Gaussian Vector Process Research Articles

Simulation algorithms for stationary sequences with distributions in the form of a mixture of Gaussian distributions

Abstract In this paper, we present three algorithms for simulation of intervals of stationary vector and scalar sequences with partial distributions of their subsequences of fixed length in the form of a mixture of Gaussian distributions. The first algorithm is based on superposition of two Gaussian vector processes and the second and third ones use the method of conditional distributions and the method of superpositions to simulate the mixtures and to select realizations for approximate construction of conditional realizations. Some properties of these algorithms are presented.

Extreme value analysis of high-dimensional Gaussian vector processes

The analysis of the extreme value pertaining to first-passage failure is a key consideration for engineering systems undergoing random vibration. Most of the literature focus on a univariate stochastic process, however for system reliability analysis, the extreme value of multiple dependent processes is required. Due to the problem complexity, analytical methods are scarce, and they are limited to low-dimensional problems (dimension refers to the number of component processes). This paper presents an analytical method for extreme analysis of multivariate stationary Gaussian processes. The method can efficiently solve high-dimensional problems, thus it has diverse applications such as analysing dynamic systems with many degrees-of-freedom, or evaluating space-time extremes of random fields by discretizing the spatial domain. A maximum process is defined, representing the maxima of the components at each time instant. The exact upcrossing rate of the maximum process is derived, and the Poisson approximation is invoked to estimate the extreme value distribution. The method is fast as it involves multiple one-dimensional integrals, which can be computed using Gaussian quadrature. Moreover, an accurate approximation is proposed that obviates the need for numerical integration. The proposed method is implemented on two examples, specifically a dynamic system and spreading ocean waves. Monte Carlo simulations, which are computationally demanding, are performed to verify the accuracy and correctness of the proposed method.

High Excursions of a Quadratic form for a Gaussian Stationary Vector Process

Exact asymptotic behavior is given for high excursion probabilities of a quadratic form for a zero-mean Gaussian stationary vector process with Pickands’ type covariance matrix in the vicinity of zero. The case of a quadratic form with a positive maximum eigenvalue of order 1 is considered.

High excursions of Bessel and related random processes

Asymptotic behavior of large excursions probabilities is evaluated for Euclidean norm of a wide class of Gaussian non-stationary vector processes with independent identically distributed components. It is assumed that the components have means zero and variances reaching its absolute maximum at only one point of the considered time interval. The Bessel process is an important example of such processes.

A modified spectral representation method to simulate non-Gaussian random vector process considering wave-passage effect

Wave-passage effect is always of great importance when performing the dynamic analysis of lager scale structures. This effect can be easily considered in the simulation of Gaussian vector process, but there is not an efficient way to consider it in simulating non-Gaussian vector process. In this paper, a method is proposed to generate stationary non-Gaussian vector process considering wave-passage effect. In the simulation of vector process, wave-passage effect leads to a complex-defined cross spectral density matrix (CSDM). To separate the phase part from the CSDM, an improved form of spectral representation method (SRM) is proposed, in which the phase part representing the wave-passage effect can be directly expressed in an explicit form in the simulation formula. Then the characteristic of phase part representing the wave-passage effect during the nonlinear translation (translating underlying Gaussian vector process to the prescribed non-Gaussian vector process) is investigated. Interestingly, it is found that, the phase part is not changed. Based on this good feature, the modified SRM and the iterative scheme can be combined to simulate non-Gaussian vector process considering wave-passage effect. Two examples, involving the simulation of seismic ground motions and wind velocity time histories, are presented to show the characteristics of the simulated vector process and to verify the capabilities of the proposed method.

High excursions of Bessel process and other processes of Bessel type

A high excursion probability for the modulus of a Gaussian vector process with independent identically distributed components is evaluated. It is assumed that the components have means zero and variances reaching its absolute maximum at a single point of the considered time interval. An important example of such processes is the Bessel process.

High Excursions of Bessel Process and Other Processes of Bessel Type

A high excursion probability for the modulus of a Gaussian vector process with independent identically distributed components is evaluated. It is assumed that the components have means zero and variances reaching its absolute maximum at a single point of the considered time interval. An important example of such processes is the Bessel process.

Extremes of Gaussian chaos processes with trend

Let X ( t ) = ( X 1 ( t ) , … , X d ( t ) ) , t ∈ [ 0 , S ] be a Gaussian vector process and let g ( x ) , x ∈ R d be a continuous homogeneous function. We are concerned with the exact tail asymptotic of the chaos process g ( X ( t ) ) , t ∈ [ 0 , S ] with a trend function h ( t ) . Both scenarios X ( t ) is locally-stationary and X ( t ) is non-stationary are considered. Important examples include the product of Gaussian processes and chi-processes.

