Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process

Lorna R. M. Wilson, Keith I. Hopcraft

Research output: Contribution to journalArticle

3 Citations (Scopus)
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Abstract

The problem of zero crossings is of great historical prevalence and promises extensive application. The challenge is to establish precisely how the autocorrelation function or power spectrum of a one-dimensional continuous random process determines the density function of the intervals between the zero crossings of that process. This paper investigates the case where periodicities are incorporated into the autocorrelation function of a smooth process. Numerical simulations, and statistics about the number of crossings in a fixed interval, reveal that in this case the zero crossings segue between a random and deterministic point process depending on the relative time scales of the periodic and nonperiodic components of the autocorrelation function. By considering the Laplace transform of the density function, we show that incorporating correlation between successive intervals is essential to obtaining accurate results for the interval variance. The same method enables prediction of the density function tail in some regions, and we suggest approaches for extending this to cover all regions. In an ever-more complex world, the potential applications for this scale of regularity in a random process are far reaching and powerful.
Original languageEnglish
Pages (from-to)062129
JournalPhysical Review E
Volume96
Issue number6
DOIs
Publication statusPublished - 18 Dec 2017

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Zero-crossing
roots of equations
Extreme Values
Autocorrelation Function
Gaussian Process
Periodicity
autocorrelation
periodic variations
Density Function
Interval
Random process
intervals
random processes
Point Process
Power Spectrum
Laplace transform
Tail
Time Scales
Regularity
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Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process. / Wilson, Lorna R. M.; Hopcraft, Keith I.

In: Physical Review E, Vol. 96, No. 6, 18.12.2017, p. 062129.

Research output: Contribution to journalArticle

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