White noise and Gaussian random fields

2015 ◽  
Author(s):  
Takeyuki Hida
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Gaussian Random Fields

Multilevel approximation of Gaussian random fields: Fast simulation

2019 ◽  
Vol 30 (01) ◽  
pp. 181-223 ◽  
Author(s):  
Lukas Herrmann ◽  
Kristin Kirchner ◽  
Christoph Schwab
Keyword(s):  
White Noise ◽  
Fractional Order ◽  
Random Fields ◽  
Computational Cost ◽  
Gaussian Random Fields ◽  
Fast Simulation ◽  
Covariance Operators ◽  
Multilevel Approximation ◽  
Sinc Quadrature ◽  
Consistency Error

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

On the comparison of the distribution of the supremum of random fields represented by stochastic integrals

1989 ◽  
Vol 21 (4) ◽  
pp. 770-780 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Bassan
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Upper Bounds ◽  
Gaussian Random Fields ◽  
Stochastic Integrals ◽  
Response Functions ◽  
Noise Field ◽  
Fixed Radius ◽  
Different Response

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = ∫CpG(P, P′) dW(P′) and U(P) = ∫CpdW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions.The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P).The comparison of the suprema is carried out also, when CP = ℝ2, between fields with different response functions.

scholarly journals White noise approach to Gaussian random fields

1990 ◽  
Vol 119 ◽  
pp. 93-106 ◽  
Author(s):  
Ke-Seung Lee
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Integrable Function ◽  
Smooth Boundary ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Main Tool ◽  
Gaussian Random Fields ◽  
Dimensional Euclidean Space ◽  
Square Integrable Function

The purpose of this paper is to investigate way of dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd. Our main tool is variational calculus, where the boundary of a domain varies and deforms and we appeal to the white noise analysis. We therefore assume that X(D) is expressed white noise integral of the form(0.1) X(D) = X(D, W)=∫D F(D, u)W(u)du,where W is the Rd-parameter white noise and the kernel F(D, u) is a square integrable function over Rd, and where D is a bounded domain with smooth boundary.

On the comparison of the distribution of the supremum of random fields represented by stochastic integrals

1989 ◽  
Vol 21 (04) ◽  
pp. 770-780
Author(s):  
Enzo Orsingher ◽  
Bruno Bassan
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Upper Bounds ◽  
Gaussian Random Fields ◽  
Stochastic Integrals ◽  
Response Functions ◽  
Noise Field ◽  
Fixed Radius ◽  
Different Response

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = ∫ Cp G(P, P′) dW(P′) and U(P) = ∫ Cp dW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions. The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P). The comparison of the suprema is carried out also, when C P = ℝ2, between fields with different response functions.

Comparison of Nonlinear Spatial Correlation Models by the Influence of the Data Augmentation to the Classification Risk

2002 ◽  
Vol 7 (1) ◽  
pp. 31-42
Author(s):  
J. Šaltytė ◽  
K. Dučinskas
Keyword(s):  
Spatial Correlation ◽  
Random Fields ◽  
Data Augmentation ◽  
Gaussian Random Fields ◽  
Classification Rule ◽  
Numerical Comparison ◽  
First Order ◽  
Bayesian Risk ◽  
Correlation Models

The Bayesian classification rule used for the classification of the observations of the (second-order) stationary Gaussian random fields with different means and common factorised covariance matrices is investigated. The influence of the observed data augmentation to the Bayesian risk is examined for three different nonlinear widely applicable spatial correlation models. The explicit expression of the Bayesian risk for the classification of augmented data is derived. Numerical comparison of these models by the variability of Bayesian risk in case of the first-order neighbourhood scheme is performed.

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