The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

1989 ◽  
Vol 21 (2) ◽  
pp. 315-333 ◽  
Author(s):  
H. E. Daniels
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Covariance Function ◽  
Brownian Bridge ◽  
Breaking Load ◽  
Plastic Yield ◽  
Analogous Process ◽  
The Mean

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t1(1 + t2), .

The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

1989 ◽  
Vol 21 (02) ◽  
pp. 315-333 ◽  
Author(s):  
H. E. Daniels
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Covariance Function ◽  
Brownian Bridge ◽  
Breaking Load ◽  
Plastic Yield ◽  
Analogous Process ◽  
The Mean

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t 1(1 + t 2), .

Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

2007 ◽  
Vol 44 (04) ◽  
pp. 1056-1067 ◽  
Author(s):  
Andreas Lindell ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Simple Random Walk ◽  
Marginal Distributions

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (2) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (02) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

scholarly journals Asymptotic distribution of sum and maximum for Gaussian processes

1999 ◽  
Vol 36 (4) ◽  
pp. 1031-1044 ◽  
Author(s):  
Hwai-Chung Ho ◽  
William P. McCormick
Keyword(s):  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Joint Distribution ◽  
Covariance Function ◽  
Random Variables ◽  
Regularity Conditions ◽  
Gaussian Sequence ◽  
Stationary Gaussian Sequence ◽  
Standard Normal

Let {Xn, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = EX0Xn. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

2007 ◽  
Vol 44 (4) ◽  
pp. 1056-1067
Author(s):  
Andreas Lindell ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Simple Random Walk ◽  
Marginal Distributions

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

Computation with Infinite Neural Networks

1998 ◽  
Vol 10 (5) ◽  
pp. 1203-1216 ◽  
Author(s):  
Christopher K. I. Williams
Keyword(s):  
Neural Networks ◽  
Gaussian Process ◽  
Infinite Number ◽  
Wide Class ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Bayesian Prediction ◽  
Infinite Networks

For neural networks with a wide class of weight priors, it can be shown that in the limit of an infinite number of hidden units, the prior over functions tends to a gaussian process. In this article, analytic forms are derived for the covariance function of the gaussian processes corresponding to networks with sigmoidal and gaussian hidden units. This allows predictions to be made efficiently using networks with an infinite number of hidden units and shows, somewhat paradoxically, that it may be easier to carry out Bayesian prediction with infinite networks rather than finite ones.

scholarly journals Stochastic Analysis of Gaussian Processes via Fredholm Representation

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Tommi Sottinen ◽  
Lauri Viitasaari
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Differential Equations ◽  
Gaussian Process ◽  
Stochastic Differential Equations ◽  
Gaussian Processes ◽  
Stochastic Analysis ◽  
Series Expansions ◽  
Transfer Principle ◽  
Maximum Likelihood Estimations

We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (02) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

scholarly journals Detecting periodicities with Gaussian processes

2016 ◽  
Vol 2 ◽  
pp. e50 ◽  
Author(s):  
Nicolas Durrande ◽  
James Hensman ◽  
Magnus Rattray ◽  
Neil D. Lawrence
Keyword(s):  
Hilbert Space ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Gaussian Process Regression ◽  
Inner Product ◽  
Input Output ◽  
Periodic Data

We consider the problem of detecting and quantifying the periodic component of a function given noise-corrupted observations of a limited number of input/output tuples. Our approach is based on Gaussian process regression, which provides a flexible non-parametric framework for modelling periodic data. We introduce a novel decomposition of the covariance function as the sum of periodic and aperiodic kernels. This decomposition allows for the creation of sub-models which capture the periodic nature of the signal and its complement. To quantify the periodicity of the signal, we derive a periodicity ratio which reflects the uncertainty in the fitted sub-models. Although the method can be applied to many kernels, we give a special emphasis to the Matérn family, from the expression of the reproducing kernel Hilbert space inner product to the implementation of the associated periodic kernels in a Gaussian process toolkit. The proposed method is illustrated by considering the detection of periodically expressed genes in thearabidopsisgenome.

Sign in / Sign up

Export Citation Format

Share Document