Level-Crossing Problem of a Gaussian Process Having Gaussian Power Spectrum Density

Mimaki, Tadashi; Myoken, Hiroshi; Kawabata, Tsutomu

doi:10.1143/jjap.24.l278

Cited by 5 publications

(2 citation statements)

References 8 publications

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“…The tradition with experimental testing of the dependence assumptions, including the Markov assumption, was continued by Mimaki (1973) and co-workers, (Mimaki et al, 1981(Mimaki et al, , 1984(Mimaki et al, , 1985. Munakata (1997) listed solved and unsolved crossing problems, focusing on experimental evidence and practical application of the available traditional methods to noise in signals.…”

Section: The Problem and Some Of Its Early Historymentioning

confidence: 99%

Effective computations of joint excursion times for stationary Gaussian processes

Lindgren,

Podgorski,

Rychlik

2020

Preprint

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This work is to popularize the method of computing the distribution of the excursion times for a Gaussian process that involves extended and multivariate Rice's formula. The approach was used in numerical implementations of the high-dimensional integration routine and in earlier work it was shown that the computations are more effective and thus more precise than those based on Rice expansions.The joint distribution of successive excursion times is clearly related to the distribution of the number of level crossings, a problem that can be attacked via the Rice series expansion, based on the moments of the number of crossings. Another point of attack is the "Independent Interval Approximation" (IIA) intensively studied for the persistency of physical systems. It treats the lengths of successive crossing intervals as statistically independent. Under IIA, a renewal type argument leads to an expression that provides the approximate interval distribution via its Laplace transform.However, the independence is not valid in most typical situations. Even if it leads to acceptable results for the persistency exponent of the long excursion time distribution or some classes of processes, rigorous assessment of the approximation error is not readily available. Moreover, we show that the IIA approach cannot deliver properly defined probability distributions and thus the method is limited only to persistence studies.The ocean science community favours a third approach, in which a class of parametric marginal distributions, either fitted to excursion data or derived from a narrow band approximation, is extended by a copula technique to bivariate and higher order distributions.This paper presents an alternative approach that is both more general, more accurate and relatively unknown. It is based on exact expressions for the probability density for one and for two successive excursion lengths. The numerical routine RIND

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Section: The Problem and Some Of Its Early Historymentioning

confidence: 99%

Effective computations of joint excursion times for stationary Gaussian processes

Lindgren,

Podgorski,

Rychlik

2020

Preprint

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“…In fact, already proved that consecutive zero-crossing distances cannot be independent. We refer to Mimaki (1973) and (Mimaki et al, 1985) for early experiments on interval dependence and its relation to spectral width, from a Gaussian processes is acknowledged, Sire (2008).…”

Section: Joint Distribution Of Successive Zero-crossing Distancesmentioning

confidence: 99%

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

Lindgren¹

2019

Statist. Sci.

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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 compute the statistical distribution of the maximum over a xed interval, the length of excursions above a level, and the geometry of oscillations, and other 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 nite interval and the waiting time to rst crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.We also treat the notoriously dicult 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.MSC 2010 subject classications: Primary 60G17; secondary 60-08, 60G10, 60G70, 65C50.

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Improved microstructure, dielectric and ferroelectric properties of microwave-sintered Sr0.5Ba0.5Nb2O6

Patro

Kulkarni

Gupta

et al. 2007

Physica B: Condensed Matter

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Level-Crossing Problem of a Gaussian Process Having Gaussian Power Spectrum Density

Cited by 5 publications

References 8 publications

Effective computations of joint excursion times for stationary Gaussian processes

Effective computations of joint excursion times for stationary Gaussian processes

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

Improved microstructure, dielectric and ferroelectric properties of microwave-sintered Sr0.5Ba0.5Nb2O6

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