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.

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.


Consider a bounded time series of length , where , and let its mean value be denoted . Integration or summation converts this into an unbounded process :

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

Next, is divided into time windows of length samples each, and a local least squares straight-line fit (the local trend) is calculated by minimising the squared errors within each time window. Let indicate the resulting piecewise sequence of straight-line fits. Then, the root-mean-square deviation from the trend, the fluctuation, is calculated:

Finally, this process of detrending followed by fluctuation measurement is repeated over a range of different window sizes , and a log-log graph of against is constructed.[2][3]

A straight line on this log-log graph indicates statistical self-affinity expressed as . The scaling exponent is calculated as the slope of a straight line fit to the log-log graph of against using least-squares. This exponent is a generalization of the Hurst exponent. Because the expected displacement in an uncorrelated random walk of length N grows like , an exponent of would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is fractional Gaussian noise, with the precise value giving information about the series self-correlations:

  • : anti-correlated
  • : uncorrelated, white noise
  • : correlated
  • : 1/f-noise, pink noise
  • : non-stationary, unbounded
  • : Brownian noise

Generalization to supralinear trends[edit]

Trends of higher order can be removed by higher order DFA, in which a linear fit is replaced by a polynomial fit.[4] In the described case, linear fits () are applied to the profile, thus it is called DFA1. To remove trends of higher order, DFA, uses polynomial fits of order .

Owing to the summation (integration) from to , 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 . In general, DFA of order removes (polynomial) trends of order . For linear trends in the mean of at least DFA2 is needed.

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[edit]

Since in the fluctuation function the square (root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means . The multifractal generalization (MF-DFA)[5] uses a variable moment and provides . 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 and to the second moment minus 1 for nonstationary cases .[6][7][5]

Essentially, the scaling exponents need not be independent of the scale of the system. In the case depends on the power extracted from

where the previous DFA is . Multifractal systems scale as a function . To uncover multifractality, Multifractal Detrended Fluctuation Analysis is one possible method.[8]

Applications and study[edit]

The DFA method has been applied to many systems, e.g. DNA sequences,[9][10] neuronal oscillations,[7] speech pathology detection,[11] heartbeat fluctuation in different sleep stages,[12] and animal behavior pattern analysis.[13]

The effect of trends on DFA has been studied.[14]

Relations to other methods, for specific types of signal[edit]

For signals with power-law-decaying autocorrelation[edit]

In the case of power-law decaying auto-correlations, the correlation function decays with an exponent : . In addition the power spectrum decays as . The three exponents are related by:[9]

  • and
  • .

The relations can be derived using the Wiener–Khinchin theorem. The relation of DFA to the power spectrum method has been well studied.[15]

Thus, is tied to the slope of the power spectrum and is used to describe the color of noise by this relationship: .

For fractional Gaussian noise[edit]

For fractional Gaussian noise (FGN), we have , and thus , and , where is the Hurst exponent. for FGN is equal to .[16]

For fractional Brownian motion[edit]

For fractional Brownian motion (FBM), we have , and thus , and , where is the Hurst exponent. for FBM is equal to .[6] 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[edit]

As with most methods that depend upon line fitting, it is always possible to find a number 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 . Furthermore, a combination of techniques including MLE, rather than least-squares has been shown to better approximate the scaling, or power-law, exponent.[17]

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 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.

See also[edit]


