Extreme values of the cyclostationary Gaussian random process

1993 ◽  
Vol 30 (01) ◽  
pp. 82-97 ◽  
Author(s):  
D. G. Konstant ◽  
V.I. Piterbarg
Keyword(s):  
Gaussian Process ◽  
Random Process ◽  
Gaussian Processes ◽  
Extreme Values ◽  
Random Processes ◽  
Mean Square ◽  
Gaussian Random Process ◽  
Covariance Functions ◽  
Limit Formula ◽  
Image Position

In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula is established for and all x > 0.

Extreme values of the cyclostationary Gaussian random process

1993 ◽  
Vol 30 (1) ◽  
pp. 82-97 ◽  
Author(s):  
D. G. Konstant ◽  
V.I. Piterbarg
Keyword(s):  
Gaussian Process ◽  
Random Process ◽  
Gaussian Processes ◽  
Extreme Values ◽  
Random Processes ◽  
Mean Square ◽  
Gaussian Random Process ◽  
Covariance Functions ◽  
Limit Formula ◽  
Image Position

In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula is established for and all x > 0.

scholarly journals Alternative estimate of curve exceeding probability of sub-Gaussian random process

2020 ◽  
pp. 37-39
Author(s):  
O. Kollie ◽  
R. Yamnenko
Keyword(s):  
Stochastic Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Special Interest ◽  
Queuing Theory ◽  
Random Processes ◽  
Financial Mathematics ◽  
Continuous Linear ◽  
Gaussian Random Process ◽  
Overflow Probability

Investigation of sub-gaussian random processes are of special interest since obtained results can be applied to Gaussian processes. In this article the properties of trajectories of a sub-Gaussian process drifted by a curve a studied. The following functionals of extremal type from stochastic process are studied: $\sup_{t\in B}(X(t)-f(t))$, $\inf{t\in B}(X(t)-f(t))$ and $\sup_{t\in B}|X(t)-f(t)|$. An alternative estimate of exceeding by sub-Gaussian process a level, given by a continuous linear curve is obtained. The research is based on the results obtained in the work \cite{yamnenko_vasylyk_TSP_2007}. The results can be applied to such problems of queuing theory and financial mathematics as an estimation of buffer overflow probability and bankruptcy

Extreme Values of Waves and Ship Responses Subject to the Markov Chain Condition

1979 ◽  
Vol 23 (03) ◽  
pp. 188-197
Author(s):  
Michel K. Ochi
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Probability Distribution ◽  
Random Process ◽  
Probability Distribution Function ◽  
Extreme Values ◽  
Random Processes ◽  
Chain Condition ◽  
Extreme Value ◽  
Spectral Bandwidth

This paper discusses the effect of statistical dependence of the maxima (peak values) of a stationary random process on the magnitude of the extreme values. A theoretical analysis of the extreme values of a stationary normal random process is made, assuming the maxima are subject to the Markov chain condition. For this, the probability distribution function of maxima as well as the joint probability distribution function of two successive maxima of a normal process having an arbitrary spectral bandwidth are applied to Epstein's theorem for evaluating the extreme values in a given sample under the Markov chain condition. A numerical evaluation of the extreme values is then carried out for a total of 14 random processes, including nine ocean wave records, with various spectral bandwidth parameters ranging from 0.11 to 0.78. From the results of the computations, it is concluded that the Markov concept is applicable to the maxima of random processes whose spectral bandwidth parameter, ɛ, is less than 0.5, and that the extreme values with and without the Markov concept are constant irrespective of the e-value, and the former is approximately 10 percent greater than the latter. It is also found that the sample size for which the extreme value reaches a certain level with the Markov concept is much less than that without the Markov concept. For example, the extreme value will reach a level of 4.0 (nondimensional value) in 1100 observations of the maxima with the Markov concept, while the extreme value will reach the same level in 3200 observations of the maxima without the Markov concept.

scholarly journals Wavelet-based simulation of random processes from certain classes with given accuracy and reliability

2019 ◽  
Vol 25 (3) ◽  
pp. 217-225
Author(s):  
Ievgen Turchyn
Keyword(s):  
Stochastic Processes ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Random Processes

Abstract We consider stochastic processes {Y(t)} which can be represented as {Y(t)=(X(t))^{s}} , {s\in\mathbb{N}} , where {X(t)} is a stationary strictly sub-Gaussian process, and build a wavelet-based model that simulates {Y(t)} with given accuracy and reliability in {L_{p}([0,T])} . A model for simulation with given accuracy and reliability in {L_{p}([0,T])} is also built for processes {Z(t)} which can be represented as {Z(t)=X_{1}(t)X_{2}(t)} , where {X_{1}(t)} and {X_{2}(t)} are independent stationary strictly sub-Gaussian processes.

Extreme values and crossings for the X2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

1980 ◽  
Vol 12 (3) ◽  
pp. 746-774 ◽  
Author(s):  
Georg Lindgren
Keyword(s):  
Gaussian Processes ◽  
Survival Probability ◽  
Extreme Values ◽  
Extreme Points ◽  
Linear Functions ◽  
Degree Polynomial ◽  
Engineering Sciences ◽  
Multivariate Gaussian ◽  
Safe Region ◽  
Image Position

Extreme values of non-linear functions of multivariate Gaussian processes are of considerable interest in engineering sciences dealing with the safety of structures. One then seeks the survival probability where X(t) = (X1(t), …, Xn(t)) is a stationary, multivariate Gaussian load process, and S is a safe region. In general, the asymptotic survival probability for large T-values is the most interesting quantity.By considering the point process formed by the extreme points of the vector process X(t), and proving a general Poisson convergence theorem, we obtain the asymptotic survival probability for a large class of safe regions, including those defined by the level curves of any second- (or higher-) degree polynomial in (x1, …, xn). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, β = inf>x∉S ||x||, i.e. the smallest distance from the origin to an unsafe point.

