Detrended Fluctuation Analysis
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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 2020[1] and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.

Calculation

Given a bounded time series ${\displaystyle x_{t}}$ of length ${\displaystyle N}$, where ${\displaystyle t\in \mathbb {N} }$, integration or summation first converts this into an unbounded process ${\displaystyle X_{t}}$:

${\displaystyle X_{t}=\sum _{i=1}^{t}(x_{i}-\langle x\rangle )}$

where ${\displaystyle \langle x\rangle }$ denotes the mean value of the time series. ${\displaystyle X_{t}}$ is called cumulative sum or profile. This process converts, for example, an i.i.d. white noise process into a random walk.

Next, ${\displaystyle X_{t}}$ is divided into time windows of length ${\displaystyle n}$ 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 ${\displaystyle Y_{t}}$ indicate the resulting piecewise sequence of straight-line fits. Then, the root-mean-square deviation from the trend, the fluctuation, is calculated:

${\displaystyle F(n)={\sqrt {{\frac {1}{N}}\sum _{t=1}^{N}\left(X_{t}-Y_{t}\right)^{2}}}.}$

Finally, this process of detrending followed by fluctuation measurement is repeated over a range of different window sizes ${\displaystyle n}$, and a log-log graph of ${\displaystyle F(n)}$ against ${\displaystyle n}$ is constructed.[2][3]

A straight line on this log-log graph indicates statistical self-affinity expressed as ${\displaystyle F(n)\propto n^{\alpha }}$. The scaling exponent ${\displaystyle \alpha }$ is calculated as the slope of a straight line fit to the log-log graph of ${\displaystyle n}$ against ${\displaystyle F(n)}$ 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 ${\displaystyle {\sqrt {N}}}$, an exponent of ${\displaystyle {\tfrac {1}{2}}}$ 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:

• ${\displaystyle \alpha <1/2}$: anti-correlated
• ${\displaystyle \alpha \simeq 1/2}$: uncorrelated, white noise
• ${\displaystyle \alpha >1/2}$: correlated
• ${\displaystyle \alpha \simeq 1}$: 1/f-noise, pink noise
• ${\displaystyle \alpha >1}$: non-stationary, unbounded
• ${\displaystyle \alpha \simeq 3/2}$: Brownian noise

Trends of higher order can be removed by higher order DFA, where a linear fit is replaced by a polynomial fit.[4] In the described case, linear fits (${\displaystyle i=1}$) are applied to the profile, thus it is called DFA1. To remove trends of higher order, DFA${\displaystyle i}$, uses polynomial fits of order ${\displaystyle i}$. Due to the summation (integration) from ${\displaystyle x_{i}}$ to ${\displaystyle X_{t}}$, 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 ${\displaystyle x_{i}}$. In general DFA of order ${\displaystyle i}$ removes (polynomial) trends of order ${\displaystyle i-1}$. For linear trends in the mean of ${\displaystyle x_{i}}$ 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. The DFA method was applied to many systems; e.g., DNA sequences,[5][6] neuronal oscillations,[7] speech pathology detection,[8] and heartbeat fluctuation in different sleep stages.[9] Effect of trends on DFA were studied in[10] and relation to the power spectrum method is presented in.[11]

Since in the fluctuation function ${\displaystyle F(n)}$ the square (root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means ${\displaystyle \alpha =\alpha (2)}$. The multifractal generalization (MF-DFA)[12] uses a variable moment ${\displaystyle q}$ and provides ${\displaystyle \alpha (q)}$. 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 ${\displaystyle H=\alpha (2)}$ and to the second moment minus 1 for nonstationary cases ${\displaystyle H=\alpha (2)-1}$.[13][7][12]

Relations to other methods

In the case of power-law decaying auto-correlations, the correlation function decays with an exponent ${\displaystyle \gamma }$: ${\displaystyle C(L)\sim L^{-\gamma }\!\ }$. In addition the power spectrum decays as ${\displaystyle P(f)\sim f^{-\beta }\!\ }$. The three exponent are related by:[5]

• ${\displaystyle \gamma =2-2\alpha }$
• ${\displaystyle \beta =2\alpha -1}$ and
• ${\displaystyle \gamma =1-\beta }$.

The relations can be derived using the Wiener-Khinchin theorem.

Thus, ${\displaystyle \alpha }$ is tied to the slope of the power spectrum ${\displaystyle \beta }$ and is used to describe the color of noise by this relationship: ${\displaystyle \alpha =(\beta +1)/2}$.

For fractional Gaussian noise (FGN), we have ${\displaystyle \beta \in [-1,1]}$, and thus ${\displaystyle \alpha \in [0,1]}$, and ${\displaystyle \beta =2H-1}$, where ${\displaystyle H}$ is the Hurst exponent. ${\displaystyle \alpha }$ for FGN is equal to ${\displaystyle H}$.[14]

For fractional Brownian motion (FBM), we have ${\displaystyle \beta \in [1,3]}$, and thus ${\displaystyle \alpha \in [1,2]}$, and ${\displaystyle \beta =2H+1}$, where ${\displaystyle H}$ is the Hurst exponent. ${\displaystyle \alpha }$ for FBM is equal to ${\displaystyle H+1}$.[13] 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 ${\displaystyle \alpha }$ 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 ${\displaystyle L}$. Furthermore, a combination of techniques including MLE, rather than least-squares has been shown to better approximate the scaling, or power-law, exponent.[15]

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 ${\displaystyle \alpha }$ 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.

Multifractality and Multifractal Detrended Fluctuation Analysis

It isn't always the case that the scaling exponents are independent of the scale of the system. In the case ${\displaystyle \alpha }$ depends on the power ${\displaystyle q}$ extracted from

${\displaystyle F_{q}(n)=\left({\frac {1}{N}}\sum _{t=1}^{N}\left(X_{t}-Y_{t}\right)^{q}\right)^{1/q},}$

where the previous DFA is ${\displaystyle q=2}$. Multifractal systems scale as a function ${\displaystyle F_{q}(n)\propto n^{\alpha (q)}}$. To uncover multifractality, Multifractal Detrended Fluctuation Analysis is one possible method.[16]

References

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.
5. ^ 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.
6. ^ Bunde A, Havlin S (1996). "Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York". Cite journal requires |journal= (help)
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. ^ 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. 2. pp. II-1080-II-1083. doi:10.1109/ICASSP.2006.1660534. ISBN 1-4244-0469-X.
9. ^ 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.
10. ^ 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.
11. ^ 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.
12. ^ 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.
13. ^ a b Movahed, M. Sadegh; et al. (2006). "Multifractal detrended fluctuation analysis of sunspot time series". Journal of Statistical Mechanics: Theory and Experiment. 02.
14. ^ Taqqu, Murad S.; et al. (1995). "Estimators for long-range dependence: an empirical study". Fractals. 3 (4): 785-798. doi:10.1142/S0218348X95000692.
15. ^ 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.
16. ^ 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.