non-Gaussian Random Fields Research Articles

Model selection for a class of stochastic processes or random fields with bounded range

Methods are developed for finding an optimal model for a non-Gaussian stationary stochastic process or homogeneous random field under limited information. The available information consists of: (i) one or more finite length samples of the process or field; and (ii) knowledge that the process or field takes values in a bounded interval of the real line whose ends may or may not be known. The methods are developed and applied to the special case of non-Gaussian processes or fields belonging to the class of beta translation processes. Beta translation processes provide a flexible model for representing physical phenomena taking values in a bounded range, and are therefore useful for many applications. Numerical examples are presented to illustrate the utility of beta translation processes and the proposed methods for model selection.

Effects of spatial variation of soil properties on seismic performance of port structures

From past seismic events such as the 1995 Kobe (Hyogoken-Nanbu) and 2001 Nisqually earthquakes, it was found that liquefaction-induced lateral spread has caused significant damage to structures such as buildings and bridges, in addition to underground utility facilities like pipelines. In this respect, seaport facilities are particularly vulnerable to these liquefaction-related damages, because they are usually constructed on poorly consolidated natural deposit or fills. This study investigates the effect of the liquefaction and lateral spread on the seismic response of caisson type quay walls. For this purpose, 2D nonlinear dynamic analyses of soil–structure system are carried out with the aid of finite difference software, FLAC. The unique feature of this study lies in the fact that the 2D soil system is idealized as homogeneous non-Gaussian random field. A simulation algorithm is then used to generate a set of digital realizations of 2D random field sample. Each realization is used for the dynamic analysis to generate a unique response of the soil–structure system. Repeating this analysis for the entire set of realizations, the probabilistic nature of the response is characterized in the Monte Carlo sense. This result based on random field is compared with the response obtained under the uniform field assumption with the mean value of soil property. The comparison shows that in general the uniform model provides unconservative result compared with the response from the random field model due to nonlinear behavior of the soil–structure system. It is also found that the consideration of spatial variation of soil can capture the dispersion of observed response of quay walls.

Construction of non-Gaussian random fields with any given correlation structure

A positive definite function can be thought of as the covariance function of a Gaussian random field, according to the celebrated Kolmogorov existence theorem. A question of great theoretical and practical interest is: how could one construct a non-Gaussian random field with the given positive definite function as its covariance function? In this paper we demonstrate a novel and simple method for constructing many such non-Gaussian random fields, with the corresponding finite-dimensional distributions identified. Also, we show how to construct a non-Gaussian random field with a given negative definite function as its variogram.

Some new random field tools for spatial analysis

This is a brief review, in relatively non-technical terms, of recent rather technical advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric characteristics of excursion sets of random fields, such as their volume, surface area and Euler characteristics. What is particularly important in these formulae is that whereas the previous theory covered only stationary, Gaussian random fields, the new theory requires neither stationarity nor, a fortiori, isotropy. Furthermore, it covers a wide class of non-Gaussian random fields. The formulae provided by these advances have a wide range of potential applications, including new techniques of parameter estimation, model testing, and thresholding for spatial and space-time functions. The paper reviews this theory, and provides brief descriptions of some of the applications.

Critical review and latest developments of a class of simulation algorithms for strongly non-Gaussian random fields

A methodology is presented for simulation of strongly non-Gaussian random fields. It involves an iterative scheme that produces sample functions that match a prescribed non-Gaussian marginal distribution and a prescribed Spectral Density Function (SDF). The simulated field possesses all the properties of translation fields. The methodology also determines the SDF of an underlying Gaussian field according to translation field theory. This is the latest development in a class of simulation algorithms that are presented and critically reviewed. Several numerical examples are provided demonstrating the capabilities of the methodology, comparing it with three previous algorithms, and determining the limits of its applicability. Compared to earlier algorithms, the proposed methodology provides increased accuracy at a fraction of the computational cost.

