Effective persistency evaluation via exact excursion distributions for random processes and fields
Lindgren, Georg LU
; Podgórski, Krzysztof
LU
and Rychlik, Igor
LU
(2022)
In Journal of Physics Communications
6.
- Abstract
- Finding the probability that a stochastic system stays in a certain region of its state space over a specified time—a long-standing problem both in computational physics and in applied and theoretical mathematics—is approached through the extended and multivariate Rice formula. In principle, it applies to any smooth process multivariate both in argument and in value given that efficient numerical implementations of the high-dimensional integration are available. The computational method offers an exact integral representation yielding remarkably accurate results and provides an alternative method of computing persistency probability and exponent for a physical system. It can be viewed as an implementation of path integration for a smooth... (More)
- Finding the probability that a stochastic system stays in a certain region of its state space over a specified time—a long-standing problem both in computational physics and in applied and theoretical mathematics—is approached through the extended and multivariate Rice formula. In principle, it applies to any smooth process multivariate both in argument and in value given that efficient numerical implementations of the high-dimensional integration are available. The computational method offers an exact integral representation yielding remarkably accurate results and provides an alternative method of computing persistency probability and exponent for a physical system. It can be viewed as an implementation of path integration for a smooth Gaussian process with an arbitrary covariance. Its high accuracy is due to efficient computation of expectations with respect to high-dimensional nearly singular Gaussian distributions. For Gaussian processes, the computations are effective and more precise than those based on the Rice series expansions and the independent interval approximation. For the benchmark diffusion process, it produces the persistency exponent that is essentially the same as the recently obtained analytical value and surpasses accuracy, interpretability as well as control of the error, previous methods including the independent or Markovian approximation. The method solves the two-step excursion dependence for a stationary differentiable Gaussian process, in both theoretical and numerical sense. The solution is based on exact expressions for the probability density for one and two successive excursion lengths. The numerical routine RIND computes the densities using recent advances in scientific computing and is easily accessible for a general covariance function, via a simple numerical interface. The work offers also analytical results that explain the effectiveness of the implemented methodology and elaborates its utilization for non-Gaussian processes. (Less)
- Abstract (Swedish)
- Finding the probability that a stochastic system stays in a certain region of its state space over a specified time -- a long-standing problem both in computational physics and in applied and theoretical mathematics -- is approached through the extended and multivariate Rice formula.
In principle, it applies to any smooth process multivariate both in argument and in value given that efficient numerical implementations of the high-dimensional integration are available. The method offers an exact integral representation yielding remarkably accurate results and provides an alternative method of computing persistency probability and exponent for a physical system.
The computational methodology can be viewed as an implementation of... (More) - Finding the probability that a stochastic system stays in a certain region of its state space over a specified time -- a long-standing problem both in computational physics and in applied and theoretical mathematics -- is approached through the extended and multivariate Rice formula.
In principle, it applies to any smooth process multivariate both in argument and in value given that efficient numerical implementations of the high-dimensional integration are available. The method offers an exact integral representation yielding remarkably accurate results and provides an alternative method of computing persistency probability and exponent for a physical system.
The computational methodology can be viewed as an implementation of path integration for a smooth Gaussian process with an arbitrary covariance.
The high accuracy of the proposed method is due to efficient computation of expectations with respect to high-dimensional nearly singular Gaussian distributions.
For Gaussian processes, the computations are effective and more precise than those based on the Rice series expansions and the independent interval approximation.
For the benchmark diffusion process, it produces the persistency exponent that is essentially the same as the recently obtained analytical value and surpasses in accuracy, interpretability as well as control of the error, previous methods including the independent or Markovian excursion approximation.
The method solves the two-step excursion dependence for a general stationary differentiable Gaussian process, in both theoretical and practical numerical sense.
The solution is based on exact expressions for the probability density for one and two successive excursion lengths. The numerical routine {\sf RIND} computes the densities using recent advances in scientific computing and is easily accessible for a general covariance function, via a simple numerical interface.
The work offers also some analytical results that explain the effectiveness of the implemented methodology and elaborates how it can be utilized for non-Gaussian processes. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/7df77c6e-bb24-4d21-bc15-63d52562233f
- author
- Lindgren, Georg
LU
; Podgórski, Krzysztof
LU
and Rychlik, Igor
LU
- organization
- publishing date
- 2022-03-28
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- diffusion, Persistency exponent, Gaussian Process, Slepian models, diffusion, persistency exponent, Gaussian process, Slepian model
- in
- Journal of Physics Communications
- volume
- 6
- article number
- 035007
- pages
- 16 pages
- publisher
- IOP Publishing
- external identifiers
-
- scopus:85128347479
- ISSN
- 2399-6528
- DOI
- 10.1088/2399-6528/ac5e24
- language
- English
- LU publication?
- yes
- id
- 7df77c6e-bb24-4d21-bc15-63d52562233f
- date added to LUP
- 2022-04-06 10:34:36
- date last changed
- 2022-06-21 13:47:39
@article{7df77c6e-bb24-4d21-bc15-63d52562233f,
abstract = {{Finding the probability that a stochastic system stays in a certain region of its state space over a specified time—a long-standing problem both in computational physics and in applied and theoretical mathematics—is approached through the extended and multivariate Rice formula. In principle, it applies to any smooth process multivariate both in argument and in value given that efficient numerical implementations of the high-dimensional integration are available. The computational method offers an exact integral representation yielding remarkably accurate results and provides an alternative method of computing persistency probability and exponent for a physical system. It can be viewed as an implementation of path integration for a smooth Gaussian process with an arbitrary covariance. Its high accuracy is due to efficient computation of expectations with respect to high-dimensional nearly singular Gaussian distributions. For Gaussian processes, the computations are effective and more precise than those based on the Rice series expansions and the independent interval approximation. For the benchmark diffusion process, it produces the persistency exponent that is essentially the same as the recently obtained analytical value and surpasses accuracy, interpretability as well as control of the error, previous methods including the independent or Markovian approximation. The method solves the two-step excursion dependence for a stationary differentiable Gaussian process, in both theoretical and numerical sense. The solution is based on exact expressions for the probability density for one and two successive excursion lengths. The numerical routine RIND computes the densities using recent advances in scientific computing and is easily accessible for a general covariance function, via a simple numerical interface. The work offers also analytical results that explain the effectiveness of the implemented methodology and elaborates its utilization for non-Gaussian processes.}},
author = {{Lindgren, Georg and Podgórski, Krzysztof and Rychlik, Igor}},
issn = {{2399-6528}},
keywords = {{diffusion; Persistency exponent; Gaussian Process; Slepian models; diffusion; persistency exponent; Gaussian process; Slepian model}},
language = {{eng}},
month = {{03}},
publisher = {{IOP Publishing}},
series = {{Journal of Physics Communications}},
title = {{Effective persistency evaluation via exact excursion distributions for random processes and fields}},
url = {{http://dx.doi.org/10.1088/2399-6528/ac5e24}},
doi = {{10.1088/2399-6528/ac5e24}},
volume = {{6}},
year = {{2022}},
}