Gaussian integrals and Rice series in crossing distributions : to compute the distribution of maxima and other features of Gaussian processes
Lindgren, Georg LU
(2019)
In Statistical Science
34(1).
p.100-128
- Abstract
- We describe and compare how methods based on the classical Rice’s formula for the expected number, and higher moments, of level crossings by a Gaussian process stand up to contemporary numerical methods to accurately deal with crossing related characteristics of the sample paths.
We illustrate the relative merits in accuracy and computing time of the Rice moment methods and the exact numerical method, developed since the late 1990s, on three groups of distribution problems, the maximum over a finite interval and the waiting time to first crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.
We also treat the notoriously difficult problem of dependence between successive zero... (More) - We describe and compare how methods based on the classical Rice’s formula for the expected number, and higher moments, of level crossings by a Gaussian process stand up to contemporary numerical methods to accurately deal with crossing related characteristics of the sample paths.
We illustrate the relative merits in accuracy and computing time of the Rice moment methods and the exact numerical method, developed since the late 1990s, on three groups of distribution problems, the maximum over a finite interval and the waiting time to first crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.
We also treat the notoriously difficult problem of dependence between successive zero crossing distances. The exact solution has been known since at least 2000, but it has remained largely unnoticed outside the ocean science community.
Extensive simulation studies illustrate the accuracy of the numerical methods. As a historical introduction an attempt is made to illustrate the relation between Rice’s original formulation and arguments and the exact numerical methods.
(Less) - Abstract (Swedish)
- We describe and compare how methods based on the classical
Rice’s formula for the expected number, and higher moments, of level crossings
by a Gaussian process stand up to contemporary numerical methods to
accurately deal with crossing related characteristics of the sample paths.
We illustrate the relative merits in accuracy and computing time of the Rice
moment methods and the exact numerical method, developed since the late
1990s, on three groups of distribution problems, the maximum over a finite
interval and the waiting time to first crossing, the length of excursions over a
level, and the joint period/amplitude of oscillations.
We also treat the notoriously difficult problem of dependence... (More) - We describe and compare how methods based on the classical
Rice’s formula for the expected number, and higher moments, of level crossings
by a Gaussian process stand up to contemporary numerical methods to
accurately deal with crossing related characteristics of the sample paths.
We illustrate the relative merits in accuracy and computing time of the Rice
moment methods and the exact numerical method, developed since the late
1990s, on three groups of distribution problems, the maximum over a finite
interval and the waiting time to first crossing, the length of excursions over a
level, and the joint period/amplitude of oscillations.
We also treat the notoriously difficult problem of dependence between successive
zero crossing distances. The exact solution has been known since at
least 2000, but it has remained largely unnoticed outside the ocean science
community.
Extensive simulation studies illustrate the accuracy of the numerical methods.
As a historical introduction an attempt is made to illustrate the relation
between Rice’s original formulation and arguments and the exact numerical
methods. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/e4f7936e-1f0f-4223-a59e-17af53ca3b73
- author
- Lindgren, Georg
LU
- organization
- publishing date
- 2019
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- computational statistics, distribution of maximum, Durbin’s formula, excursion length distribution, first passage, independent interval assumption, level crossings, multivariate normal probabilities, pe- riod/amplitude distribution, Rice’s formula, RIND program, statistical computation, stochastic process, successive crossing distance distribution, truncated normal moments, Wafo toolbox
- in
- Statistical Science
- volume
- 34
- issue
- 1
- pages
- 100 - 128
- publisher
- IMS
- external identifiers
-
- scopus:85065490257
- ISSN
- 0883-4237
- DOI
- 10.1214/18-STS662
- language
- English
- LU publication?
- yes
- id
- e4f7936e-1f0f-4223-a59e-17af53ca3b73
- date added to LUP
- 2018-12-13 12:33:30
- date last changed
- 2022-04-02 05:01:54
@article{e4f7936e-1f0f-4223-a59e-17af53ca3b73,
abstract = {{We describe and compare how methods based on the classical Rice’s formula for the expected number, and higher moments, of level crossings by a Gaussian process stand up to contemporary numerical methods to accurately deal with crossing related characteristics of the sample paths.<br/><br/>We illustrate the relative merits in accuracy and computing time of the Rice moment methods and the exact numerical method, developed since the late 1990s, on three groups of distribution problems, the maximum over a finite interval and the waiting time to first crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.<br/><br/>We also treat the notoriously difficult problem of dependence between successive zero crossing distances. The exact solution has been known since at least 2000, but it has remained largely unnoticed outside the ocean science community.<br/><br/>Extensive simulation studies illustrate the accuracy of the numerical methods. As a historical introduction an attempt is made to illustrate the relation between Rice’s original formulation and arguments and the exact numerical methods.<br/>}},
author = {{Lindgren, Georg}},
issn = {{0883-4237}},
keywords = {{computational statistics; distribution of maximum; Durbin’s formula; excursion length distribution; first passage; independent interval assumption; level crossings; multivariate normal probabilities; pe- riod/amplitude distribution; Rice’s formula; RIND program; statistical computation; stochastic process; successive crossing distance distribution; truncated normal moments; Wafo toolbox}},
language = {{eng}},
number = {{1}},
pages = {{100--128}},
publisher = {{IMS}},
series = {{Statistical Science}},
title = {{Gaussian integrals and Rice series in crossing distributions : to compute the distribution of maxima and other features of Gaussian processes}},
url = {{https://lup.lub.lu.se/search/files/55464881/RiceSeries_r1.pdf}},
doi = {{10.1214/18-STS662}},
volume = {{34}},
year = {{2019}},
}