An invariance principle for estimating the correlation function of a homogeneous random field

1981 ◽  
Vol 32 (3) ◽  
pp. 213-219
Author(s):  
A. V. Ivanov ◽  
N. N. Leonenko
Keyword(s):  
Correlation Function ◽  
Random Field ◽  
Invariance Principle ◽  
Homogeneous Random Field

Invariance principle for estimating the correlation function of a homogeneous isotropic random field

1982 ◽  
Vol 33 (3) ◽  
pp. 241-249
Author(s):  
A. V. Ivanov ◽  
N. N. Leonenko
Keyword(s):  
Correlation Function ◽  
Random Field ◽  
Invariance Principle ◽  
Isotropic Random Field ◽  
Isotropic Random

Invariance principle for estimates of regression coefficients of a random field

1982 ◽  
Vol 33 (6) ◽  
pp. 580-586
Author(s):  
N. N. Leonenko
Keyword(s):  
Random Field ◽  
Invariance Principle ◽  
Regression Coefficients

A note on the estimation of the mean of a homogeneous random field

1971 ◽  
Vol 8 (3) ◽  
pp. 626-629
Author(s):  
Michael Skalsky
Keyword(s):  
Finite Number ◽  
Random Field ◽  
Random Process ◽  
Mathematical Expectation ◽  
Covariance Function ◽  
Arithmetic Mean ◽  
Stationary Random Process ◽  
Homogeneous Random Field ◽  
The Mean

An important problem, arising in connection with the estimation of mathematical expectation of a homogeneous random field X(x1, ···, xn) in Rn by means of the arithmetic mean of observed values, is to determine the number of observations for which the variance of the estimate attains its minimum. Vilenkin [2] has shown, that in the case of a stationary random process X(x) such a finite number exists, provided that the covariance function satisfies certain conditions.

scholarly journals CPT-Based Probabilistic Characterization of Undrained Shear Strength of Clay

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Mi Tian ◽  
Xiaotao Sheng
Keyword(s):  
Shear Strength ◽  
Correlation Function ◽  
Random Field ◽  
Site Investigation ◽  
Undrained Shear Strength ◽  
Bayesian Approaches ◽  
Probabilistic Characterization ◽  
Undrained Shear

Applying random field theory involves two important issues: the statistical homogeneity (or stationarity) and determination of random field parameters and correlation function. However, the profiles of soil properties are typically assumed to be statistically homogeneous or stationary without rigorous statistical verification. It is also a challenging task to simultaneously determine random field parameters and the correlation function due to a limited amount of direct test data and various uncertainties (e.g., transformation uncertainties) arising during site investigation. This paper presents Bayesian approaches for probabilistic characterization of undrained shear strength using cone penetration test (CPT) data and prior information. Homogeneous soil units are first identified using CPT data and subsequently assessed for weak stationarity by the modified Bartlett test to reject the null hypothesis of stationarity. Then, Bayesian approaches are developed to determine the random field parameters and simultaneously select the most probable correlation function among a pool of candidate correlation functions within the identified statistically homogeneous layers. The proposed approaches are illustrated using CPT data at a clay site in Shanghai, China. It is shown that Bayesian approaches provide a rational tool for proper determination of random field model for probabilistic characterization of undrained shear strength with consideration of transformation uncertainty.

SELF-INTERSECTION LOCAL TIME FOR ${\mathcal S}' ({\mathbb R}^d)$-WIENER PROCESSES AND RELATED ORNSTEIN–UHLENBECK PROCESSES

1999 ◽  
Vol 02 (04) ◽  
pp. 569-615 ◽  
Author(s):  
TOMASZ BOJDECKI ◽  
LUIS G. GOROSTIZA
Keyword(s):  
Random Field ◽  
Local Time ◽  
Fourier Transforms ◽  
Stable Process ◽  
Local Times ◽  
Spherically Symmetric ◽  
Homogeneous Random Field ◽  
Intersection Local Time ◽  
Wiener Processes ◽  
First Results

Existence and continuity results are obtained for self-intersection local time of [Formula: see text]-valued Ornstein–Uhlenbeck processes [Formula: see text], where X0 is Gaussian, Wt is an [Formula: see text]-Wiener process (independent of X0), and T't is the adjoint of a semigroup Tt on [Formula: see text]. Two types of covariance kernels for X0 and for W are considered: square tempered kernels and homogeneous random field kernels. The case where Tt corresponds to the spherically symmetric α-stable process in ℝd, α∈(0,2], is treated in detail. The method consists in proving first results for self-intersection local times of the ingredient processes: Wt, T't X0 and [Formula: see text], from which the results for Xt are derived. As a by-product, a class of non-finite tempered measures on ℝd whose Fourier transforms are functions is identified. The tools are mostly analytical.

Simulation of stochastic waves in a non-homogeneous random field

1995 ◽  
Vol 14 (5) ◽  
pp. 387-396 ◽  
Author(s):  
Junji Kiyono ◽  
Kenzo Toki ◽  
Tadanobu Sato ◽  
Haruhiro Mizutani
Keyword(s):  
Random Field ◽  
Homogeneous Random Field ◽  
Stochastic Waves

scholarly journals Gaussian Random Particles with Flexible Hausdorff Dimension

2015 ◽  
Vol 47 (02) ◽  
pp. 307-327
Author(s):  
Linda V. Hansen ◽  
Thordis L. Thorarinsdottir ◽  
Evgeni Ovcharov ◽  
Tilmann Gneiting ◽  
Donald Richards
Keyword(s):  
Correlation Function ◽  
Hausdorff Dimension ◽  
Random Field ◽  
Kernel Smoothing ◽  
Three Dimensional ◽  
Random Sets ◽  
Closed Form Expression ◽  
Spherical Cap ◽  
Fractal Index ◽  
Von Mises

Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises-Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. The Hausdorff dimension of the surface of the Gaussian particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3.

Intensity of Crossings of a Given Level by a Homogeneous Random Field

2017 ◽  
Vol 53 (5) ◽  
pp. 785-792
Author(s):  
D. V. Yevgrafov
Keyword(s):  
Random Field ◽  
Homogeneous Random Field

Asymptotic efficiency of the arithmetic-mean estimator of the unknown mean of a homogeneous random field

1993 ◽  
Vol 66 (4) ◽  
pp. 2438-2441 ◽  
Author(s):  
S. Bektashov ◽  
Dang Dyk Hau ◽  
M. I. Yadrenko
Keyword(s):  
Random Field ◽  
Asymptotic Efficiency ◽  
Arithmetic Mean ◽  
Homogeneous Random Field
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