlaw 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 nonstationary (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^{[1]} and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by nonstationarities.
Contents
Calculation
Given a bounded time series x_{t}, , integration or summation first converts this into an unbounded process X_{t}:
X_{t} is called cumulative sum or profile. This process converts, for example, an i.i.d. white noise process into a random walk.
Next, X_{t} is divided into time windows Y_{j} of length L samples, and a local least squares straightline fit (the local trend) is calculated by minimising the squared error E^{2} with respect to the slope and intercept parameters a,b:
Trends of higher order, can be removed by higher order DFA, where the linear function ai + b is replaced by a polynomial of order n^{[2]}. Next, the rootmeansquare deviation from the trend, the fluctuation, is calculated over every window at every time scale:
This detrending followed by fluctuation measurement process is repeated over the whole signal at a range of different window sizes L, and a loglog graph of L against F(L) is constructed.
A straight line on this loglog graph indicates statistical selfaffinity expressed as . The scaling exponent α is calculated as the slope of a straight line fit to the loglog graph of L against F(L) using leastsquares. This exponent is a generalization of the Hurst exponent. Because the expected displacement in an uncorrelated random walk of length L grows like , an exponent of would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is Fractional Brownian motion, with the precise value giving information about the series selfcorrelations:
 α < 1 / 2: anticorrelated
 : uncorrelated, white noise
 α > 1 / 2: correlated
 : 1/fnoise, pink noise
 α > 1: nonstationary, random walk like, unbounded
 : 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 x_{i} to 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 x_{i}. In general DFA of order n removes (polynomial) trends of order n − 1. For linear trends in the mean of x_{i} 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. The DFA method was applied to many systems, ranging from DNA sequences^{[3]}^{[4]} and speech pathology detection^{[5]} to heartbeat fluctuation different sleep stages^{[6]} . Effect of trends on DFA were studied in^{[7]} and relation to the power spectrum method is presented in ^{[8]}.
Since in the fluctuation function F(L) the square(root) is used, DFA measures the scalingbehavior of the second momentfluctuations, this means α = α(2). The multifractal generalization (MFDFA)^{[9]} uses a variable moment q and provides α(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 H = α(2) and to the second moment minus 1 for nonstationary cases H = α(2) − 1.^{[citation needed]}
Relations to other methods
In the case of powerlaw decaying autocorrelations, the correlation function decays with an exponent γ: . In addition the power spectrum decays as . The three exponent are related by^{[3]}:
 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), we have , 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), we have , 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, thus, the exponents of their power spectra differ by 2.
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 selfsimilar. Selfsimilarity requires that the points on the loglog graph are sufficiently collinear across a very wide range of window sizes L.
Also, there are many scaling exponentlike quantities that can be measured for a selfsimilar time series, including the divider dimension and Hurst exponent. Therefore, the DFA scaling exponent α is not a fractal dimension sharing all the desirable properties of the Hausdorff dimension, for example, although in certain special cases it can be shown to be related to the boxcounting dimension for the graph of a time series.
Related topics
 Random walks
 Long range correlations
 Self affinity
 time series analysis
 Hurst exponent
 Fluctuations analysis
 Long term persistence
References
 ^ Peng, C.K. et al. (1994). "Mosaic organization of DNA nucleotides". Phys Rev E 49: 1685–1689. http://prola.aps.org/pdf/PRE/v49/i2/p1685_1.
 ^ Kantelhardt J.W. et al. (2001). "Detecting longrange correlations with detrended fluctuation analysis". Physica A 295: 441–454.
 ^ ^{a} ^{b} Buldyrev et al (1995). "LongRange CorrelationProperties of Coding And Noncoding DnaSequences Genbank Analysis". Phys Rev E 51: 5084–5091. http://prola.aps.org/abstract/PRE/v51/i5/p5084_1.
 ^ Bunde A, Havlin S (1996). Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York.
 ^ Little et al (2006). "Nonlinear, BiophysicallyInformed Speech Pathology Detection". 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings.: Toulouse, France. pp. II1080II1083. http://www.physics.ox.ac.uk/users/littlem/publications/dfafullpath.pdf.
 ^ Bunde A. et al (2000). "Correlated and uncorrelated regions in heartrate fluctuations during sleep". Phys Rev E 85(17): 3736–3739.
 ^ Hu, K. et al (2001). "Effect of trends on detrended fluctuation analysis". Phys Rev E 64(1): 011114.
 ^ Heneghan et al (2000). "Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes". Phys Rev E 62(5): 6103–6110. http://prola.aps.org/abstract/PRE/v62/i5/p6103_1.
 ^ H.E. Stanley, J.W. Kantelhardt; S.A. Zschiegner, E. KoscielnyBunde, S. Havlin, A. Bunde (2002). "Multifractal detrended fluctuation analysis of nonstationary time series". Physica A 316: 87. http://havlin.biu.ac.il/Publications.php?keyword=Multifractal+detrended+fluctuation+analysis+of+nonstationary+time+series++&year=*&match=all.
External links
 Tutorial on how to calculate detrended fluctuation analysis 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.
Categories: Stochastic processes
 Time series analysis
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