Relations To Other Methods
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 exponent are related by:
- and
- .
The relations can be derived using the Wiener–Khinchin theorem.
Thus, is tied to the slope of the power spectrum 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 .
For fractional Brownian motion (FBM), we have, and thus, and, where is the Hurst exponent. for FBM is equal to . In this context, FBM is the cumulative sum or the integral of FGN, thus, the exponents of their power spectra differ by 2.
Read more about this topic: Detrended Fluctuation Analysis
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