Method of Moments for Exit Probabilities of Gaussian Vector Processes From a Large Region

Asymptotic behavior of the exit probability from homothetically unboundedly expanding region is evaluated for twice continuously differentiable stationary Gaussian vector processes. The evaluation ...

Method of moments for exit probabilities of Gaussian vector processes from a large region

Для гауссовского дважды непрерывно дифференцируемого стационарного векторного процесса, заданного на конечном временном интервале, найдена асимптотика вероятности попадания его траекторий в бесконечно гомотетически удаляющуюся область. Для нахождения этой асимптотики применен метод моментов Райса: вводится точечный процесс входов траекторий в эту область и с помощью метода Лапласа изучается асимптотика среднего числа входов и оценивается второй факториальный момент этого числа.

Error Analysis of Spatially Varying Seismic Ground Motion Simulation by Spectral Representation Method

AbstractThis paper investigates statistical errors in the simulation of spatially varying seismic ground motions modeled by evolutionary Gaussian vector processes and simulated by the spectral repr...

Large extremes of Gaussian chaos processes

We study probabilities of large extremes of Gaussian chaos processes, that is, homogeneous functions of Gaussian vector processes. Important examples are products of Gaussian processes and quadratic forms of them. Exact asymptotic behaviors of the probabilities are found. To this aim, we use joint results of E. Hashorva, D. Korshunov and the author on Gaussian chaos, as well as a substantially modified asymptotical Double Sum Method.

High extrema of Gaussian chaos processes

Let ξ(t)=(ξ1(t),…,ξd(t)) be a Gaussian stationary vector process. Let \(g:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}\) be a homogeneous function. We study probabilities of large extrema of the Gaussian chaos process g(ξ(t)). Important examples include \(g(\mathbf {\boldsymbol {\xi }}(t))={\prod }_{i=1}^{d}\xi _{i}(t)\) and \(g(\mathbf {\boldsymbol {\xi }}(t))={\sum }_{i=1}^{d}a_{i}{\xi _{i}^{2}}(t)\). We review existing results partially obtained in collaboration with E. Hashorva, D. Korshunov, and A. Zhdanov. We also present the principal methods of our investigations which are the Laplace asymptotic method and other asymptotic methods for probabilities of high excursions of Gaussian vector process’ trajectories.

Piterbarg's max-discretization theorem for stationary vector Gaussian processes observed on different grids

In this paper, we derive Piterbarg's max-discretization theorem for two different grids considering centred stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over and over a grid . In this paper, we extend the recent findings by considering additionally the maximum over another grid . We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting . As a by-product we find that the joint limiting distribution of the maximum over different grids, which we refer to as the Piterbarg distribution, is in the case of weakly dependent Gaussian processes a max-stable distribution.

On probability of high extremes for product of two independent Gaussian stationary processes

Let X(t), Y(t), t≥0, be two independent zero-mean stationary Gaussian processes, whose covariance functions are such that ri(t)=1−|t|ai+o(|t|ai) as \(t\rightarrow 0\), with 0 u)\) as \(u\rightarrow \infty \). We discuss possibilities generalizing obtained results to other Gaussian chaos processes h(X(t)), with a Gaussian vector process X(t) and a homogeneous function h of positive order.

Likelihood ratio test for a piecewise continuous Weibull model with an unknown change point

In this paper, we consider the likelihood ratio test for the scale and shape parameters in a piecewise continuous Weibull model with an unknown change point. Under the null hypothesis of no change in scale and shape parameters, we derive that the likelihood ratio process converges weakly to the squared Euclidian norm of a weighted mean zero Gaussian vector process.

Conditional Simulation of Nonstationary Fluctuating Wind Speeds for Long-Span Bridges

This paper presents a computationally efficient method for the conditional simulation of nonstationary fluctuating wind speeds for the buffeting analysis of long-span bridges. The fluctuating wind speeds, with part of their time histories measured at some points, are regarded as a zero-mean nonstationary Gaussian vector process characterized by an evolutionary power spectral density (EPSD) matrix. The Kriging method is then applied to the random vector of the Fourier coefficients of the evolutionary vector process for conditional simulation. A mathematical proof of this conditional simulation is provided. A fast algorithm of the conditional simulation is further derived, by converting the Cholesky decomposition of a large size covariance matrix of the Fourier coefficients into the Cholesky decomposition of a series of small-size coherence matrices with much less computational time. The procedure for implementing the fast algorithm-based conditional simulation method is detailed. Finally, this method is applied for the conditional simulation of typhoon-induced nonstationary wind speed time histories for the buffeting analysis of Stonecutters Bridge in Hong Kong. The time domain buffeting responses of the bridge obtained using the conditionally simulated wind speed time histories indicate that the proposed simulation method is feasible and applicable.

