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.