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| In [[stochastic processes]], [[chaotic dynamical systems|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]].
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| 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 [[stationary process|non-stationary]] (changing with time). It is related to measures based upon spectral techniques such as [[autocorrelation]] and [[Fourier transform]].
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| Peng et al. introduced DFA in 1994 in a paper that has been cited over 2000 times as of 2013<ref>{{cite journal|last=Peng|first=C.K. et al.|title=Mosaic organization of DNA nucleotides|journal=Phys. Rev. E|year=1994|volume=49|pages=1685–1689|url=http://prola.aps.org/pdf/PRE/v49/i2/p1685_1}}</ref> and represents an extension of the (ordinary) [[fluctuation analysis]] (FA), which is affected by non-stationarities.
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| ==Calculation==
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| Given a bounded [[time series]] <math>x_t</math>, <math>t \in \mathbb{N}</math>, integration or summation first converts this into an unbounded process <math>X_t</math>:
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| :<math>X_t=\sum_{i=1}^t (x_i-\langle x_i\rangle)</math>
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| <math>X_t</math> is called cumulative sum or profile. This process converts, for example, an [[i.i.d.]] [[white noise]] process into a [[random walk]].
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| Next, <math>X_t</math> is divided into time windows <math>Y_j</math> of length <math>L</math> samples, and a local [[least squares]] straight-line fit (the local trend) is calculated by minimising the squared error <math>E^2</math> with respect to the slope and intercept parameters <math>a, b</math>:
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| :<math>E^2 = \sum_{j = 1}^L \left( Y_j - j a - b \right)^2.</math>
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| Trends of higher order, can be removed by higher order DFA, where the linear function <math>j a +b</math> is replaced by a polynomial of order <math>n</math>.<ref name = "Kantelhardt">{{cite journal|last=Kantelhardt J.W. et al.|title=Detecting long-range correlations with detrended fluctuation analysis|journal=Physica A|year=2001|volume=295|pages=441–454}}</ref>
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| Next, the root-mean-square deviation from the trend, the '''fluctuation''', is calculated over every window at every time scale:
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| :<math>F( L ) = \left[ \frac{1}{L}\sum_{j = 1}^L \left( Y_j - j a - b \right)^2 \right]^{\frac{1}{2}}.</math>
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| This detrending followed by fluctuation measurement process is repeated over the whole signal at a range of different window sizes <math>L</math>, and a [[log-log graph]] of <math>L</math> against <math>F(L)</math> is constructed.
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| A straight line on this log-log graph indicates statistical [[self-affinity]] expressed as <math>F(L) \propto L^{\alpha}</math>. The scaling exponent <math>\alpha</math> is calculated as the slope of a straight line fit to the log-log graph of <math>L</math> against <math>F(L)</math> using least-squares. This exponent is a generalization of the [[Hurst exponent]]. Because the expected displacement in an [[Random_walk#Correlated_steps|uncorrelated random walk]] of length L grows like <math>\sqrt{L}</math>, an exponent of <math>\tfrac{1}{2}</math> 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 self-correlations:
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| * <math>\alpha<1/2</math>: anti-correlated
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| * <math>\alpha \simeq 1/2</math>: uncorrelated, [[white noise]]
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| * <math>\alpha>1/2</math>: correlated
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| * <math>\alpha\simeq 1</math>: 1/f-noise, [[pink noise]]
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| * <math>\alpha>1</math>: non-stationary, unbounded
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| * <math>\alpha\simeq 3/2</math>: [[Brownian noise]]
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| There are different orders of DFA. In the described case, linear fits (<math>n=1</math>) are applied to the profile, thus it is called DFA1. In general, DFA<math>n</math>, uses polynomial fits of order <math>n</math>. Due to the summation (integration) from <math>x_i</math> to <math>X_t</math>, 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 <math>x_i</math>. In general DFA of order <math>n</math> removes (polynomial) trends of order <math>n-1</math>. For linear trends in the mean of <math>x_i</math> at least DFA2 is needed. The Hurst [[Rescaled range|R/S analysis]] removes constants trends in the original sequence and thus, in its detrending it is equivalent to DFA1.