ARMA Identification of Graphical Models

Consider a Gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components. Such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence. This leads to a sparsity pattern in the inverse of the matrix-valued spectral density. Such graphical models find applications in speech, bioinformatics, image processing, econometrics and many other fields, where the problem to fit an autoregressive (AR) model to such a process has been considered. In this paper we take this problem one step further, namely to fit an autoregressive moving-average (ARMA) model to the same data. We develop a theoretical framework and an optimization procedure which also spreads further light on previous approaches and results. This procedure is then applied to the identification problem of estimating the ARMA parameters as well as the topology of the graph from statistical data.

Cox limit theorem for large excursions of a norm of a Gaussian vector process

Cox limit theorem for a number of large excursions ( a -points) of a norm of a Gaussian vector process with stationary, centered and i.i.d. components is established.

Limit Theorem for High Levela-Upcrossings by $\chi$-Process

Previous article Next article Limit Theorem for High Level a-Upcrossings by $\chi$-ProcessV. Piterbarg and S. StamatovicV. Piterbarg and S. Stamatovichttps://doi.org/10.1137/S0040585X97980786PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis paper proves the Poisson limit theorem for a number of large excursions of the modulus of the Gaussian vector process with independent identically distributed increments.[1] J. Adler, On excursion sets, tube formulas and maxima of random fields, Ann. Appl. Probab., 10 (2000), pp. 1–74. 9lb ZZZZZZ 1050-5164 Ann. Appl. Probab. CrossrefGoogle Scholar[2] Google Scholar[3] R. Illsley, The excursions of a stationary Gaussian process outside a large two‐dimensional region, Adv. in Appl. Probab., 33 (2001), pp. 141–159. aaz AAPBBD 0001-8678 Adv. Appl. Probab. CrossrefGoogle Scholar[4] Google Scholar[5] Google Scholar[6] Google Scholar[7] Google ScholarKeywordsGaussian processlarge excursion$\chi$-square process Previous article Next article FiguresRelatedReferencesCited ByDetails Almost sure central limit theorems for the maxima of Gaussian functionsCommunications in Statistics - Theory and Methods, Vol. 51, No. 20 | 11 January 2021 Cross Ref High Excursions of a Quadratic form for a Gaussian Stationary Vector ProcessJournal of Mathematical Sciences, Vol. 262, No. 4 | 24 May 2022 Cross Ref High excursions of Bessel and related random processesStochastic Processes and their Applications, Vol. 130, No. 8 | 1 Aug 2020 Cross Ref Method of Moments for Exit Probabilities of Gaussian Vector Processes From a Large RegionA. O. Kleban and V. I. PiterbargTheory of Probability & Its Applications, Vol. 63, No. 4 | 5 February 2019AbstractPDF (200 KB)Большие выбросы процессов гауссовского хаоса. Аппроксимация в дискретном времениТеория вероятностей и ее применения, Vol. 63, No. 1 | 1 Jan 2018 Cross Ref Теория вероятностей и ее применения, Vol. 63, No. 4 | 2018 Cross Ref High Extremes of Gaussian Chaos Processes: A Discrete Time Approximation ApproachA. I. Zhdanov and V. I. PiterbargTheory of Probability & Its Applications, Vol. 63, No. 1 | 24 October 2018AbstractPDF (311 KB)High extrema of Gaussian chaos processesExtremes, Vol. 19, No. 2 | 2 February 2016 Cross Ref Large extremes of Gaussian chaos processesDoklady Mathematics, Vol. 93, No. 2 | 20 May 2016 Cross Ref On Probability of High Extremes for Product of Two Gaussian Stationary ProcessesTheory of Probability & Its Applications, Vol. 60, No. 3 | 15 September 2016AbstractPDF (165 KB)On probability of high extremes for product of two independent Gaussian stationary processesExtremes, Vol. 18, No. 1 | 31 August 2014 Cross Ref Limit laws for the maxima of stationary chi-processes under random indexTEST, Vol. 23, No. 4 | 8 April 2014 Cross Ref Exact asymptotics and limit theorems for supremum of stationary χ -processes over a random intervalStochastic Processes and their Applications, Vol. 123, No. 8 | 1 Aug 2013 Cross Ref Limit theorems for extremes of strongly dependent cyclo-stationary χ-processesExtremes, Vol. 16, No. 2 | 9 February 2013 Cross Ref Cox limit theorem for large excursions of a norm of a Gaussian vector processStatistics & Probability Letters, Vol. 80, No. 19-20 | 1 Oct 2010 Cross Ref Volume 48, Issue 4| 2004Theory of Probability & Its Applications575-752 History Published online:25 July 2006 InformationCopyright © 2004 Society for Industrial and Applied MathematicsKeywordsGaussian processlarge excursion$\chi$-square processPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X97980786Article page range:pp. 734-741ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics