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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 (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)
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)
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author
organization
publishing date
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
2019-11-13 05:22:43
@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},
  keyword      = {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          = {http://dx.doi.org/10.1214/18-STS662},
  volume       = {34},
  year         = {2019},
}