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.
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):
(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]
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:
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.
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 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.
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.
DFA can be generalized by computing
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]
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]
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]
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 (FGN), we have , and thus , and , where is the Hurst exponent. for FGN is equal to .[18]
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.