Stationary Gaussian Random Field Research Articles

Stochastic modeling of periodic beams under uncertain boundary conditions and environmental fluctuations

Periodic structures have been attracting a great deal of academic and industrial interest lately, due to their distinctive vibration and wave propagation behavior, which can be explored for the development of innovative solutions to structural dynamics and vibroacoustic problems. Although such a potential has been demonstrated in a large number of studies, the investigation of detrimental effects, which can be present in practical applications, is still necessary. This paper reports investigations on the combined influence of uncertainties affecting ambient temperature — which alters material properties and induces stress-stiffening due to constrained thermal dilatation — and boundary conditions (BCs) on the bandgap characteristics of periodic beams. The space-dependent temperature fluctuations are represented as a one-dimensional stationary Gaussian random field, discretized using the Karhunen-Loève expansion, while non-ideal BCs, represented as springs, are modeled as discrete random variables. Sampling-based stochastic analyses of the central frequency and bandwidth of the beam’s attenuation bands are performed using Monte Carlo simulations. The results demonstrate that the variability in the attenuation band features is influenced not only by the coefficients of variation (CVs) of the input random quantities, but also by the correlation length of the random temperature fluctuations. Numerical simulations reveal that the bandgap central frequency is primarily affected by the temperature random field, while the BCs govern the bandwidth. Although low CV and standard deviation values are obtained for the dispersion of the bandgap features, reliability analyses indicate that some designs exhibit low reliability. Increased variability in both the bandgap central frequency and bandwidth is observed for greater temperature correlation lengths and CVs. The contributions of the study include the proposal of a comprehensive stochastic modeling procedure duly accounting for relevant random influences, and evidencing that those influences can be significant, requiring consideration in the design of robust periodic structures.

Functional Convergence of Berry’s Nodal Lengths: Approximate Tightness and Total Disorder

We consider Berry’s random planar wave model (J Phys A 10(12):2083–2092, 1977), and prove spatial functional limit theorems—in the high-energy limit—for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study of the projection of nodal lengths onto the so-called second Wiener chaos, whose high-energy fluctuations are given by a Gaussian total disorder field indexed by polygonal curves. Such an exact characterization is then combined with moment estimates for suprema of stationary Gaussian random fields, and with a tightness criterion by Davydov and Zitikis (Ann Inst Stat Math 60(2):345–365, 2008).

Point process of clusters for a stationary Gaussian random field on a lattice

Point process of clusters for a stationary Gaussian random field on a lattice

Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields

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.

Point processes of exceedances by Gaussian random fields with applications to asymptotic locations of extreme order statistics

In this paper, we studied the limit properties of the point processes of exceedances by stationary dependent Gaussian random fields. Applying the obtained results, we investigated the asymptotic relations between the locations and heights of extreme order statistics for stationary dependent Gaussian random fields. It is shown that these asymptotic relations are not sensitive to the dependence of the Gaussian random fields.

A stochastic fractional Laplace equation driven by colored noise on bounded domain, and its covariance functional

The paper provides conditions for the fractional Laplacian and its spectral representation on stationary Gaussian random fields to be well-defined. In addition, we study existence and uniqueness of the weak solution for a stochastic fractional elliptic equation driven by an additive colored noise over an open bounded set. Both spectral and variational approaches are used to provide a solution. Further, the functional covariance associated with the solution is derived.

On fixed-domain asymptotics, parameter estimation and isotropic Gaussian random fields with Matérn covariance functions

A method is proposed for estimating the microergodic parameters (including the smoothness parameter) of stationary Gaussian random fields on Rd with isotropic Matérn covariance functions using irregularly spaced data. This approach uses higher-order quadratic variations and is applied to three designs, namely stratified sampling design, randomized sampling design and deformed lattice design. Microergodic parameter estimators are constructed for each of the designs. Under mild conditions, these estimators are shown to be consistent with respect to fixed-domain asymptotics. Upper bounds to the convergence rate of the estimators are also established. A simulation study is conducted to gauge the accuracy of the proposed estimators.

Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R

In this paper, we consider isotropic and stationary real Gaussian random fields defined on S2×R and we investigate the asymptotic behavior, as T→+∞, of the empirical measure (excursion area) in S2×[0,T] at any threshold, covering both cases when the field exhibits short and long memory, that is, integrable and nonintegrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry’s cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws.

Directional phantom distribution functions for stationary random fields

We give necessary and sufficient conditions for the existence of a phantom distribution function for a stationary random field on a regular lattice. We also introduce a less demanding notion of a directional phantom distribution, with potentially broader area of applicability. Such approach leads to sectorial limit properties, a phenomenon well-known in limit theorems for random fields. An example of a stationary Gaussian random field is provided showing that the two notions do not coincide. Criteria for the existence of the corresponding notions of the extremal index and the sectorial extremal index are also given.

