Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process
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 z...
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| Format: | Article |
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American Physical Society
2017
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| Online Access: | https://eprints.nottingham.ac.uk/49870/ |
| _version_ | 1848798098940231680 |
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| author | Wilson, Lorna R. M. Hopcraft, Keith I. |
| author_facet | Wilson, Lorna R. M. Hopcraft, Keith I. |
| author_sort | Wilson, Lorna R. M. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:14:23Z |
| format | Article |
| id | nottingham-49870 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:14:23Z |
| publishDate | 2017 |
| publisher | American Physical Society |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-498702020-05-04T19:23:09Z https://eprints.nottingham.ac.uk/49870/ 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. 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. American Physical Society 2017-12-18 Article PeerReviewed Wilson, Lorna R. M. and Hopcraft, Keith I. (2017) Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process. Physical Review E, 96 (6). 062129. ISSN 1079-7114 https://journals.aps.org/pre/abstract/10.1103/PhysRevE.96.062129 doi:10.1103/PhysRevE.96.062129 doi:10.1103/PhysRevE.96.062129 |
| spellingShingle | Wilson, Lorna R. M. Hopcraft, Keith I. Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process |
| title | Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process |
| title_full | Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process |
| title_fullStr | Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process |
| title_full_unstemmed | Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process |
| title_short | Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process |
| title_sort | periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a gaussian process |
| url | https://eprints.nottingham.ac.uk/49870/ https://eprints.nottingham.ac.uk/49870/ https://eprints.nottingham.ac.uk/49870/ |