Detrended fluctuation analysis (DFA)

Detrended fluctuation analysis (DFA) is a scaling analysis method proposed to detect long-range power-law correlations in signals. This variability analysis is useful in developing an intuition for what represents the intrinsic nature of the system dynamics.
DFA was originally introduced in 1994 by Peng et al. [1].
This approach, applied in many processes (DNA sequence, physiological signals (heart rate, walking, breathing and blood pressure), physics data and geography time series), addresses correlations in signals which are non-stationary.
This scaling analysis was further popularized by Hausdorff et al. [2] to evaluate the dynamics of human gait and was specifically used to quantify stride-to-stride fluctuations.
DFA quantifies the scale-free fluctuations in time series by a scaling exponent (α). This scaling component may identify alternations of the system’s behavior. e.g., α for heart-rate interval is different in healthy individual and patient with heart failure [3].
The scale-free characteristics, which are a property of fractal correlation, have self-affinity and self-similarity features. Self-similarity refers to an object or a signal whose parts are similar to the whole one. Self affinity is a feature of the self similarity so that the similarity parts have to be rescaled. In the statistical essence, a rescaled part of the objects or time series has the same statistical distribution.
α, an emergent property of a system, may reveal that any event in time series has self-correlations or it is random:

  • α<1/2: anti-correlated
  • α~1/2: uncorrelated, white noise
  • α>1/2: correlated
  • α~1: 1/f-noise, pink noise
  • α>1: non-stationary, unbounded
  • α~3/2: Brownian noise

In our lab, we try to understand fractal behaviors in time series recorded from human activities to find possible answers to rehabilitation problems.


  1. Peng CK, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic organization of DNA nucleotides. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1994 Feb; 49(2):1685-9.
  2. Hausdorff JM, Purdon PL, Peng CK, Ladin Z, Wei JY, Goldberger AL. Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations. J Appl Physiol. 1996 May; 80(5):1448-57.
  3. Goldberger AL, Amaral LA, Hausdorff JM, Ivanov P, Peng CK, Stanley HE. Fractal dynamics in physiology: alterations with disease and aging. Proc Natl Acad Sci U S A. 2002 Feb 19; 99 Suppl 1:2466-72.