Detrended fluctuation analysis

File:Detrended fluctuation analysis, illustrated with Brownian motion.png

Given: a time series $x\_1,\; x\_2,\; ...,\; x\_N$.

Compute its average value $\backslash langle\; x\backslash rangle\; =\; \backslash frac\; 1N\; \backslash sum\_\{t=1\}^N\; x\_t$.

Sum it into a process $X\_t=\backslash sum\_\{i=1\}^t\; (x\_i-\backslash langle\; x\backslash rangle)$. 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 $T\; =\; \backslash \{n\_1,\; ...,\; n\_k\backslash \}$ of integers, such that , the smallest $n\_1\; \backslash approx\; 4$, the largest $n\_k\; \backslash approx\; N/4$, and the sequence is roughly distributed evenly in log-scale: $\backslash log(n\_2)\; -\; \backslash log(n\_1)\; \backslash approx\; \backslash log(n\_3)\; -\; \backslash log(n\_2)\; \backslash approx\; \backslash cdots$. In other words, it is approximately a geometric progression.{{Cite journal |last1=Hardstone |first1=Richard |last2=Poil |first2=Simon-Shlomo |last3=Schiavone |first3=Giuseppina |last4=Jansen |first4=Rick |last5=Nikulin |first5=Vadim |last6=Mansvelder |first6=Huibert |last7=Linkenkaer-Hansen |first7=Klaus |date=2012 |title=Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations |journal=Frontiers in Physiology |volume=3 |page=450 |doi=10.3389/fphys.2012.00450 |pmid=23226132 |pmc=3510427 |issn=1664-042X |doi-access=free }}

For each $n\; \backslash in\; T$, divide the sequence $X\_t$ into consecutive segments of length $n$. Within each segment, compute the least squares straight-line fit (the local trend). Let $Y\_\{1,n\},\; Y\_\{2,n\},\; ...,\; Y\_\{N,n\}$ be the resulting piecewise-linear fit.

Compute the root-mean-square deviation from the local trend (local fluctuation):$$F(\; n,\; i)\; =\; \backslash sqrt\{\backslash frac\{1\}\{n\}\backslash sum\_\{t\; =\; in+1\}^\{in+n\}\; \backslash left(\; X\_t\; -\; Y\_\{t,\; n\}\; \backslash right)^2\}.$$And their root-mean-square is the total fluctuation:

:$F(\; n\; )\; =\; \backslash sqrt\{\backslash frac\{1\}\{N/n\}\backslash sum\_\{i\; =\; 1\}^\{N/n\}\; F(n,\; i)^2\}.$

(If $N$ is not divisible by $n$, then one can either discard the remainder of the sequence, or repeat the procedure on the reversed sequence, then take their root-mean-square.{{Cite journal |last1=Zhou |first1=Yu |last2=Leung |first2=Yee |date=2010-06-21 |title=Multifractal temporally weighted detrended fluctuation analysis and its application in the analysis of scaling behavior in temperature series |url=https://iopscience.iop.org/article/10.1088/1742-5468/2010/06/P06021 |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=2010 |issue=6 |pages=P06021 |doi=10.1088/1742-5468/2010/06/P06021 |bibcode=2010JSMTE..06..021Z |s2cid=119901219 |issn=1742-5468}})

Make the log-log plot $\backslash log\; n\; -\; \backslash log\; F(n)$.{{cite journal|last=Peng|first=C.K.|s2cid=722880|title=Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series|journal=Chaos|year=1994|volume=49|issue=1|pages=82–87|doi=10.1063/1.166141|display-authors=etal|pmid=11538314|bibcode=1995Chaos...5...82P}}{{cite journal|last1=Bryce|first1=R.M.|last2=Sprague|first2=K.B.|title=Revisiting detrended fluctuation analysis|journal=Sci. Rep.|year=2012|volume=2|page=315|doi=10.1038/srep00315|pmc=3303145|pmid=22419991|bibcode=2012NatSR...2..315B}}

A straight line of slope $\backslash alpha$ on the log-log plot indicates a statistical self-affinity of form $F(n)\; \backslash propto\; n^\{\backslash alpha\}$. Since $F(n)$ monotonically increases with $n$, we always have $\backslash alpha\; >\; 0$.