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| The DFA method was applied to many systems; e.g., DNA sequences,<ref name ="Buldyrev">{{cite journal|last=Buldyrev et al|title=Long-Range Correlation-Properties of Coding And Noncoding Dna-Sequences- Genbank Analysis|journal=Phys. Rev. E|year=1995|volume=51|pages=5084–5091|url=http://prola.aps.org/abstract/PRE/v51/i5/p5084_1}}</ref><ref>{{cite journal|last=Bunde A|first=Havlin S|title=Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York|year=1996}}</ref> neuronal oscillations,<ref>{{cite journal|last=Hardstone|first=Richard|coauthors=Poil, Simon-Shlomo; Schiavone, Giuseppina; Jansen, Rick; Nikulin, Vadim V.; Mansvelder, Huibert D.; Linkenkaer-Hansen, Klaus|title=Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations|journal=Frontiers in Physiology|date=1 January 2012|volume=3|doi=10.3389/fphys.2012.00450}}</ref> speech pathology detection,<ref>{{cite journal|last=Little et al|title=Nonlinear, Biophysically-Informed Speech Pathology Detection|journal=2006 IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings.: Toulouse, France. pp. II-1080-II-1083|year=2006|url=http://www.physics.ox.ac.uk/users/littlem/publications/dfafullpath.pdf}}</ref> and heartbeat fluctuation in different sleep stages.<ref>{{cite journal|last=Bunde A. et al|title=Correlated and uncorrelated regions in heart-rate fluctuations during sleep|journal=Phys. Rev. E|year=2000|volume=85(17)|pages=3736–3739}}</ref> Effect of trends on DFA were studied in<ref>{{cite journal|last=Hu, K. et al|title=Effect of trends on detrended fluctuation analysis|journal=Phys. Rev. E|year=2001|volume=64(1)|pages=011114}}</ref> and relation to the power spectrum method is presented in.<ref>{{cite journal|last=Heneghan et al|title=Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes|journal=Phys. Rev. E|year=2000|volume=62(5)|pages=6103–6110|url=http://prola.aps.org/abstract/PRE/v62/i5/p6103_1}}</ref>
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| Since in the fluctuation function <math>F(L)</math> the square(root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means <math>\alpha=\alpha(2)</math>. The [[multifractal]] generalization ([[MF-DFA]])<ref name="Kantelhardt2">{{cite journal|last=H.E. Stanley|first=J.W. Kantelhardt|coauthors=S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde|title=Multifractal detrended fluctuation analysis of nonstationary time series|journal=Physica A|year=2002|volume=316|pages=87|url=http://havlin.biu.ac.il/Publications.php?keyword=Multifractal+detrended+fluctuation+analysis+of+nonstationary+time+series++&year=*&match=all}}</ref> uses a variable moment <math>q</math> and provides <math>\alpha(q)</math>. 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 <math>H=\alpha(2)</math> and to the second moment minus 1 for nonstationary cases <math>H=\alpha(2)-1</math>.<ref name="Kantelhardt2" />
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| ==Relations to other methods==
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| In the case of power-law decaying auto-correlations, the [[correlation function]] decays with an exponent <math>\gamma</math>:
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| <math>C(L)\sim L^{-\gamma}\!\ </math>.
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| In addition the [[power spectrum]] decays as <math>P(f)\sim f^{-\beta}\!\ </math>.
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| The three exponent are related by:<ref name="Buldyrev" />
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| * <math>\gamma=2-2\alpha</math>
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| * <math>\beta=2\alpha-1</math> and
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| * <math>\gamma=1-\beta</math>.
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| The relations can be derived using the [[Wiener–Khinchin theorem]].
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| Thus, <math>\alpha</math> is tied to the slope of the power spectrum <math>\beta</math> used to describe the [[color of noise]] by this relationship: <math>\alpha = (\beta+1)/2</math>.
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| For [[Fractional Brownian motion|fractional Gaussian noise]] (FGN), we have <math> \beta \in [-1,1] </math>, and thus <math>\alpha = [0,1]</math>, and <math>\beta = 2H-1</math>, where <math>H</math> is the [[Hurst exponent]]. <math>\alpha</math> for FGN is equal to <math>H</math>.
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| For [[fractional Brownian motion]] (FBM), we have <math> \beta \in [1,3] </math>, and thus <math>\alpha = [1,2]</math>, and <math>\beta = 2H+1</math>, where <math>H</math> is the [[Hurst exponent]]. <math>\alpha</math> for FBM is equal to <math>H+1</math>. In this context, FBM is the cumulative sum or the [[integral]] of FGN, thus, the exponents of their
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| power spectra differ by 2.
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| ==Pitfalls in interpretation==
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| As with most methods that depend upon line fitting, it is always possible to find a number <math>\alpha</math> 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 <math>L</math>.
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| 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 <math>\alpha</math> 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 [[box-counting dimension]] for the graph of a time series.
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| ==See also==
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| * [[Self-organized criticality]]
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| * [[Self-affinity]]
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| * [[Time series analysis]]
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| * [[Hurst exponent]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| * [http://www.nbtwiki.net/doku.php?id=tutorial:detrended_fluctuation_analysis_dfa Tutorial on how to calculate detrended fluctuation analysis] in Matlab using the [[Neurophysiological Biomarker Toolbox]].
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| * [http://www.maxlittle.net/software/ FastDFA] [[MATLAB]] code for rapidly calculating the DFA scaling exponent on very large datasets.
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| * [http://www.physionet.org/physiotools/dfa Physionet] A good overview of DFA and C code to calculate it.
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| [[Category:Stochastic processes]]
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| [[Category:Time series analysis]]
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| [[Category:Fractals]]
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