Two-dimensional Random Field Research Articles

Stochastic analysis model of uncertain temperature characteristics for embankment in warm permafrost regions

For embankments in cold regions, the soil properties and the upper boundary conditions are stochastic because of complex geological processes and changeable atmospheric environment. In this study, we model the soil properties as random fields and the upper boundary conditions as stochastic processes. A triangular local average (TLA) method is used to discretize the two-dimensional (2D) random fields. The random temperature fields of an embankment in a cold region are investigated by Neumann stochastic finite element method (NSFEM), and the computational formulas of mean and standard deviation are developed. In the calculation flow chart, a stochastic finite element (FE) program has been compiled by Matrix Laboratory (MATLAB) software. The results show that TLA method perfectly matches with triangular FE method. The randomness of soil properties and boundary conditions lead to the randomness of temperature. The results will improve our understanding of the random temperature field of embankments in cold regions. The proposed method can be used to solve other uncertain thermodynamic problems.

Application of the time-invariant linear filter approximation to parametrization of surface metrology with high-quality x-ray optics

We investigate the time-invariant linear filter (TILF) approach to optimally parameterize the surface metrology of high-quality x-ray optics considered as a result of a stationary uniform random process. The approach is a generalization of autoregressive moving average (ARMA) modeling of one-dimensional slope measurements with x-ray mirrors considered. We show that the suggested TILF approximation has all the advantages of one-sided autoregressive and ARMA modeling, allowing a high degree of confidence when fitting the metrology data with a limited number of parameters. Compared to ARMA modeling, the TILF approximation gains in terms of better fitting accuracy and the absence of the causality limitation. Moreover, the TILF approach can be directly generalized to two-dimensional random fields. With the determined model parameters, the surface topography of prospective beamline optics can be reliably forecast before they are fabricated. These forecast metrology data, containing essential and reliable statistical information about the existing optics which are fabricated by the same vendor and technology, but generally, have different sizes, and slope and height root-mean-square variations, are vitally needed for numerical simulations of the performance of new x-ray beamlines and those under upgrade.

Multivariate Spartan spatial random field models

This paper introduces a family of stationary multivariate spatial random fields with D scalar components that extend the scalar model of Gibbs random fields with local interactions (i.e., Spartan spatial random fields). We derive permissibility conditions for Spartan multivariate spatial random fields with a specific structure of local interactions. We also present explicit expressions for the respective matrix covariance functions obtained at the limit of infinite spectral cutoff in one, two and three spatial dimensions. Finally, we illustrate the proposed covariance models by means of simulated bivariate time series and two-dimensional random fields.

The Sequential Generation of Gaussian Random Fields for Applications in the Geospatial Sciences

This paper presents practical methods for the sequential generation or simulation of a Gaussian two-dimensional random field. The specific realizations typically correspond to geospatial errors or perturbations over a horizontal plane or grid. The errors are either scalar, such as vertical errors, or multivariate, such as , , and errors. These realizations enable simulation-based performance assessment and tuning of various geospatial applications. Both homogeneous and non-homogeneous random fields are addressed. The sequential generation is very fast and compared to methods based on Cholesky decomposition of an a priori covariance matrix and Sequential Gaussian Simulation. The multi-grid point covariance matrix is also developed for all the above random fields, essential for the optimal performance of many geospatial applications ingesting data with these types of errors.

Characterization of kinetic coarsening in a random-field Ising model.

We report a study of nonequilibrium relaxation in a two-dimensional random field Ising model at a nonzero temperature. We attempt to observe the coarsening from a different perspective with a particular focus on three dynamical quantities that characterize the kinetic coarsening. We provide a simple generalized scaling relation of coarsening supported by numerical results. The excellent data collapse of the dynamical quantities justifies our proposition. The scaling relation corroborates the recent observation that the average linear domain size satisfies different scaling behavior in different time regimes.

