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 2,000 times as of 2013 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.
Given a bounded time series of length , where , integration or summation first converts this into an unbounded process :
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:
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 Brownian motion, 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
Trends of higher order can be removed by higher order DFA, where a linear fit is replaced by a polynomial fit. 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 . Due 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 constants trends in the original sequence and thus, in its detrending it is equivalent to DFA1. The DFA method was applied to many systems; e.g., DNA sequences, neuronal oscillations, speech pathology detection, and heartbeat fluctuation in different sleep stages. Effect of trends on DFA were studied in and relation to the power spectrum method is presented in.
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) 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 .
Relations to other methods
The relations can be derived using the Wiener–Khinchin theorem.
Thus, is tied to the slope of the power spectrum and is used to describe the color of noise by this relationship: .
For fractional Brownian motion (FBM), we have , and thus , and , where is the Hurst exponent. for FBM is equal to . 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 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 .
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.
- Peng, C.K.; et al. (1994). "Mosaic organization of DNA nucleotides". Phys. Rev. E. 49: 1685–1689. Bibcode:1994PhRvE..49.1685P. doi:10.1103/physreve.49.1685.
- Peng, C.K.; et al. (1994). "Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series". Chaos. 49: 82–87. Bibcode:1995Chaos...5...82P. doi:10.1063/1.166141. PMID 11538314.
- Bryce, R.M.; Sprague, K.B. (2012). "Revisiting detrended fluctuation analysis". Sci. Rep. 2: 315. Bibcode:2012NatSR...2E.315B. doi:10.1038/srep00315. PMC . PMID 22419991.
- Kantelhardt J.W.; et al. (2001). "Detecting long-range correlations with detrended fluctuation analysis". Physica A. 295: 441–454. arXiv: . Bibcode:2001PhyA..295..441K. doi:10.1016/s0378-4371(01)00144-3.
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- Bunde A, Havlin S (1996). "Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York".
- 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. doi:10.3389/fphys.2012.00450.
- Little; et al. (2006). "Nonlinear, Biophysically-Informed Speech Pathology Detection" (PDF). 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings.: Toulouse, France. pp. II-1080-II-1083.[permanent dead link]
- 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.
- Hu, K.; et al. (2001). "Effect of trends on detrended fluctuation analysis". Phys. Rev. E. 64 (1): 011114. arXiv: . Bibcode:2001PhRvE..64a1114H. doi:10.1103/physreve.64.011114.
- 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. Bibcode:2000PhRvE..62.6103H. doi:10.1103/physreve.62.6103.
- 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: 87. arXiv: . Bibcode:2002PhyA..316...87K. doi:10.1016/s0378-4371(02)01383-3.
- Clauset, Aaron; Rohilla Shalizi, Cosma; Newman, M. E. J. (2009). "Power-Law Distributions in Empirical Data". SIAM Review. 51(4): 661–703. doi:10.1137/070710111.
- 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.