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. 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 x t {displaystyle x_{t}} of length N {displaystyle N} , where t ∈ N {displaystyle tin mathbb {N} } , integration or summation first converts this into an unbounded process X t {displaystyle X_{t}} : where ⟨ x ⟩ {displaystyle langle x angle } denotes the mean value of the time series. X t {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, X t {displaystyle X_{t}} is divided into time windows of length n {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 Y t {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: Finally, this process of detrending followed by fluctuation measurement is repeated over a range of different window sizes n {displaystyle n} , and a log-log graph of F ( n ) {displaystyle F(n)} against n {displaystyle n} is constructed. A straight line on this log-log graph indicates statistical self-affinity expressed as F ( n ) ∝ n α {displaystyle F(n)propto n^{alpha }} . The scaling exponent α {displaystyle alpha } is calculated as the slope of a straight line fit to the log-log graph of n {displaystyle n} against F ( n ) {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 N {displaystyle {sqrt {N}}} , an exponent of 1 2 {displaystyle { frac {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: 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 ( i = 1 {displaystyle i=1} ) are applied to the profile, thus it is called DFA1. To remove trends of higher order, DFA i {displaystyle i} , uses polynomial fits of order i {displaystyle i} . Due to the summation (integration) from x i {displaystyle x_{i}} to X t {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 x i {displaystyle x_{i}} . In general DFA of order i {displaystyle i} removes (polynomial) trends of order i − 1 {displaystyle i-1} . For linear trends in the mean of x i {displaystyle x_{i}} 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.