Detrended fluctuation analysis
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
[edit]![](http://up.wiki.x.io/wikipedia/commons/thumb/5/57/Detrended_fluctuation_analysis%2C_illustrated_with_Brownian_motion.png/220px-Detrended_fluctuation_analysis%2C_illustrated_with_Brownian_motion.png)
Algorithm
[edit]Given: a time series .
Compute its average value .
Sum it into a process . 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 of integers, such that , the smallest , the largest , and the sequence is roughly distributed evenly in log-scale: . In other words, it is approximately a geometric progression.[2]
For each , divide the sequence into consecutive segments of length . Within each segment, compute the least squares straight-line fit (the local trend). Let be the resulting piecewise-linear fit.
Compute the root-mean-square deviation from the local trend (local fluctuation):And their root-mean-square is the total fluctuation:
(If is not divisible by , 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])
Make the log-log plot .[4][5]
Interpretation
[edit]A straight line of slope on the log-log plot indicates a statistical self-affinity of form . Since monotonically increases with , we always have .
The scaling exponent is a generalization of the Hurst exponent, 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
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.
Pitfalls in interpretation
[edit]Though the DFA algorithm always produces a positive number 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 . 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 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
[edit]Generalization to polynomial trends (higher order DFA)
[edit]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 is a cumulative sum of , a linear trend in is a constant trend in , which is a constant trend in (visible as short sections of "flat plateaus"). In this regard, DFA1 removes the mean from segments of the time series before quantifying the fluctuation.
Similarly, a degree n trend in is a degree (n-1) trend in . For example, DFA1 removes linear trends from segments of the time series before quantifying the fluctuation, DFA1 removes parabolic trends from , 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)
[edit]DFA can be generalized by computingthen making the log-log plot of , If there is a strong linearity in the plot of , then that slope is .[8] DFA is the special case where .
Multifractal systems scale as a function . 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 for stationary cases, and for nonstationary cases.[8][9][10]
Applications
[edit]The DFA method has been applied to many systems, e.g. DNA sequences,[11][12] neuronal oscillations,[10] 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
[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:[11]
- and
- .
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, 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 .[18]
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 .[9] In this context, FBM is the cumulative sum or the integral of FGN, thus, the exponents of their power spectra differ by 2.
See also
[edit]References
[edit]- ^ 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.
- ^ 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. doi:10.3389/fphys.2012.00450. ISSN 1664-042X. PMC 3510427. PMID 23226132.
- ^ 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. doi:10.1088/1742-5468/2010/06/P06021. ISSN 1742-5468. S2CID 119901219.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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. Archived from the original on 2018-08-28. Retrieved 2011-07-20.
- ^ a b Movahed, M. Sadegh; et al. (2006). "Multifractal detrended fluctuation analysis of sunspot time series". Journal of Statistical Mechanics: Theory and Experiment. 02.
- ^ 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.
- ^ 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.
- ^ Bunde A, Havlin S (1996). "Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York".
{{cite journal}}
: Cite journal requires|journal=
(help) - ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Taqqu, Murad S.; et al. (1995). "Estimators for long-range dependence: an empirical study". Fractals. 3 (4): 785–798. doi:10.1142/S0218348X95000692.
External links
[edit]- Tutorial on how to calculate detrended fluctuation analysis Archived 2019-02-03 at the Wayback Machine 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.