Almost sure limit theorem for stationary Gaussian random fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

Image Classification Based on Bayes Discriminant Functions

An efficient stochastic simulation method is presented for generating sample time histories of multidimensional, spatially correlated, stationary Gaussian random fields. A key feature of the method is the use of filtering functions in directly expressing the conditional mean value of any component of the random field. The general formulation and an updating algorithm for determining the required filtering functions for simulation of any general multidimensional random field are established. Also, emphasis is placed on the efficiencies, which can be achieved when relatively simple restrictions are placed on the cross-spectral densities and the incoherence structure of the random field. These restrictions, which are commonly used in modeling seismic ground motion, are shown to allow the computational effort to be reduced by a factor of more than 100 for a three-dimensional random field. Numerical results are presented to illustrate the application of the direct simulation method to the generation of time histories of seismic ground motion. The method is shown to be both efficient and accurate.

In this paper we implement a method for the 3D-rigid motion invariant texture discrimination and binary classification for discrete 3D-textures that are spatially homogeneous by modelling them as stationary Gaussian random fields. We use a novel 'distance' between 3D-textures that remains invariant under all 3D-rigid motions of the texture to develop rules for 3D-rigid motion invariant texture discrimination and binary classification of textures.

Strong regularity for stationary discrete random fields is discussed. An extension of the classical Beurling's Theorem to functions of several variables is given. Necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. A recipe formula for the best linear extrapolator is also given.

Bayesian multiscale analysis of images modeled as Gaussian Markov random fields

Two kinds of estimation variance functions for estimating a local average (LA) of a stationary (homogeneous) random field (RF) are derived. One is local estimation (LE) and the other is general estimation (GE) of LA. The former is for estimating LA at the observation location, and the latter is for obtaining LA at any arbitrary location within the RF. The geotechnical implications of these two estimations are that LE is for estimating LA of geotechnical parameters at the spot where the investigations are made, whereas GE is for estimating LA at any arbitrary location within the same layer. The behavior of the two estimation variance functions differs greatly, controlled by the number of observations (i.e. sample size) and the normalized layer thickness (layer thickness divided by autocorrelation distance of RF). Based on the derived estimation variance functions, methods for determining reliable characteristic values of geotechnical parameters and necessary sample size are proposed. The methods are based on the same framework as that of the traditional statistical theory, i.e. confidence interval of estimated parameters. However, the assumption of independently and identically distributed (i.i.d.) samples in the traditional statistical theory is replaced by the assumption of correlated samples from a stationary RF. The results obtained from the proposed methods for LE and GE differ from each other as well as from the traditional results, which has significant implications for geotechnical parameter estimation in geotechnical engineering practice.

The problems of model specification in one‐ and two‐dimensional stationary random fields are considered, with particular reference to the class of spatial autoregressions. The role of the autocorrelation function is discussed and it is shown to have a valuable exploratory use, especially in identifying directional bias in random fields. It is recommended that this analysis be supplemented by a likelihood‐ratio test, which permits more precise tests of hypotheses about the form of the underlying model.

Circulant embedding is a powerful algorithm for fast simulation of stationary Gaussian random fields on a rectangular grid in , which works perfectly for compactly supported covariance functions. Cut‐off circulant embedding techniques have been developed for univariate random fields for dimensions up to and rely on the modification of a covariance function outside the simulation window, such that the modified covariance function is compactly supported. In this paper, we propose extensions of the cut‐off approach for univariate and bivariate Gaussian random fields. In particular, we provide a method for simulating bivariate fields with a powered exponential model and the Matérn model for certain sets of parameters. Copyright © 2018 John Wiley & Sons, Ltd.

: A particular two dimensional model in a stationary random field, which has a wide applications in statistical signal processing and in texture classifications, is considered. We prove the consistency and also obtain the asymptotic distributions of the least squares estimators of the different model parameters. It is observed that the asymptotic distribution of the least squares estimators are multivariate normal. Some numerical experiments are performed to see how the asymptotic results work for finite samples. We propose some open problems at the end.