The scaling exponent $\backslash alpha$ is a generalization of the Hurst exponent, with the precise value giving information about the series self-correlations:

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

Because the expected displacement in an uncorrelated random walk of length N grows like $\backslash sqrt\{N\}$, an exponent of $\backslash tfrac\{1\}\{2\}$ would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is fractional Gaussian noise.

Though the DFA algorithm always produces a positive number $\backslash alpha$ 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 $n$. 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.{{Cite journal |last1=Clauset |first1=Aaron |last2=Rohilla Shalizi |first2=Cosma |last3=Newman |first3=M. E. J. |year=2009 |title=Power-Law Distributions in Empirical Data |journal=SIAM Review |volume=51 |issue=4 |pages=661–703 |arxiv=0706.1062 |bibcode=2009SIAMR..51..661C |doi=10.1137/070710111 |s2cid=9155618}}

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 $\backslash alpha$ 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.

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.{{cite journal |last=Kantelhardt J.W. |display-authors=etal |year=2001 |title=Detecting long-range correlations with detrended fluctuation analysis |journal=Physica A |volume=295 |issue=3–4 |pages=441–454 |arxiv=cond-mat/0102214 |bibcode=2001PhyA..295..441K |doi=10.1016/s0378-4371(01)00144-3 |s2cid=55151698}}

Since $X\_t$ is a cumulative sum of $x\_t-\backslash langle\; x\backslash rangle$, a linear trend in $X\_t$ is a constant trend in $x\_t-\backslash langle\; x\backslash rangle$, which is a constant trend in $x\_t$ (visible as short sections of "flat plateaus"). In this regard, DFA1 removes the mean from segments of the time series $x\_t$ before quantifying the fluctuation.

Similarly, a degree n trend in $X\_t$ is a degree (n-1) trend in $x\_t$. For example, DFA1 removes linear trends from segments of the time series $x\_t$ before quantifying the fluctuation, DFA1 removes parabolic trends from $x\_t$, 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.

DFA can be generalized by computing $$F\_q(\; n\; )\; =\; \backslash left(\backslash frac\{1\}\{N/n\}\backslash sum\_\{i\; =\; 1\}^\{N/n\}\; F(n,\; i)^q\backslash right)^\{1/q\}$$ then making the log-log plot of $\backslash log\; n\; -\; \backslash log\; F\_q(n)$, If there is a strong linearity in the plot of $\backslash log\; n\; -\; \backslash log\; F\_q(n)$, then that slope is $\backslash alpha(q)$.{{cite journal |last=H.E. Stanley |first=J.W. Kantelhardt |author2=S.A. Zschiegner |author3=E. Koscielny-Bunde |author4=S. Havlin |author5=A. Bunde |year=2002 |title=Multifractal detrended fluctuation analysis of nonstationary time series |url=http://havlin.biu.ac.il/Publications.php?keyword=Multifractal+detrended+fluctuation+analysis+of+nonstationary+time+series++&year=*&match=all |journal=Physica A |volume=316 |issue=1–4 |pages=87–114 |arxiv=physics/0202070 |bibcode=2002PhyA..316...87K |doi=10.1016/s0378-4371(02)01383-3 |s2cid=18417413 |access-date=2011-07-20 |archive-date=2018-08-28 |archive-url=https://web.archive.org/web/20180828134644/http://havlin.biu.ac.il/Publications.php?keyword=Multifractal+detrended+fluctuation+analysis+of+nonstationary+time+series++&year=*&match=all |url-status=dead }} DFA is the special case where $q=2$.

