Order statistics of 1/fα signals

Area of Science:

• Statistical Physics
• Time Series Analysis
• Nonlinear Dynamics

Background:

• Understanding the statistical properties of noise is crucial in various scientific fields.
• Periodic Gaussian noise with a 1/f(α) power spectrum exhibits complex behavior relevant to signal processing and physical systems.

Purpose of the Study:

• To investigate the order statistics of periodic Gaussian noise with a 1/f(α) power spectrum.
• To identify and characterize different scaling regimes for the average gap between ordered signal values.
• To examine the spectra of average ordered values and their relationship to known physical models.

Main Methods:

• Utilizing computational simulations to analyze the noise data.
• Employing phenomenological arguments to derive scaling laws.
• Comparing derived exponents with known results for independent and identically distributed variables.

Main Results:

• Three distinct scaling regimes for the average gap d(k) were identified, dependent on the spectral exponent α.
• For 0 ≤ α < 1, the standard scaling d(k) ~ k⁻¹ holds.
• For 1 < α < 5, a novel α-dependent scaling d(k) ~ k^((α-3)/2) emerges.
• For α > 5, an α-independent scaling d(k) ~ k is observed.

Conclusions:

• The study reveals complex scaling behaviors in order statistics of 1/f(α) noise, differing significantly from standard assumptions.
• The findings suggest a connection between noise properties and phenomena in quantum mechanics, specifically particle behavior in power-law potentials.
• The derived scaling exponents provide valuable insights into the statistical nature of correlated noise signals.