The problem of sequentially scanning and predicting data arranged in a multidimensional array is considered. We introduce the notion of a scandictor, which is any scheme for the sequential scanning and prediction of such multidimensional data. The scandictability of any finite (probabilistic) data array is defined as the best achievable expected performance on that array. The scandictability of any (spatially) stationary random field on /spl Zopf//sup m/ is defined as the limit of its scandictability on finite boxes (subsets of /spl Zopf//sup m/), as their edges become large. The limit is shown to exist for any stationary field, and essentially be independent of the ratios between the box dimensions. Fundamental limitations on scandiction performance in both the probabilistic and the deterministic settings are characterized for the family of difference loss functions. We find that any stochastic process or random field that can be generated autoregressively with a maximum-entropy innovation process is optimally scandicted the way it was generated. These results are specialized for cases of particular interest. The scandictability of any stationary Gaussian field under the squared-error loss function is given a single-letter expression in terms of its spectral measure and is shown to be attained by the raster scan. For a family of binary Markov random fields (MRFs), the scandictability under the Hamming distortion measure is fully characterized.

Every two-dimensional incompressible flow follows the level lines of some scalar function $\psi$ on $\mathbb{R}^2$; transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when $\psi$ is a stationary random field. We show that under mild hypotheses there is only one possible alternative to bounded level lines: the "treelike" random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line "winding through every region of the plane." If the random field has the FKG property, then only bounded level lines are possible. For stationary $C^2$ Gaussian random fields with covariance function decaying to 0 at $\infty$, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at $\infty$ "slower than exponentially," and only bounded level lines are possible if the covariance function is nonnegative.

A central limit theorem for functions of stationary max-stable random fields on [formula omitted

The Circulant Embedding Method (CEM) is a well known technique to generate stationary Gaussian Random Fields (GRF). The main idea is to embed the covariance matrix in a larger nested block circulant matrix, whose factorization can be rapidly computed thanks to the fast Fourier transform (FFT) algorithm. The CEM requires the extended matrix to be at least positive semidefinite which is proven to be the case if the enclosing domain is sufficiently large, as proven by Theorem 2.3 in [9] for cubic domains. In this paper, we generalize this theorem to the case of rectangular parallelepipeds. Then we propose a new initialization stage of the CEM algorithm that makes it possible to quickly jump to a domain size close to the one needed for the CEM algorithm to work. These domain size estimates are based on fitting functions. Examples of fitting functions are given for the Matérn family of covariances. These functions are inspired by our numerical simulations and by the theoretical work from [9]. The parameters estimation of the fitting functions is done numerically. Several numerical tests are performed to show the efficiency of the proposed algorithms, for both isotropic and anisotropic Matérn covariances.

Slope reliability analysis has always been a research focus for geotechnical engineers. Currently, most studies use stationary random fields to simulate the spatial variability of soil parameters. However, many field test data show that soil parameters such as cohesion and friction angle often exhibit obvious nonstationary characteristics along the burial depth direction. Moreover, the slope reliability analysis method based on Monte Carlo simulation requires a combination of the FEM and a large number of computational resources, resulting in extremely low computational efficiency. Therefore, this paper proposes a convolutional neural network (CNN)-based surrogate model for slope reliability analysis, emphasizing the nonstationary spatial variability of cohesion and friction angle. First, stationary and nonstationary random fields considering the spatial variability of shear strength parameters are generated, and sample data sets are collected using an automated computational program. Next, a CNN surrogate model is established to predict the factor of safety (FOS), which is trained using nine sample data sets containing stationary and nonstationary random fields. Finally, the model performance is validated using six slope models with different shear strength parameters. The results show that the CNN surrogate model can adapt to the data characteristics of different random fields and also shows good predictive performance in handling unseen sample data sets. The predicted FOS values show a high degree of consistency with the results calculated by the FEM, with the computation time being significantly reduced.