Extreme statistics for time series: Distribution of the maximum relative to the initial value

T. W. Burkhardt, G. Györgyi, N. R. Moloney, Z. Rácz

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1 fα power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for α=0 (iid variables), α=2 (random walk), α=4 (random acceleration), and α= (single sinusoidal mode). For other, intermediate values of α, the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all α. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for α>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

Original languageEnglish
Article number041119
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume76
Issue number4
DOIs
Publication statusPublished - Oct 12 2007

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Extremes
Time series
statistics
Statistics
Identically distributed
Statistics of Extremes
Boundary conditions
Scaling Theory
Invariant Distribution
Exact Distribution
boundary conditions
Exact Results
Limiting Distribution
Power Spectrum
time signals
Random walk
Divergence
Classify
random walk
Singularity

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Extreme statistics for time series : Distribution of the maximum relative to the initial value. / Burkhardt, T. W.; Györgyi, G.; Moloney, N. R.; Rácz, Z.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 76, No. 4, 041119, 12.10.2007.

Research output: Contribution to journalArticle

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