Detrended fluctuation analysis explained

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^[1] and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.

Definition

Algorithm

x _{1,} x _{2,} ..., x _{N} .

Compute its average value

\langle x\rangle = (1 N)/(\sum) ^{N} ^{t =1} x _{t} .

Sum it into a process

X _{t=\sum} ^{t} ^{i =1} (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

T = \{n _{1,} ..., n _{k\}} of integers, such that n _{1} < n _{2} < … < n _{k} , the smallest n _{1} ≈ 4 , the largest n _{k} ≈ N , and the sequence is roughly distributed evenly in log-scale: log (n _{2)} - log (n _{1)} ≈ log (n _{3)} - log (n _{2)} ≈ … . In other words, it is approximately a geometric progression.^[2]

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):$$F(n,\; i)\; =\; \backslash sqrt.$$And their root-mean-square is the total fluctuation:

F( n ) = \sqrt{ (1)/(N/n) \sum ^{N/n} _{i =1} 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.^[3])log n - log F(n) .^{[4] [5]}

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 noiseBecause the expected displacement in an uncorrelated random walk of length N grows like \sqrt{N} , an exponent of \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

\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.^[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

\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.^[7]

Since

X _{t} is a cumulative sum of x _{t-\langle} x\rangle , a linear trend in X _{t} is a constant trend in x _{t-\langle} x\rangle , 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$$F\_q(n)\; =\; \backslash left(\backslash frac\backslash sum\_^\; F(n,\; i)^q\backslash right)^.$$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) .^[8] 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.^[9]

Applications

The DFA method has been applied to many systems, e.g. DNA sequences,^{[10] [11]} neuronal oscillations,^[12] speech pathology detection,^[13] heartbeat fluctuation in different sleep stages,^[14] and animal behavior pattern analysis.^[15]

The effect of trends on DFA has been studied.^[16]

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.^[17]

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 = 2 H -1 , where H is the Hurst exponent. \alpha for FGN is equal to H .^[18]

For fractional Brownian motion

For fractional Brownian motion (FBM), we have

\beta \in [ 1 , 3 ] , and thus \alpha \in [ 1 , 2 ] , and \beta = 2 H +1 , where H is the Hurst exponent. \alpha for FBM is equal to H +1 .^[9] In this context, FBM is the cumulative sum or the integral of FGN, thus, the exponents of theirpower spectra differ by 2.

See also

• Multifractal system
• Self-organized criticality
• Self-affinity
• Time series analysis
• Hurst exponent

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.
• MFDFA Python implementation of (Multifractal) Detrended Fluctuation Analysis.

