Zero-crossing intervals of Gaussian processes

Rainal, A. J.

doi:10.1109/tit.1962.1057783

Cited by 24 publications

(6 citation statements)

References 14 publications

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“…We conclude this part by providing an insight into the Markovian approximation (MA) for zero crossing intervals, which dates back to McFadden [19] and Rainal [20,21], who also tested the model using the sequence of correlations. Having the exact joint distribution of successive zero crossing intervals a natural next step is to test the Markov chain dependence of the full sequence of crossing intervals.…”

Section: The Joint Crossing Distribution and Markovian Approximationmentioning

confidence: 91%

“…The original Rice series was improved considerably by Longuet-Higgins who obtained a rapidly converging moment series for the probability density of zero-crossing intervals [15,16] and also compared approximations based on the initial terms in the series with experimental results [17] and with earlier alternative series, suggested by McFadden [18,19]. Some studies by McFadden and Rainal [19][20][21] are of particular interest for the present article since they present systematic theoretical as well as experimental studies of the dependence between successive crossing intervals. Three approximations were studied, independence, 'quasi'-independence, which assumes that the sum of two successive intervals is independent of the next one, and the Markov type dependence.…”

Section: The Problem and Some Of Its Early Historymentioning

confidence: 99%

“…We consider a Markov process with transition probabilities given by equation (20) and use it to obtain an approximation for the whole crossing sequence. A natural, exact, and rather strong test of the model can be based on the trivariate distribution of three consecutive intervals.…”

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confidence: 99%

“…We illustrate the technique on the WH3, WH6, WH8 processes, as examples of processes with 'almost independent', 'moderately dependent', and 'strongly dependent' successive intervals. We use the RIND applications cov2ttpdf and cov2tttpdf to compute the 2D and 3D densities in equations (20) and (21) and then compute and plot the left-hand sides conditional densities for selected values of x j and, for each x j , for different x i . Figure 6 shows the results.…”

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confidence: 99%

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Effective persistency evaluation via exact excursion distributions for random processes and fields

2022

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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 the 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 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.

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Section: The Joint Crossing Distribution and Markovian Approximationmentioning

confidence: 91%

Section: The Problem and Some Of Its Early Historymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Effective persistency evaluation via exact excursion distributions for random processes and fields

2022

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“…The studies by McFadden (1958) and Rainal (1962), with more details in (Rainal, 1963), are of particular interest for the present article, since they contain systematic theoretical as well as experimental studies of the dependence between successive crossing intervals. Three approximation candidates were studied, independence, "quasi"-independence, which i.a.…”

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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2014

Transport and Coherent Structures in Wall Turbulence

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Zero-crossing intervals of Gaussian processes

Cited by 24 publications

References 14 publications

Effective persistency evaluation via exact excursion distributions for random processes and fields

Effective persistency evaluation via exact excursion distributions for random processes and fields

Effective computations of joint excursion times for stationary Gaussian processes

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