Detrended Fluctuation Analysis Task

See: Fluctuation Analysis, Stochastic Processes, Chaotic Dynamical Systems, Time Series Analysis, Self-Affinity, Correlation Time, Autocorrelation Function, 1/f Noise, Hurst Exponent, Stationary Process, Autocorrelation, Fourier Transform.

References

(Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/detrended_fluctuation_analysis Retrieved:2016-3-31.
- 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 2000 times as of 2013 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.

(Kantelhardt et al., 2002) ⇒ Jan W. Kantelhardt, Stephan A. Zschiegner, Eva Koscielny-Bunde, Shlomo Havlin, Armin Bunde, and H. Eugene Stanley. (2002). “Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series." Physica A: Statistical Mechanics and its Applications 316, no. 1
- ABSTRACT: We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.