Stochastic eigenfrequency and buckling analyses of plates subjected to random temperature distributions

Many studies previously reported in the literature have demonstrated, both theoretically and experimentally, the influence of thermally-induced stresses on the static and dynamic behavior of structures, due to the so-called stress-stiffening effect. In most cases of practical interest, temperature variations associated to environmental and operational conditions are governed by rather complex combinations of conduction, convection and radiation mechanisms. As a result, the temperature values at different points of a structure are very difficult to control and can rationally be considered as random quantities. In this context, the present paper addresses the stochastic modeling and characterization of the influence of thermal stresses on the natural frequencies of thin rectangular plates, assuming space-dependent temperature fluctuations modeled as stationary two-dimensional Gaussian random fields. For this purpose, based on the hypotheses of the classical Kirchhoff plate theory, a Rayleigh-Ritz-based dynamic model is first derived for the bending vibrations of plates, accounting for the presence of thermal stresses. This model is combined with the Karhunen-Loève expansion (KL), which is used to discretize the temperature random field, after which the statistics of the random natural frequencies are estimated by Monte Carlo sampling. Numerical simulations are performed for plates under free boundary conditions. Simulation results, which encompass sampling-based statistics for the thermal stresses and the first six natural frequencies of the plate, are presented and discussed. In addition, since thermal stresses can induce buckling, reliability has also been estimated considering this type of failure. Results enable to conclude that space-dependent temperature uncertainty can be significant upon the vibration and buckling behavior of plates, which justifies its consideration.

The moduli of non-differentiability for Gaussian random fields with stationary increments

We establish the exact moduli of non-differentiability of Gaussian random fields with stationary increments. As an application of the result, we prove that the uniform Hölder condition for the maximum local times of Gaussian random fields with stationary increments obtained in Xiao (1997) is optimal. These results are applicable to fractional Riesz–Bessel processes and stationary Gaussian random fields in the Matérn and Cauchy classes.

Hypothesis testing for the smoothness parameter of Matérn covariance model on a regular grid

We consider the hypothesis testing problem for the smoothness parameter ν in a stationary isotropic Gaussian random field with Matérn covariance. For the data observed on a regular grid, we construct the rejection region for one-tailed tests, and starting from there, we develop a chain-like testing procedure, which can determine an interval containing the true value of ν. Such an interval can help improve the performance of various estimation methods for ν, such as restricting the parameter space or validating the assumptions for the asymptotic properties of the estimator. The test statistic is built on recursive applications of the Laplace operator to the observations. For this statistic, the fixed-domain asymptotic normality is established and the forms of asymptotic mean and variance are derived. Therefore, the proposed tests are guaranteed to have correct asymptotic size under certain conditions. Simulation studies indicate that our proposed methods are efficient for moderate sample sizes. As an application of the chain-like testing procedure, we provide a method of choosing the number of differencing for a local Whittle-likelihood type estimator of ν proposed by Wu, Lim, and Xiao, and show that it can avoid obtaining inconsistent estimates of ν via a numerical experiment.

Unified Analysis of Periodization-Based Sampling Methods for Matérn Covariances

The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Matérn covariances with smoothness index $\nu$ and correlation length $\lambda$, we analyze the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the $\nu$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $\nu$, $\lambda$ of the torus when using a smooth periodization. In doing this we arrive at a complete characterization of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.

Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets

For a smooth vectorial stationary Gaussian random field, $X:\Omega \times \mathbb {R}^{d}\to \mathbb {R}^{d}$, we provided necessary conditions to have a finite second moment for the number of roots of $X(t)-u$. Then, under a more restrictive hypothesis, some sufficient conditions were also given. The results were obtained using a method of proof inspired the one obtained by D. Geman for stationary Gaussian processes. Afterward, the same method is applied to the number of critical points of a scalar random field and to the level set of a vectorial process, $X:\Omega \times \mathbb {R}^{D}\to \mathbb {R}^{d}$, with $D>d$.

A non–stationary non–Gaussian hedonic spatial model for house selling prices

This work proposes a hedonic random field model to describe house selling prices from 2000 to 2005 in Cedar Falls, Iowa. This real estate market presents two distinctive features that are not well described by traditional stationary Gaussian random field models: (a) the city has, on its periphery, a hoglot that acts as an externality, affecting both the mean and variance of the selling prices, and (b) the distribution of house selling prices display heavy tails, even after the distance to the hoglot and house–specific covariates are accounted for in the mean structure of the model. A non–stationary and non–Gaussian random field model is constructed by multiplying two independent Gaussian random fields tailored to model the probabilistic features displayed by the Cedar Falls dataset. A Markov chain Monte Carlo algorithm that uses data augmentation is employed to fit the proposed model.