Notes and References

1. Peng. C.K.. 3498343. Mosaic organization of DNA nucleotides. Phys. Rev. E. 1994. 49. 2. 1685–1689. 10.1103/physreve.49.1685. 9961383. etal. 1994PhRvE..49.1685P. free.
2. Hardstone . Richard . Poil . Simon-Shlomo . Schiavone . Giuseppina . Jansen . Rick . Nikulin . Vadim . Mansvelder . Huibert . Linkenkaer-Hansen . Klaus . 2012 . Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations . Frontiers in Physiology . 3 . 450 . 10.3389/fphys.2012.00450 . 23226132 . 3510427 . 1664-042X . free .
3. Zhou . Yu . Leung . Yee . 2010-06-21 . Multifractal temporally weighted detrended fluctuation analysis and its application in the analysis of scaling behavior in temperature series . Journal of Statistical Mechanics: Theory and Experiment . 2010 . 6 . P06021 . 10.1088/1742-5468/2010/06/P06021 . 119901219 . 1742-5468.
4. Peng. C.K.. 722880. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos. 1994. 49. 1. 82–87. 10.1063/1.166141. etal. 11538314. 1995Chaos...5...82P.
5. Bryce. R.M.. Sprague. K.B.. Revisiting detrended fluctuation analysis. Sci. Rep.. 2012. 2. 315. 10.1038/srep00315. 3303145. 22419991. 2012NatSR...2E.315B.
6. Clauset . Aaron . Rohilla Shalizi . Cosma . Newman . M. E. J. . 2009 . Power-Law Distributions in Empirical Data . SIAM Review . 51 . 4 . 661–703 . 0706.1062 . 2009SIAMR..51..661C . 10.1137/070710111 . 9155618.
7. Kantelhardt J.W. . etal . 2001 . Detecting long-range correlations with detrended fluctuation analysis . Physica A . 295 . 3–4 . 441–454 . cond-mat/0102214 . 2001PhyA..295..441K . 10.1016/s0378-4371(01)00144-3 . 55151698.
8. H.E. Stanley . J.W. Kantelhardt . S.A. Zschiegner . E. Koscielny-Bunde . S. Havlin . A. Bunde . 2002 . Multifractal detrended fluctuation analysis of nonstationary time series . Physica A . 316 . 1–4 . 87–114 . physics/0202070 . 2002PhyA..316...87K . 10.1016/s0378-4371(02)01383-3 . 18417413 . 2011-07-20 . 2018-08-28 . https://web.archive.org/web/20180828134644/http://havlin.biu.ac.il/Publications.php?keyword=Multifractal+detrended+fluctuation+analysis+of+nonstationary+time+series++&year=*&match=all . dead .
9. Movahed . M. Sadegh . et al . 2006 . Multifractal detrended fluctuation analysis of sunspot time series . Journal of Statistical Mechanics: Theory and Experiment . 02.
10. Buldyrev. Long-Range Correlation-Properties of Coding And Noncoding Dna-Sequences- Genbank Analysis. Phys. Rev. E. 1995. 51. 5. 5084–5091. 10.1103/physreve.51.5084. 9963221. etal. 1995PhRvE..51.5084B.
11. Bunde A. Havlin S. Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York. 1996.
12. Hardstone. Richard. Poil, Simon-Shlomo . Schiavone, Giuseppina . Jansen, Rick . Nikulin, Vadim V. . Mansvelder, Huibert D. . Linkenkaer-Hansen, Klaus . Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations. Frontiers in Physiology. 1 January 2012. 3. 450. 10.3389/fphys.2012.00450. 23226132. 3510427. free .
13. Book: http://www.robots.ox.ac.uk/~sjrob/Pubs/NonlinearBiophysicalVoiceDisorderDetection.pdf . 10.1109/ICASSP.2006.1660534. Nonlinear, Biophysically-Informed Speech Pathology Detection. 2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings. 2. II-1080-II-1083. 2006. Little. M.. McSharry. P.. Moroz. I.. Irene Moroz . Roberts. S.. 1-4244-0469-X. 11068261 .
14. Bunde A.. 21568275. Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys. Rev. E. 2000. 85 . 17. 3736–3739. 10.1103/physrevlett.85.3736. etal. 11030994. 2000PhRvL..85.3736B.
15. Bogachev . Mikhail I. . Lyanova . Asya I. . Sinitca . Aleksandr M. . Pyko . Svetlana A. . Pyko . Nikita S. . Kuzmenko . Alexander V. . Romanov . Sergey A. . Brikova . Olga I. . Tsygankova . Margarita . Ivkin . Dmitry Y. . Okovityi . Sergey V. . Prikhodko . Veronika A. . Kaplun . Dmitrii I. . Sysoev . Yuri I. . Kayumov . Airat R. . March 2023 . Understanding the complex interplay of persistent and antipersistent regimes in animal movement trajectories as a prominent characteristic of their behavioral pattern profiles: Towards an automated and robust model based quantification of anxiety test data . Biomedical Signal Processing and Control . en . 81 . 104409 . 10.1016/j.bspc.2022.104409. 254206934 .
16. Hu, K.. Effect of trends on detrended fluctuation analysis. Phys. Rev. E. 2001. 64 . 1. 011114. 10.1103/physreve.64.011114. 11461232. etal. physics/0103018. 2001PhRvE..64a1114H. 2524064 .
17. Heneghan. 10791480. Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Phys. Rev. E. 2000. 62 . 5. 6103–6110. 10.1103/physreve.62.6103. 11101940. etal. 2000PhRvE..62.6103H.
18. Taqqu . Murad S. . et al . Estimators for long-range dependence: an empirical study. . Fractals . 1995 . 3 . 4 . 785–798. 10.1142/S0218348X95000692 .