Asymptotic theory of cepstral random fields

Random fields play a central role in the analysis of spatially correlated data and, as a result, have a significant impact on a broad array of scientific applications. This paper studies the cepstral random field model, providing recursive formulas that connect the spatial cepstral coefficients to an equivalent moving-average random field, which facilitates easy computation of the autocovariance matrix. We also provide a comprehensive treatment of the asymptotic theory for two-dimensional random field models: we establish asymptotic results for Bayesian, maximum likelihood and quasi-maximum likelihood estimation of random field parameters and regression parameters. The theoretical results are presented generally and are of independent interest, pertaining to a wide class of random field models. The results for the cepstral model facilitate model-building: because the cepstral coefficients are unconstrained in practice, numerical optimization is greatly simplified, and we are always guaranteed a positive definite covariance matrix. We show that inference for individual coefficients is possible, and one can refine models in a disciplined manner. Our results are illustrated through simulation and the analysis of straw yield data in an agricultural field experiment.

Generalized variability response functions for two-dimensional elasticity problems

The ability to determine probabilistic characteristics of response quantities in structural mechanics (e.g. displacements, stresses) as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters. The concept of the variability response function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). Proofs have been provided for the existence of VRFs for linear and some nonlinear statically determinate beams. For statically indeterminate structures, the Monte Carlo based generalized variability response function (GVRF) methodology has been proposed recently as a generalization of the VRF concept to indeterminate linear and some nonlinear beams. The methodology computes GVRFs, which are analogous to VRFs for statically determinate structures, and evaluates their dependence (or lack thereof) on the PDF and SDF of the random field, thereby providing an estimate of the accuracy of the GVRF. In this paper, the GVRF methodology is extended to problems involving two-dimensional, linear continua whose stochasticity is characterized by statistically homogeneous random fields. After detailing the GVRF methodology for two-dimensional random fields, two numerical examples are provided: GVRFs are computed for the displacement response and for the effective compliance of linear plane stress systems.

Analysis of spanning avalanches in the two-dimensional nonequilibrium zero-temperature random-field Ising model

We present a numerical analysis of spanning avalanches in a two-dimensional (2D) nonequilibrium zero-temperature random field Ising model. Finite-size scaling analysis, performed for distribution of the average number of spanning avalanches per single run, spanning avalanche size distribution, average size of spanning avalanche, and contribution of spanning avalanches to magnetization jump, is augmented by analysis of spanning field (i.e., field triggering spanning avalanche), which enabled us to collapse averaged magnetization curves below critical disorder. Our study, based on extensive simulations of sufficiently large systems, reveals the dominant role of subcritical 2D-spanning avalanches in model behavior below and at the critical disorder. Other types of avalanches influence finite systems, but their contribution for large systems remains small or vanish.

Influence of Spatial Variability on Pavement Responses Using Latin Hypercube Sampling on Two-Dimensional Random Fields

Although uncertainties in material properties have been addressed in the design of flexible pavements, most current modeling techniques assume that pavement layers are homogeneous. The paper addresses the influence of the spatial variability of the resilient moduli of pavement layers by evaluating the effect of the variance and correlation length on the pavement responses to loading. The integration of the spatially varying log-normal random field with the finite-difference method has been achieved through an exponential autocorrelation function. The variation in the correlation length was found to have a marginal effect on the mean values of the critical strains and a noticeable effect on the standard deviation which decreases with decreases in correlation length. This reduction in the variance arises because of the spatial averaging phenomenon over the softer and stiffer zones generated because of spatial variability. The increase in the mean value of critical strains with decreasing correlation length, although minor, illustrates that pavement performance is adversely affected by the presence of spatially varying layers. The study also confirmed that the higher the variability in the pavement layer moduli, introduced through a higher value of coefficient of variation (COV), the higher the variability in the pavement response. The study concludes that ignoring spatial variability by modeling the pavement layers as homogeneous that have very short correlation lengths can result in the underestimation of the critical strains and thus an inaccurate assessment of the pavement performance.

