A class of almost nowhere differentiable stationary Gaussian processes which are somewhere differentiable

1973 ◽  
Vol 10 (3) ◽  
pp. 682-684 ◽  
Author(s):  
P. L. Davies
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Counter Example ◽  
Sample Paths ◽  
Stationary Gaussian Processes ◽  
Almost All ◽  
Set Of Points ◽  
The Continuum

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.

A class of almost nowhere differentiable stationary Gaussian processes which are somewhere differentiable

1973 ◽  
Vol 10 (03) ◽  
pp. 682-684
Author(s):  
P. L. Davies
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Counter Example ◽  
Sample Paths ◽  
Stationary Gaussian Processes ◽  
Almost All ◽  
Set Of Points ◽  
The Continuum

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.

Occupation times of stationary gaussian processes

1970 ◽  
Vol 7 (03) ◽  
pp. 721-733
Author(s):  
Simeon M. Berman
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Occupation Times ◽  
Unit Variance ◽  
Stationary Gaussian Processes ◽  
Image Position

Let X(t), t ≧ 0, be a stationary Gaussian process with zero mean, unit variance and continuous covariance function r(t). Suppose that, for some ε > 0 so that there is a version of the process whose sample functions are continuous [1].

scholarly journals On the Continuity of Stationary Gaussian Processes

1969 ◽  
Vol 34 ◽  
pp. 89-104 ◽  
Author(s):  
Makiko Nisio
Keyword(s):  
Fourier Series ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stationary Gaussian Processes ◽  
Necessary And Sufficient ◽  
Special Case

Let us consider a stochastically continuous, separable and measurable stationary Gaussian process X = {X(t), − ∞ < t < ∞} with mean zero and with the covariance function p(t) = EX(t + s)X(s). The conditions for continuity of paths have been studied by many authors from various viewpoints. For example, Dudley [3] studied from the viewpoint of ε-entropy and Kahane [5] showed the necessary and sufficient condition in some special case, using the rather neat method of Fourier series.

Occupation times of stationary gaussian processes

1970 ◽  
Vol 7 (3) ◽  
pp. 721-733 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Occupation Times ◽  
Unit Variance ◽  
Stationary Gaussian Processes ◽  
Image Position

Let X(t), t ≧ 0, be a stationary Gaussian process with zero mean, unit variance and continuous covariance function r(t). Suppose that, for some ε > 0 so that there is a version of the process whose sample functions are continuous [1].

scholarly journals Exponential Concentration for Zeroes of Stationary Gaussian Processes

2018 ◽  
Vol 2020 (23) ◽  
pp. 9769-9796
Author(s):  
Riddhipratim Basu ◽  
Amir Dembo ◽  
Naomi Feldheim ◽  
Ofer Zeitouni
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Compact Support ◽  
Spectral Measure ◽  
Mean Value ◽  
Stationary Gaussian Process ◽  
Exponential Moments ◽  
Additional Regularity ◽  
Stationary Gaussian Processes ◽  
Exponentially Small

Abstract We show that for any centered stationary Gaussian process of absolutely integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $\eta T$ of its mean value, up to an exponentially small in $T$ probability.

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.

Wave-length and amplitude in Gaussian noise

1972 ◽  
Vol 4 (1) ◽  
pp. 81-108 ◽  
Author(s):  
Georg Lindgren
Keyword(s):  
Gaussian Process ◽  
Density Function ◽  
Gaussian Noise ◽  
Covariance Function ◽  
Wave Length ◽  
Local Maximum ◽  
Stationary Gaussian Process ◽  
Numerical Examples ◽  
Vertical Distance

We give moment approximations to the density function of the wavelength, i. e., the time between “a randomly chosen” local maximum with height u and the following minimum in a stationary Gaussian process with a given covariance function. For certain processes we give similar approximations to the distribution of the amplitude, i. e., the vertical distance between the maximum and the minimum. Numerical examples and diagrams illustrate the results.

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.

An estimate for the tails of the distribution of the supremum for a class of stationary multiparameter Gaussian processes

1981 ◽  
Vol 18 (02) ◽  
pp. 536-541 ◽  
Author(s):  
E. M. Cabaña ◽  
M. Wschebor
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Stationary Gaussian Process ◽  
Parameter Domain ◽  
Standard Of Comparison

Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.

scholarly journals Detecting periodicities with Gaussian processes

2016 ◽  
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 the arabidopsis genome.

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