Calculation
Given a bounded time series, integration or summation first converts this into an unbounded process :
is called cumulative sum or profile. This process converts, for example, an i.i.d. white noise process into a random walk.
Next, is divided into time windows of length samples, and a local least squares straight-line fit (the local trend) is calculated by minimising the squared error with respect to the slope and intercept parameters :
Trends of higher order, can be removed by higher order DFA, where the linear function is replaced by a polynomial of order . Next, the root-mean-square deviation from the trend, the fluctuation, is calculated over every window at every time scale:
This detrending followed by fluctuation measurement process is repeated over the whole signal at a range of different window sizes, and a log-log graph of against is constructed.
A straight line on this log-log graph indicates statistical self-affinity expressed as . The scaling exponent is calculated as the slope of a straight line fit to the log-log graph of against using least-squares. This exponent is a generalization of the Hurst exponent. Because the expected displacement in an uncorrelated random walk of length L grows like, an exponent of would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is Fractional Brownian motion, with the precise value giving information about the series self-correlations:
- : anti-correlated
- : uncorrelated, white noise
- : correlated
- : 1/f-noise, pink noise
- : non-stationary, random walk like, unbounded
- : Brownian noise
There are different orders of DFA. In the described case, linear fits are applied to the profile, thus it is called DFA1. In general, DFA, uses polynomial fits of order . Due to the summation (integration) from to, 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 . In general DFA of order removes (polynomial) trends of order . For linear trends in the mean of 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.
Since in the fluctuation function the square(root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means . The multifractal generalization (MF-DFA) uses a variable moment and provides . Kantelhardt et al. intended this scaling exponent as a generalization of the classical Hurst exponent. The classical Hurst exponent corresponds to the second moment for stationary cases and to the second moment minus 1 for nonstationary cases .
Read more about this topic: Detrended Fluctuation Analysis
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