Extremes of a Type of Locally Stationary Gaussian Random Fields with Applications to Shepp Statistics

Let $$\{Z(\tau ,s), (\tau ,s)\in [a,b]\times [0,T]\}$$ with some positive constants a, b, T be a centered Gaussian random field with variance function $$\sigma ^{2}(\tau ,s)$$ satisfying $$\sigma ^{2}(\tau ,s)=\sigma ^{2}(\tau )$$ . We first derive the exact tail asymptotics (as $$u \rightarrow \infty $$ ) for the probability that the maximum $$M_H(T) = \max _{(\tau , s) \in [a, b] \times [0, T]} [Z(\tau , s) / \sigma (\tau )]$$ exceeds a given level u, for any fixed $$0< a< b < \infty $$ and $$T > 0$$ ; and we further derive the extreme limit law for $$M_{H}(T)$$ . As applications of the main results, we derive the exact tail asymptotics and the extreme limit laws for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as inputs.

Wavelet Thresholding in Fixed Design Regression for Gaussian Random Fields

In this paper, we consider block thresholding wavelet estimators of spatial regression functions on stationary Gaussian random fields observed over a rectangular domain indexed with \({{\mathbb {Z}}}^2\), whose covariance function is assumed to satisfy some weak condition. We investigate their asymptotic rates of convergence under the mean integrated squared error when spatial regression functions belong to a large range of Besov function classes \(B^{s}_{p,q}({{\mathbb {R}}}^2)\). To do this, we derived a result showing the discrepancy between empirical wavelet coefficients and true wavelet coefficients is within certain small rate across above Besov function classes. Based on that, we are able to determine the rates of convergence of our estimators and the supremum norm error over above function classes. The obtained rates of convergence correspond to those established in the standard univariate nonparametric regression with short-range dependence. Therefore, those rates could be considered as sharp as possible. A mild simulation study is carried out to examine the finite sample performance of the proposed estimates.

Anisotropic Gaussian wave models

Let $d$ be an integer greater or equal to 2 and let $\mathbf k$ be a $d$-dimensional random vector. We call Gaussian wave model with random wavevector $\mathbf k$ any stationary Gaussian random field defined on $\mathbb{R}^d$ with covariance function $t\mapsto \mathbb{E}[\cos(\mathbf k.t)]$. Any stationary Gaussian random field on $\mathbb{R}^d$ can be studied as a random wave. The purpose of the present paper is to link properties of the random wave with the distribution of the random wavevector, with a focus on geometric properties. We mainly concentrate on random waves such that the distribution of the norm of the wavevector and the one of its direction are independent. In the planar case, we prove that the expected length of the nodal lines is decreasing as the anisotropy of the wavevector is increasing, and we study the direction that maximizes the expected length of the crest lines. We illustrate our results on two specific models: a generalization of Berry's monochromatic planar waves and a spatiotemporal sea wave model whose random wavevector is supported by the Airy surface in $\mathbb{R}^3$. According to a general theorem, these two Gaussian fields are anisotropic almost sure solutions of partial differential equations that involve the Laplacian operator: $\Delta f+\kappa^2f=0$ (where $\kappa=\|\mathbf k\|$) for the former, $\Delta f+\partial^4_tf=0$ for the latter.

Singularity among selfsimilar Gaussian random fields with different scaling parameters and others

It is shown in this paper that the probability measures generated by selfsimilar Gaussian random fields are mutually singular, whenever they have different scaling parameters. So are those generated from a selfsimilar Gaussian random field and a stationary Gaussian random field. Certain conditions are also given for the singularity of the probability measures generated from two Gaussian random fields whose covariance functions are Schoenberg–Lévy kernels, and for those from stationary Gaussian random fields with spectral densities.

A fast algorithm for multifield representation and multiscale simulation of high‐quality 3D stochastic aggregate microstructures by concurrent coupling of stationary Gaussian and fractional Brownian random fields

SummaryThis study presents a multiscale computational framework for the representation and generation of concrete aggregate microstructures on the basis of the multifield theory, which couples the stationary Gaussian random field with the fractional Brownian random field. Specifically, the stationary Gaussian field is utilized to simulate the morphological shape of an aggregate on the coarse scale, whereas the surface topography of the aggregate on the fine scale is represented by the fractional Brownian field. To bridge the 2 scales, a concurrent coupling formula is proposed. This coupled technique allows for smooth transition between the coarse and fine scales and permits the rapid generation of highly realistic concrete aggregates that can be tailored to the desired quality and requirements, making the algorithm computationally appealing. In the generation of the random fields on the 2 scales, the Fourier representation of block circulant covariance matrices with circulant blocks is exploited, which yields substantial efficiency advantages over the conventional Cholesky decomposition approach in factorizing covariance matrices as well as simulating random fields. Meanwhile, a microsurface postprocessing and reconstruction procedure is also developed to convert the generated random fields into realistic 3D shapes. The numerical methodology proposed in this study offers tremendous potential for a plethora of applications in cement‐based materials.