Random Field-Based Approach for Strength Evaluation of Suspension Bridge Cables

A methodology is introduced to estimate the strength of suspension bridge cables using results of tensile strength tests performed on wire samples extracted from the bridge’s main cables. The innovation of the proposed methodology is to consider the spatial correlation of the wire strength over the wire’s length, a real and experimentally measured property of ductile steel wires that is disregarded in the current standard approach. The wire strength is modeled as a nonGaussian random field along its length. The number of parallel wires in the cable’s cross section is then considered to estimate the strength of the entire cable. The capabilities of the proposed methodology are demonstrated through an application involving an experimental data set of wire segments extracted from the Williamsburg Bridge. Results of the proposed methodology are compared to corresponding results of the current standard approach. It is explained why the former results are relatively more accurate than the latter. A procedure fo...

Statistics of Fourier Modes in Non‐Gaussian Fields

A theory for statistical distributions of Fourier modes in non-Gaussian fields is established in this paper. In astrophysics and cosmological physics, Fourier analyses of random fields are useful and widely used. While distributions of the Fourier modes in Gaussian fields are well-known, there was no well-established theory for those in non-Gaussian fields. In non-Gaussian random fields, polyspectra, higher order counterparts of the power spectrum, are usually considered to characterize statistical information contained in the fields. The distribution functions of Fourier modes are directly considered in this paper and their explicit relations to the polyspectra are given. Under the condition that any of the polyspectra does not diverge, the distribution function is expanded by dimensionless combinations of polyspectra and a total volume in which the Fourier transforms are performed. The expression up to second order is generally given, and higher order results are also derived in some cases. A concept of the N-point distribution function of Fourier modes is introduced and explicitly calculated. Among them, the one-point distribution function is completely given in a closed form up to arbitrary order. As an application, statistics of Fourier phases are explored in detail. A number of aspects regarding statistical properties of phases are found. It is clarified, for the first time, how phase correlations arise in general non-Gaussian fields. Some of our analytic results are tested against numerical realizations of non-Gaussian fields, showing good agreements.

Adaptive Random Field Mesh Refinements in Stochastic Finite Element Reliability Analysis of Structures

A technique for adaptive random field refinement for stochastic finite element reliability analysis of structures is presented in this paper. Refinement indicator based on global importance measures are proposed and used for carrying out adaptive random field mesh refinements. Reliability index based error indicator is proposed and used for assessing the percentage error in the estimation of notional failure probability. Adaptive mesh refinement is carried out using hierarchical graded mesh obtained through bisection of elements. Spatially varying stochastic system parameters (such as Young's modulus and mass density) and load parameters are modeled in general as non-Gaussian random fields with prescribed marginal distributions and covariance functions in conjunction with Nataf's models. Expansion optimum linear estimation method is used for random field discretisation. A framework is developed for spatial discretisation of random fields for system/load parameters considering Gaussian/non-Gaussian nature of random fields, multidimensional random fields, and multiple-random fields. Structural reliability analysis is carried out using first order reliability method with a few refined features, such as, treatment of multiple design points and/or multiple regions of comparable importance. The gradients of the performance function are computed using direct differentiation method. Problems of multiple performance functions, either in series, or in parallel, are handled using the method based on product of conditional marginals. The efficacy of the proposed adaptive technique is illustrated by carrying out numerical studies on a set of examples covering linear static, free vibration and forced vibration problems.

A stochastic micromechanical model for elastic properties of functionally graded materials

A stochastic micromechanical model is presented for predicting probabilistic characteristics of elastic mechanical properties of an isotropic functionally graded material (FGM) subject to statistical uncertainties in material properties of constituents and their respective volume fractions. The model involves non-homogeneous, non-Gaussian random field representation of phase volume fractions and random variable description of constituent material properties, a three-phase Mori–Tanaka model for underlying micromechanics and homogenization, and a novel dimensional decomposition method for obtaining probabilistic descriptors of effective FGM properties. Four numerical examples involving statistical properties of input random fields, limited experimental validation, and the second-moment characteristics and probability density functions of effective mechanical properties of FGM illustrate the proposed stochastic model. The results indicate that the model provides both accurate and computationally efficient estimates of probabilistic characteristics of effective FGM properties.