  1. ^ Peng, C.K.; et al. (1994). "Mosaic organization of DNA nucleotides". Phys. Rev. E. 49 (2): 1685–1689. Bibcode:1994PhRvE..49.1685P. doi:10.1103/physreve.49.1685. PMID 9961383. S2CID 3498343.
  2. ^ Peng, C.K.; et al. (1994). "Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series". Chaos. 49 (1): 82–87. Bibcode:1995Chaos...5...82P. doi:10.1063/1.166141. PMID 11538314. S2CID 722880.
  3. ^ Bryce, R.M.; Sprague, K.B. (2012). "Revisiting detrended fluctuation analysis". Sci. Rep. 2: 315. Bibcode:2012NatSR...2E.315B. doi:10.1038/srep00315. PMC 3303145. PMID 22419991.
  4. ^ Kantelhardt J.W.; et al. (2001). "Detecting long-range correlations with detrended fluctuation analysis". Physica A. 295 (3–4): 441–454. arXiv:cond-mat/0102214. Bibcode:2001PhyA..295..441K. doi:10.1016/s0378-4371(01)00144-3. S2CID 55151698.
  5. ^ a b 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. arXiv:physics/0202070. Bibcode:2002PhyA..316...87K. doi:10.1016/s0378-4371(02)01383-3. S2CID 18417413.
  6. ^ a b Movahed, M. Sadegh; et al. (2006). "Multifractal detrended fluctuation analysis of sunspot time series". Journal of Statistical Mechanics: Theory and Experiment. 02.
  7. ^ a b Hardstone, Richard; Poil, Simon-Shlomo; Schiavone, Giuseppina; Jansen, Rick; Nikulin, Vadim V.; Mansvelder, Huibert D.; Linkenkaer-Hansen, Klaus (1 January 2012). "Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations". Frontiers in Physiology. 3: 450. doi:10.3389/fphys.2012.00450. PMC 3510427. PMID 23226132.
  8. ^ Kantelhardt, J.W.; et al. (2002). "Multifractal detrended fluctuation analysis of nonstationary time series". Physica A: Statistical Mechanics and Its Applications. 316 (1–4): 87–114. arXiv:physics/0202070. Bibcode:2002PhyA..316...87K. doi:10.1016/S0378-4371(02)01383-3. S2CID 18417413.
  9. ^ a b Buldyrev; et al. (1995). "Long-Range Correlation-Properties of Coding And Noncoding Dna-Sequences- Genbank Analysis". Phys. Rev. E. 51 (5): 5084–5091. Bibcode:1995PhRvE..51.5084B. doi:10.1103/physreve.51.5084. PMID 9963221.
  10. ^ Bunde A, Havlin S (1996). "Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York". {{cite journal}}: Cite journal requires |journal= (help)
  11. ^ Little, M.; McSharry, P.; Moroz, I.; Roberts, S. (2006). "Nonlinear, Biophysically-Informed Speech Pathology Detection" (PDF). 2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings. Vol. 2. pp. II-1080–II-1083. doi:10.1109/ICASSP.2006.1660534. ISBN 1-4244-0469-X. S2CID 11068261.
  12. ^ Bunde A.; et al. (2000). "Correlated and uncorrelated regions in heart-rate fluctuations during sleep". Phys. Rev. E. 85 (17): 3736–3739. Bibcode:2000PhRvL..85.3736B. doi:10.1103/physrevlett.85.3736. PMID 11030994. S2CID 21568275.
  13. ^ 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. 81: 104409. doi:10.1016/j.bspc.2022.104409. S2CID 254206934.
  14. ^ Hu, K.; et al. (2001). "Effect of trends on detrended fluctuation analysis". Phys. Rev. E. 64 (1): 011114. arXiv:physics/0103018. Bibcode:2001PhRvE..64a1114H. doi:10.1103/physreve.64.011114. PMID 11461232. S2CID 2524064.
  15. ^ 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. Bibcode:2000PhRvE..62.6103H. doi:10.1103/physreve.62.6103. PMID 11101940. S2CID 10791480.
  16. ^ Taqqu, Murad S.; et al. (1995). "Estimators for long-range dependence: an empirical study". Fractals. 3 (4): 785–798. doi:10.1142/S0218348X95000692.
  17. ^ Clauset, Aaron; Rohilla Shalizi, Cosma; Newman, M. E. J. (2009). "Power-Law Distributions in Empirical Data". SIAM Review. 51 (4): 661–703. arXiv:0706.1062. Bibcode:2009SIAMR..51..661C. doi:10.1137/070710111. S2CID 9155618.

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