Level Sets and Extrema of Random Processes and Fields

Azaı̈s, Jean-Marc; Wschebor, Mario

doi:10.1002/9780470434642

Cited by 358 publications

(149 citation statements)

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“…In the sequel, we will need assumptions on the kernel function g and on the impulse β of the shot noise process X defined by (1). These assumptions are grouped together into the following condition denoted by (C):…”

Section: General Resultsmentioning

confidence: 99%

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A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

Biermé

Desolneux

2012

Journal of Applied Probability

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We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

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Section: General Resultsmentioning

confidence: 99%

“…In this paper, we will consider a shot noise process which is a real-valued random process given by (1) X(t) = i∈Z…”

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confidence: 99%

A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

Biermé

Desolneux

2012

Journal of Applied Probability

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“…Therefore, assume y has a covariance function ρ(s, t) that is differentiable in both arguments for 0 ≤ s ≤ t ≤ 1. Note that this is weaker than full differentiability of ρ, because it is not necessary that ρ be differentiable on the diagonal (for such processes, other methods are available for the computation of boundary crossing probabilities -see Azais and Wschebor (2009), for example).…”

Section: 1mentioning

confidence: 99%

A Comparison of Alternative Approaches to Supremum-Norm Goodness of Fit Tests with Estimated Parameters

Parker

2012

SSRN Journal

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Goodness of fit tests based on empirical processes have nonstandard limiting distributions when the null hypothesis is composite -that is, when parameters of the null model are estimated. Several solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985) can be applied to compute appropriate critical values for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact which has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin's approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981) through the score function of the parametric model. Simulation experiments suggest that these two testing strategies are roughly comparable to one another and more powerful than a simple bootstrap procedure.

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“…In this study, a new model (named as the hybrid model) for wind speed variability is proposed. The proposed model is used to estimate the frequencies of extreme winds, i.e., winds exceeding a fixed high threshold, by means of the so-called Rice's method, which was presented in details in, e.g., Azais and Wschebor (2009). Particular applications of this method were presented in, e.g., Baxevani and Rychlik (2006), Rychlik et al (2011).…”

Section: Introductionmentioning

confidence: 99%

Spatio-temporal modelling of wind speed variations and extremes in the Caribbean and the Gulf of Mexico

Rychlik

Mao

2018

Theor Appl Climatol

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The wind speed variability in the North Atlantic has been successfully modelled using a spatio-temporal transformed Gaussian field. However, this type of model does not correctly describe the extreme wind speeds attributed to tropical storms and hurricanes. In this study, the transformed Gaussian model is further developed to include the occurrence of severe storms. In this new model, random components are added to the transformed Gaussian field to model rare events with extreme wind speeds. The resulting random field is locally stationary and homogeneous. The localized dependence structure is described by time-and space-dependent parameters. The parameters have a natural physical interpretation. To exemplify its application, the model is fitted to the ECMWF ERA-Interim reanalysis data set. The model is applied to compute longterm wind speed distributions and return values, e.g., 100-or 1000-year extreme wind speeds, and to simulate random wind speed time series at a fixed location or spatio-temporal wind fields around that location.

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Level Sets and Extrema of Random Processes and Fields

Cited by 358 publications

References 0 publications

A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

A Comparison of Alternative Approaches to Supremum-Norm Goodness of Fit Tests with Estimated Parameters

Spatio-temporal modelling of wind speed variations and extremes in the Caribbean and the Gulf of Mexico

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