Multifractal systems scale as a function $F\_q(n)\; \backslash propto\; n^\{\backslash alpha(q)\}$. 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 $H=\backslash alpha(2)$ for stationary cases, and $H=\backslash alpha(2)-1$ for nonstationary cases.{{cite journal |last1=Movahed |first1=M. Sadegh |display-authors=et al |date=2006 |title=Multifractal detrended fluctuation analysis of sunspot time series |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=02}}

The DFA method has been applied to many systems, e.g. DNA sequences;{{cite journal|last=Buldyrev|title=Long-Range Correlation-Properties of Coding And Noncoding Dna-Sequences- Genbank Analysis|journal=Phys. Rev. E|year=1995|volume=51|issue=5|pages=5084–5091|doi=10.1103/physreve.51.5084|pmid=9963221|display-authors=etal|bibcode=1995PhRvE..51.5084B}}{{cite journal|last=Bunde A|first=Havlin S|title=Fractals and Disordered Systems, Springer, Berlin, Heidelberg, New York|year=1996}} heartbeat dynamics in sleep and wake,{{Cite journal |last1=Ivanov |first1=P. Ch |last2=Bunde |first2=A |last3=Amaral |first3=L. A. N |last4=Havlin |first4=S |last5=Fritsch-Yelle |first5=J |author5-link=Janice Meck|last6=Baevsky |first6=R. M |last7=Stanley |first7=H. E |last8=Goldberger |first8=A. 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In the case of power-law decaying auto-correlations, the correlation function decays with an exponent $\backslash gamma$:

$C(L)\backslash sim\; L^\{-\backslash gamma\}\backslash !\backslash$.

In addition the power spectrum decays as $P(f)\backslash sim\; f^\{-\backslash beta\}\backslash !\backslash$.

The three exponents are related by:

• $\backslash gamma=2-2\backslash alpha$
• $\backslash beta=2\backslash alpha-1$ and
• $\backslash gamma=1-\backslash beta$.

The relations can be derived using the Wiener–Khinchin theorem. The relation of DFA to the power spectrum method has been well studied.{{cite journal|last=Heneghan|s2cid=10791480|title=Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes|journal=Phys. Rev. E|year=2000|volume=62 | issue = 5|pages=6103–6110|doi=10.1103/physreve.62.6103|pmid=11101940|display-authors=etal|bibcode=2000PhRvE..62.6103H}}

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

For fractional Gaussian noise (FGN), we have $\backslash beta\; \backslash in\; [-1,1]$, and thus $\backslash alpha\; \backslash in\; [0,1]$, and $\backslash beta\; =\; 2H-1$, where $H$ is the Hurst exponent. $\backslash alpha$ for FGN is equal to $H$.{{cite journal |last1=Taqqu |first1=Murad S. |display-authors=et al |title=Estimators for long-range dependence: an empirical study. |journal=Fractals |date=1995 |volume=3 |issue=4 |pages=785–798|doi=10.1142/S0218348X95000692 }}

For fractional Brownian motion (FBM), we have $\backslash beta\; \backslash in\; [1,3]$, and thus $\backslash alpha\; \backslash in\; [1,2]$, and $\backslash beta\; =\; 2H+1$, where $H$ is the Hurst exponent. $\backslash alpha$ for FBM is equal to $H+1$. In this context, FBM is the cumulative sum or the integral of FGN, thus, the exponents of their

power spectra differ by 2.

• {{annotated link|Multifractal system}}
• {{annotated link|Self-organized criticality}}
• {{annotated link|Self-affinity}}
• {{annotated link|Time series analysis}}
• {{annotated link|Hurst exponent}}

{{reflist}}

• [https://www.nbtwiki.net/doku.php?id=tutorial:detrended_fluctuation_analysis_dfa Tutorial on how to calculate detrended fluctuation analysis] {{Webarchive|url=https://web.archive.org/web/20190203115306/https://www.nbtwiki.net/doku.php?id=tutorial:detrended_fluctuation_analysis_dfa |date=2019-02-03 }} in Matlab using the Neurophysiological Biomarker Toolbox.
• [http://www.maxlittle.net/software/ FastDFA] MATLAB code for rapidly calculating the DFA scaling exponent on very large datasets.
• [http://www.physionet.org/physiotools/dfa Physionet] A good overview of DFA and C code to calculate it.
• [https://github.com/LRydin/MFDFA MFDFA] Python implementation of (Multifractal) Detrended Fluctuation Analysis.

Category:Autocorrelation

Category:Fractals