Predictive Approaches for Choosing Hyperparameters in Gaussian Processes

2001 ◽  
Vol 13 (5) ◽  
pp. 1103-1118 ◽  
Author(s):  
S. Sundararajan ◽  
S. S. Keerthi
Keyword(s):  
Gaussian Processes ◽  
Regression Models ◽  
Cross Validation ◽  
Maximum A Posteriori ◽  
Mean Square ◽  
A Posteriori ◽  
Covariance Functions ◽  
Predictive Probability ◽  
Predictive Sample Reuse ◽  
Predictive Approaches

Gaussian processes are powerful regression models specified by parameterized mean and covariance functions. Standard approaches to choose these parameters (known by the name hyperparameters) are maximum likelihood and maximum a posteriori. In this article, we propose and investigate predictive approaches based on Geisser's predictive sample reuse (PSR) methodology and the related Stone's cross-validation (CV) methodology. More specifically, we derive results for Geisser's surrogate predictive probability (GPP), Geisser's predictive mean square error (GPE), and the standard CV error and make a comparative study. Within an approximation we arrive at the generalized cross-validation (GCV) and establish its relationship with the GPP and GPE approaches. These approaches are tested on a number of problems. Experimental results show that these approaches are strongly competitive with the existing approaches.

Spatial Prediction With Mobile Sensor Networks Using Gaussian Process Regression Based on Gaussian Markov Random Fields

2011 ◽  
Author(s):  
Yunfei Xu ◽  
Jongeun Choi
Keyword(s):  
Sensor Networks ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Markov Random Fields ◽  
Mobile Sensor Networks ◽  
Spatial Prediction ◽  
Prediction Algorithm ◽  
Mobile Sensor ◽  
Covariance Functions ◽  
Wide Range

In this paper, a new class of Gaussian processes is proposed for resource-constrained mobile sensor networks. Such a Gaussian process builds on a GMRF with respect to a proximity graph over a surveillance region. The main advantages of using this class of Gaussian processes over standard Gaussian processes defined by mean and covariance functions are its numerical efficiency and scalability due to its built-in GMRF and its capability of representing a wide range of non-stationary physical processes. The formulas for Bayesian posterior predictive statistics such as prediction mean and variance are derived and a sequential field prediction algorithm is provided for sequentially sampled observations. For a special case using compactly supported kernels, we propose a distributed algorithm to implement field prediction by correctly fusing all observations in Bayesian statistics. Simulation results illustrate the effectiveness of our approach.

Extreme values and crossings for the X 2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

1980 ◽  
Vol 12 (03) ◽  
pp. 746-774
Author(s):  
Georg Lindgren
Keyword(s):  
Gaussian Processes ◽  
Survival Probability ◽  
Extreme Values ◽  
Extreme Points ◽  
Linear Functions ◽  
Degree Polynomial ◽  
Engineering Sciences ◽  
Multivariate Gaussian ◽  
Safe Region ◽  
Image Position

Extreme values of non-linear functions of multivariate Gaussian processes are of considerable interest in engineering sciences dealing with the safety of structures. One then seeks the survival probability where X(t) = (X 1(t), …, X n (t)) is a stationary, multivariate Gaussian load process, and S is a safe region. In general, the asymptotic survival probability for large T-values is the most interesting quantity. By considering the point process formed by the extreme points of the vector process X(t), and proving a general Poisson convergence theorem, we obtain the asymptotic survival probability for a large class of safe regions, including those defined by the level curves of any second- (or higher-) degree polynomial in (x 1, …, x n ). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, β = inf>x∉S ||x||, i.e. the smallest distance from the origin to an unsafe point.

scholarly journals Duration Distribution and Up-crossings Rate of Generalized Hyperbolic Processes

2016 ◽  
Vol 36 (3) ◽  
Author(s):  
Moh’d T. Alodat ◽  
Khalid M. Aludaat
Keyword(s):  
Gaussian Process ◽  
Random Process ◽  
Alternative Model ◽  
Sea Surface ◽  
Surface Elevation ◽  
Gaussian Random Process ◽  
Duration Distribution ◽  
Non Gaussian

A Gaussian process is usually used to model the sea surface elevation in the oceanography. As the depth of the water decreases or the sea severity increases, the sea surface elevation departs from symmetry and Gaussianity. In this paper, a stationary non-Gaussian random process called the generalized hyperbolic process is used as an alternative model. The process generates a family of processes. We derive the rate of up-crossings for this process and the distribution of the height of the process. We also derive the duration distribution of an excursion for the generalized hyperbolic process.

scholarly journals Extremes of Gaussian processes with a smooth random trend

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2267-2279 ◽  
Author(s):  
Vladimir Piterbarg ◽  
Goran Popivoda ◽  
Sinisa Stamatovic
Keyword(s):  
Gaussian Process ◽  
Random Process ◽  
Gaussian Processes ◽  
Stationary Process ◽  
Exact Result ◽  
Extremes Of Gaussian Processes

Let ?(t), t ? R, be a Gaussian zero mean stationary process, and ?(t) another random process, smooth enough, being independent of ?(t). We will consider the process ?(t) + ?(t) such that conditioned on ?(t) it is a Gaussian process. We want to establish an asymptotic exact result for P (t?[o,T] sup (?(t) + ?(t)) > u), as u ? ?, where T > 0.

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