Lyapunov Exponents for Particles Advected in Compressible Random Velocity Fields at Small and Large Kubo Numbers

We calculate the Lyapunov exponents describing spatial clustering of particles advected in one- and two-dimensional random velocity fields at finite Kubo numbers Ku (a dimensionless parameter characterising the correlation time of the velocity field). In one dimension we obtain accurate results up to Ku ~ 1 by resummation of a perturbation expansion in Ku. At large Kubo numbers we compute the Lyapunov exponent by taking into account the fact that the particles follow the minima of the potential function corresponding to the velocity field. The Lyapunov exponent is always negative. In two spatial dimensions the sign of the maximal Lyapunov exponent \lambda_1 may change, depending upon the degree of compressibility of the flow and the Kubo number. For small Kubo numbers we compute the first four non-vanishing terms in the small-Ku expansion of the Lyapunov exponents. By resumming these expansions we obtain a precise estimate of the location of the path-coalescence transition (where \lambda_1 changes sign) for Kubo numbers up to approximately Ku = 0.5. For large Kubo numbers we estimate the Lyapunov exponents for a partially compressible velocity field by assuming that the particles sample those stagnation points of the velocity field that have a negative real part of the maximal eigenvalue of the matrix of flow-velocity gradients.

An overview of a unified theory of dynamics of vehicle–pavement interaction under moving and stochastic load

This article lays out a unified theory for dynamics of vehicle–pavement interaction under moving and stochastic loads. It covers three major aspects of the subject: pavement surface, tire–pavement contact forces, and response of continuum media under moving and stochastic vehicular loads. Under the subject of pavement surface, the spectrum of thermal joints is analyzed using Fourier analysis of periodic function. One-dimensional and two-dimensional random field models of pavement surface are discussed given three different assumptions. Under the subject of tire–pavement contact forces, a vehicle is modeled as a linear system. At a constant speed of travel, random field of pavement surface serves as a stationary stochastic process exciting vehicle vibration, which, in turn, generates contact force at the interface of tire and pavement. The contact forces are analyzed in the time domain and the frequency domains using random vibration theory. It is shown that the contact force can be treated as a nonzero mean stationary process with a normal distribution. Power spectral density of the contact force of a vehicle with walking-beam suspension is simulated as an illustration. Under the subject of response of continuum media under moving and stochastic vehicular loads, both time-domain and frequency-domain analyses are presented for analytic treatment of moving load problem. It is shown that stochastic response of linear continuum media subject to a moving stationary load is a nonstationary process. Such a nonstationary stochastic process can be converted to a stationary stochastic process in a follow-up moving coordinate.

BETTI NUMBERS OF GAUSSIAN FIELDS

We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three- and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; $\beta_0$ is the number of connected regions, $\beta_1$ is the number of circular holes, and $\beta_2$ is the number of three-dimensional voids. Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. $\beta_0$ dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). $\beta_1$ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and $\beta_2$ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum $n$ in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as $n$ decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even though the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.

Sparse spectrum model for a turbulent phase

Monte Carlo (MC) simulation of phase front perturbations by atmospheric turbulence finds numerous applications for design and modeling of the adaptive optics systems, laser beam propagation simulations, and evaluating the performance of the various optical systems operating in the open air environment. Accurate generation of two-dimensional random fields of turbulent phase is complicated by the enormous diversity of scales that can reach five orders of magnitude in each coordinate. In addition there is a need for generation of the long "ribbons" of turbulent phase that are used to represent the time evolution of the wave front. This makes it unfeasible to use the standard discrete Fourier transform-based technique as a basis for the MC simulation algorithm. We propose a new model for turbulent phase: the sparse spectrum (SS) random field. The principal assumption of the SS model is that each realization of the random field has a discrete random spectral support. Statistics of the random amplitudes and wave vectors of the SS model are arranged to provide the required spectral and correlation properties of the random field. The SS-based MC model offers substantial reduction of computer costs for simulation of the wide-band random fields and processes, and is capable of generating long aperiodic phase "ribbons." We report the results of model trials that determine the number of sparse components, and the range of wavenumbers that is necessary to accurately reproduce the random field with a power-law spectrum.

Dynamical properties of random-field Ising model

Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter, and the spin-spin correlation functions are studied in the nonequilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two-dimensional random field Ising model possesses long-range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long-range order in the system.