Spherical harmonic-based random fields for aggregates used in concrete

Two non-Gaussian random field models are developed for characterizing the geometry of star-like inclusions in two-phase or multi-phase materials. For illustration, the models are calibrated to the geometrical features of a population of 128 similar ASTM C57 aggregates used in concrete. The first model is a sum of spherical harmonic functions with random non-Gaussian coefficients. The second model is a nonlinear memoryless mapping of a sum of spherical harmonic functions with Gaussian coefficients. The two random field models are used to develop Monte Carlo algorithms for generating virtual aggregates. By construction, the virtual aggregates capture essential statistics of the target population of aggregates. The performance of the proposed models is assessed by quantitative and qualitative metrics.

Spectral Properties of Burgers and KPZ Turbulence

This paper presents the higher-order spectral densities of non-Gaussian random fields arising as scaling limits in the Burgers and KPZ turbulence problems with strongly dependent non-Gaussian initial conditions.

Non-Gaussian Complex Random Fields, their Skeletons and Path Measures

This work investigates complex random fields Z, which have a rotation invariant path measure. Fields of this type are constructed and analyzed in terms of (pathwise convergent) L 2-expansions, and quasi invariance properties of their path measures are studied. The results are used to investigate ℋL 2(Z), the space of holomorphic L 2-functionals of Z. Conditions are given such that every F∈ℋL 2(Z) admits an L 2-power series expansion, and a general skeleton theorem is proved, which justifies the notion ‘holomorphic’.

Reliability Assessment on Thickness Effect in Fatigue Crack Growth

The evaluation of specimen thickness effect of fatigue crack growth life by the simulation of probabilistic fatigue crack growth is presented. In this paper, the material resistance to fatigue crack growth is treated as a spatial stochastic process, which varies randomly on the crack surface. Using the previous experimental data, the non-Gaussian (eventually Weibull, in this report) random fields simulation method is applied. This method is useful to estimate the probability distribution of fatigue crack growth life and the variability due to specimen thickness by simulating material resistance to fatigue crack growth along a crack path.

Correlated mixture between adiabatic and isocurvature fluctuations and recent CMB observations

This work presents a reduced ${\ensuremath{\chi}}_{\ensuremath{\nu}}^{2}$ test to search for non-Gaussian signals in the cosmic microwave background radiation (CMBR) TT power spectrum of recent CMBR data, Wilkinson Anisotropy Microwave Probe, Arcminute Cosmology Bolometer Array Receiver, and Cosmic Background Imager data sets, assuming a mixed density field including adiabatic and isocurvature fluctuations. We assume a skew positive mixed model with adiabatic inflation perturbations plus additional isocurvature perturbations possibly produced by topological defects. The joint probability distribution used in this context is a weighted combination of Gaussian and non-Gaussian random fields. Results from simulations of CMBR temperature for the mixed field show a distinct signature in CMB power spectrum for very small deviations ($\ensuremath{\sim}0.1%$) from a pure Gaussian field, and can be used as a direct test for the nature of primordial fluctuations. A reduced ${\ensuremath{\chi}}_{\ensuremath{\nu}}^{2}$ test applied on the most recent CMBR observations reveals that an isocurvature fluctuations field is not ruled out and indeed permits a very good description for a flat geometry $\ensuremath{\Lambda}$-CDM Universe, ${\ensuremath{\chi}}_{930}^{2}\ensuremath{\sim}1.5$, rather than the simple inflationary standard model with ${\ensuremath{\chi}}_{930}^{2}\ensuremath{\sim}2.3$. This result may looks is particular discrepant with the reduced ${\ensuremath{\chi}}^{2}$ of 1.07 obtained with the same model in Spergel et al. [Astrophys. J. 148, 175 (2003)] for temperature only, however, our work is restricted to a region of the parameter space that does not include the best fit model for TT only of Spergel et al..

