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Maximal height statistics for 1/f(alpha) signals.

G Györgyi1, N R Moloney, K Ozogány

  • 1Institute for Theoretical Physics - HAS Research Groups, Eötvös University, Pázmány sétány 1/a, 1117 Budapest, Hungary. gyorgyi@glu.elte.hu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 16, 2007
PubMed
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This study analyzes the height distribution of 1/f(alpha) signals. For strong correlations (alpha>1), the distribution rapidly approaches a limit, unlike decaying correlations (0<=alpha<1).

Area of Science:

  • Statistical Physics
  • Signal Processing
  • Complex Systems

Background:

  • Characterizing the statistical properties of signals with power-law spectra (1/f(alpha)) is crucial in various scientific fields.
  • Understanding the maximal relative height distribution of such signals, particularly one-dimensional interfaces, provides insights into their underlying dynamics and correlations.
  • The Fisher-Tippett-Gumbel distribution is a known limiting distribution for extreme values, but its applicability and convergence rate for 1/f(alpha) signals require investigation.

Purpose of the Study:

  • To investigate the maximal relative height distribution of stationary periodic Gaussian signals with a 1/f(alpha) power spectrum.
  • To analyze the influence of the spectral exponent (alpha) on the convergence rate to the limiting distribution.
  • To develop and validate analytical methods for describing this distribution across different correlation regimes.

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Main Methods:

  • Numerical simulations were employed to compute the height distribution for various alpha values.
  • Analytical techniques, including perturbation expansion and path integral methods, were used to derive theoretical results.
  • A trace formula was derived for even integer alpha, enabling extraction of asymptotic behavior.

Main Results:

  • For decaying correlations (0<=alpha<1), the convergence to the Fisher-Tippett-Gumbel distribution is slow with increasing sample size.
  • For strong correlations (alpha>1), convergence is rapid, allowing for accurate numerical construction of distribution functions.
  • Analytical results for large alpha and asymptotic behavior were derived and found to agree with numerical simulations.

Conclusions:

  • The correlation strength (alpha) significantly impacts the convergence rate of the maximal relative height distribution.
  • Analytical methods provide accurate descriptions of the distribution, especially in the strong correlation regime and for asymptotic cases.
  • Similarities in distribution function asymptotes suggest underlying commonalities in extreme and roughness statistics of these interfaces.