Prediction of protein-RNA residue-base contacts using two-dimensional conditional random field with the lasso

BackgroundTo uncover molecular functions and networks in biological cellular systems, it is important to dissect interactions between proteins and RNAs. Many studies have been performed to investigate and analyze interactions between protein amino acid residues and RNA bases. In terms of interactions between residues in proteins, it is generally accepted that an amino acid residue at interacting sites has coevolved together with the partner residue in order to keep the interaction between residues in proteins. Based on this hypothesis, in our previous study to identify residue-residue contact pairs in interacting proteins, we made calculations of mutual information (M I) between amino acid residues from some multiple sequence alignment of homologous proteins, and combined it with a discriminative random field (DRF) approach, which is a special type of conditional random fields (CRFs) and has been proved useful for the purpose of extracting distinguishing areas from a photograph in the image processing field. Recently, the evolutionary correlation of interactions between residues and DNA bases has also been found in certain transcription factors and the DNA-binding sites.ResultsIn this paper, we employ more generic two-dimensional CRFs than such DRFs to predict interactions between protein amino acid residues and RNA bases. In addition, we introduce labels representing kinds of amino acids and bases as local features of a CRF. Furthermore, we examine the utility of L1-norm regularization (lasso) for the CRF. For evaluation of our method, we use residue-base interactions between several Pfam domains and Rfam entries, conduct cross-validation, and calculate the average AUC (Area under ROC Curve) score. The results suggest that our CRF-based method using mutual information and labels with the lasso is useful for further improving the performance, especially provided that the features of CRF are successfully reduced by the lasso approach.ConclusionsWe propose simple and generic two-dimensional CRF models using labels and mutual information with the lasso. Use of the CRF-based method in combination with the lasso is particularly useful for predicting the residue-base contacts in protein-RNA interactions.

투수계수의 공간적 변동성을 고려한 층상지반에 대한 확률론적 침투해석

본 연구에서는 수리구조물이 설치된 2층으로 이루어진 포화 기초지반에서의 구속흐름(confined flow)에 대하여 확률론적 침투해석을 수행하였다. 투수계수는 지반의 층상구조에 따라 명확한 변동성을 보일 뿐 아니라 각각의 층 내에서도 공간적인 변동성을 보인다. 따라서 기존의 결정론적 침투해석기법을 층상지반에서의 투수계수의 불확실성과 공간적 변동성을 고려할 수 있도록 확률론적 해석으로 확장하였다. 각 층에 지정된 입력 확률분포함수와 자기상관함수(autocorrelation function)를 따르는 2차원의 랜덤필드를 생성하기 위하여 Karhunen-Lo<TEX>$\grave{e}$</TEX>ve 전개법을 사용하였다. 제안된 절차의 적용성을 검토하고 수리구조물 하부의 2층 지반을 통한 흐름에 공간적 불균질성이 미치는 영향을 연구하기 위해 생성된 랜덤필드를 이용하여 Monte Carlo 시뮬레이션을 수행하였다. 해석결과는 층상지반에서의 침투거동 평가에서 지반의 층상구조와 지층내에서의 투수계수의 공간적 변동성에 의한 지반에서의 다양한 침투패턴을 확률론적 해석기법을 통하여 효율적으로 고려할 수 있음을 보여주었다. In this study, probabilistic analysis of seepage through a two-layered soil foundation was performed. The hydraulic conductivity of soil shows significant spatial variations in different layers because of stratification; further, it varies on a smaller scale within each individual layer. Therefore, the deterministic seepage analysis method was extended to develop a probabilistic approach that accounts for the uncertainties and spatial variation of the hydraulic conductivity in a layered soil profile. Two-dimensional random fields were generated on the basis of the Karhunen-Lo<TEX>$\grave{e}$</TEX>ve expansion in a manner consistent with a specified marginal distribution function and an autocorrelation function for each layer. A Monte Carlo simulation was then used to determine the statistical response based on the random fields. A series of analyses were performed to verify the application potential of the proposed method and to study the effects of uncertainty due to the spatial heterogeneity on the seepage behavior of two-layered soil foundation beneath water retaining structure. The results showed that the probabilistic framework can be used to efficiently consider the various flow patterns caused by the spatial variability of the hydraulic conductivity in seepage assessment for a layered soil foundation.