Applications of a simplified non-Gaussian field model to wave propagation in turbulent media

In order to study the optical effects of turbulent intermittency,a simplified non-Gaussian random field model has been constructed,which can model atmospheric dielectric permittivity(or refractive-index) fluctuation approximately.Four arbitrary functions enter into the characteristic functional of this model.Using the properties of the atmospheric turbulence,such as the statistical homogeneity,the single-point probability distribution and the spectrum of the refractive-index fluctuations,these functions can be selected or determined.With these given functions,we have studied the statistical properties,especially the mean field adn the mutual coherence function,of a plane partially coherent incident wave.A simple example of numerical simulation has also been presented.

On model equations for particle dispersion in inhomogeneous turbulence

Comparisons are made between the Advection–Diffusion Equation (ADE) approach for particle transport and the two-fluid model approach based on the PDF method. In principle, the ADE approach offers a much simpler way of calculating the inertial deposition of particles in a turbulent boundary layer than that based on the PDF approach. However the ADE equations that have recently been used are only strictly valid for a simple Gaussian process when particle inertia is small. Using a prescribed, but in general non-Gaussian random particle velocity field, it is shown that the net particle mass flux contains a drift term in addition to that from the mean velocity of the particle velocity field, associated with the compressibility of the velocity field. Furthermore the diffusive flux in general depends not only upon the gradient of the mean concentration (true only for a Gaussian random flow field) but also upon higher order derivatives whose relative contribution depends on diffusion coefficients D ijk… etc. These coefficients depend upon the statistical moments associated with random displacements and compressibility of the particle flow field along particle trajectories which in turn depend upon particle inertia. In contrast the PDF approach offers the advantage of using a simple gradient (Gaussian) approximation in particle phase space which can lead to a non-Gaussian spatial dispersion process when particle inertia is important. Conditions based on the particle mean free path are derived for which a simple ADE is appropriate. Some of the features of particle transport in an inhomogeneous turbulent flow are illustrated by examining particle dispersion in a random flow field composed of pairs of counter rotating vortices which has an rms velocity which increase linearly from a stagnation point.

White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media

Starting with the Wigner distribution formulation for beam wave propagation in Hölder continuous non-Gaussian random refractive index fields we show that the wave beam regime naturally leads to the white-noise scaling limit and converges to a Gaussian white-noise model which is characterized by the martingale problem associated to a stochastic differential-integral equation of the Itô type. In the simultaneous geometrical optics the convergence to the Gaussian white-noise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian white-noise model is that its n-point correlation functions are governed by closed form equations.

On Characteristics of Weibull Distribution Parameters of Fatigue Crack Growth Lives

This paper illustrates the characteristics of parameters for three-parameter Weibull probability distribution of fatigue crack growth lives under constant stress intensity factor control experiments. Applying the statistical properties of the previous experimental results, the fatigue crack curves were simulated by using the non-Gaussian random fields simulation method and analyzed for the different specimen thickness and stress intensity level to determine the probability distributions of the fatigue crack growth life.

Geostatistics for Power Models of Gaussian Fields

This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions.

Elastic microcracked bodies with random properties

The stochastic properties of a ‘dense’ distribution of microcracks in an elastic body are analyzed by using a multifield continuum model describing the influence of the microcracks on the gross mechanical behavior of the body. Numerical examples are presented: the strain localization phenomena that have been already found in deterministic bodies are confirmed, and similar patterns are shown to exist also for the stochastic moments of the displacements. In particular, the patterns in the portraits of skewness and kurtosis become stronger when the correlation of the distance between neighboring microcracks increases. Such a distance is considered as a uni-variate non-Gaussian random field. Strain localization is an indicator toward the irreversible growth and coalescence of microcracks.