Influence of Spatially Variable Geotechnical Properties of MSW on Stability of Landfill Slopes

AbstractProbabilistic slope stability analysis of the municipal solid waste (MSW) landfill slope (height = 30 m and 1V∶3H) with due consideration to the spatial variation of geotechnical properties of MSW (single depth layer versus multilayered depth) is performed. The published data on geotechnical properties of MSW is used to define statistics of spatial variation of geotechnical parameters. Random field theory combined with finite difference numerical code, fast Lagrangian analysis of continua (FLAC) is used. Two-dimensional non-Gaussian homogeneous random field is generated by Cholesky decomposition technique. Monte Carlo simulations are performed to determine the statistics of the stability of the MSW landfill slope, evaluated in terms of factor of safety, and information is utilized to assess the performance in probabilistic framework, i.e., reliability index (β). The results of the analysis are compared and discussed in the light of conventional factor of safety approach, in which the geotechnical ...

Simulation of geometric imperfections in cold-formed steel members using spectral representation approach

In this paper, a summary of the available imperfection measurements for cold-formed steel members is presented. Three methods to simulate imperfection fields are introduced: the first is the classical approach employing a superposition of eigenmode imperfections, but scaled to match peaks in the measured physical measurements. The second is a method based on the multi-dimensional spectral representation method, in which imperfections are considered as a two-dimensional random field and simulations are performed taking a spectra-based approach. The third is a novel combination of modal approaches and spectral representation that directly considers the frequency content of the imperfection field, but employs a spectral representation method driven by the cross-sectional eigenmode shapes to generate the imperfection fields. The effect of these different approaches on the simulated strength and collapse behavior of members is investigated using material and geometric nonlinear finite element collapse modeling. The third imperfection generation method, termed the 1D Modal Spectra Method, provides an intriguing new tool in the simulation of thin-walled members.

Digital simulation of two-dimensional random fields with arbitrary power spectra and non-Gaussian probability distribution functions

Methods for simulation of two-dimensional signals with arbitrary power spectral densities and signal amplitude probability density functions are disclosed. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In most cases the method provides satisfactory results and can thus be considered an engineering approach. Several illustrative examples with relevance for optics are given.

The bispectrum covariance beyond Gaussianity

To investigate and specify the statistical properties of cosmological fields with particular attention to possible non-Gaussian features, accurate formulae for the bispectrum and the bispectrum covariance are required. The bispectrum is the lowest-order statistic providing an estimate for non-Gaussianities of a distribution, and the bispectrum covariance depicts the errors of the bispectrum measurement and their correlation on different scales. Currently, there do exist fitting formulae for the bispectrum and an analytical expression for the bispectrum covariance, but the former is not very accurate and the latter contains several intricate terms and only one of them can be readily evaluated from the power spectrum of the studied field. Neglecting all higher-order terms results in the Gaussian approximation of the bispectrum covariance. We study the range of validity of this Gaussian approximation for two-dimensional non-Gaussian random fields. For this purpose, we simulate Gaussian and non-Gaussian random fields, the latter represented by log-normal fields and obtained directly from the former by a simple transformation. From the simulated fields, we calculate the power spectra, the bispectra, and the covariance from the sample variance of the bispectra, for different degrees of non-Gaussianity \alpha, which is equivalent to the skewness on a given angular scale \theta g. We find that the Gaussian approximation provides a good approximation for \alpha<0.6 and a reasonably accurate approximation for \alpha< 1, both on scales >8\theta g. Using results from cosmic shear simulations, we estimate that the cosmic shear convergence fields are described by \alpha<0.7 at \theta g~4". We therefore conclude that the Gaussian approximation for the bispectrum covariance is likely to be applicable in ongoing and future cosmic shear studies.