Two-dimensional Random Field Research Articles

Two-Dimensional Random Field Conditioned by CPT Data: Case Study

The presented geotechnical random field (GRF) algorithm is utilized to realize the soil inherent variability based on in situ site investigation results. This paper aims at investigating the effect of cone penetration test (CPT) borehole arrangements in horizontal plane on slope stability reliability analysis considering stochastic fields of soil design parameters via the conventional stationary along with the non-stationary random field adopted in the GRF approach. An advantage of the GRF algorithm is that it yields conservative results in comparison to the conventional stationary random field theory; therefore, using non-stationary random field underestimates slope stability reliability. The current study tried to highlight the importance of the availability of in situ profiles of soil strength parameters when stability analysis of geotechnical structures is concerned. Slope stability was chosen to interpret the role of the CPT profiles in the adjacent site in determining the factor of safety of the slope against shear failure. Furthermore, the results of the case under study showed that when the geometric dimensions of the slope are too small compared to the horizontal correlation length, more CPT borehole data availability leads only to very slightly conservative results in comparison to the case with single available CPT profile.

Numerical simulation of random fields using correlated random vector and the Karhunen-Loève expansion

This paper presents an approach of one- and two-dimensional random field simulation methods using a correlated random vector and the Karhunen–Loeve expansion. Comparison of the authors’ analytical solution of the Fredholm integral equation of the second kind with the numerical solution using the finite element method and the inverse vector iteration technique is presented. Numerical approach and sample realizations of one- and two-dimensional random fields are presented using described techniques as well as generated probability distribution functions for chosen point of the analysed domain.

Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields

In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.

A Note on Exponential Decay in the Random Field Ising Model

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (1) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (2) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kert\'{e}sz line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.

A separable two-dimensional random field model of binary response data from multi-day behavioral experiments

BackgroundThe study of learning in populations of subjects can provide insights into the changes that occur in the brain with aging, drug intervention, and psychiatric disease. New methodWe introduce a separable two-dimensional (2D) random field (RF) model for analyzing binary response data acquired during the learning of object-reward associations across multiple days. The method can quantify the variability of performance within a day and across days, and can capture abrupt changes in learning. ResultsWe apply the method to data from young and aged macaque monkeys performing a reversal-learning task. The method provides an estimate of performance within a day for each age group, and a learning rate across days for each monkey. We find that, as a group, the older monkeys require more trials to learn the object discriminations than do the young monkeys, and that the cognitive flexibility of the younger group is higher. We also use the model estimates of performance as features for clustering the monkeys into two groups. The clustering results in two groups that, for the most part, coincide with those formed by the age groups. Simulation studies suggest that clustering captures inter-individual differences in performance levels. Comparison with existing method(s)In comparison with generalized linear models, this method is better able to capture the inherent two-dimensional nature of the data and find between group differences. ConclusionsApplied to binary response data from groups of individuals performing multi-day behavioral experiments, the model discriminates between-group differences and identifies subgroups.

Polarization monotones of two-dimensional and three-dimensional random electromagnetic fields

We propose a formal resource-theoretic approach to quantify the degree of polarization of two and three-dimensional random electromagnetic fields. This endows the space of spectral polarization matrices with the orders induced by majorization or convex mixing that naturally recover the best-known polarization measures.

Estimating a Separably Markov Random Field from Binary Observations.

A fundamental problem in neuroscience is to characterize the dynamics of spiking from the neurons in a circuit that is involved in learning about a stimulus or a contingency. A key limitation of current methods to analyze neural spiking data is the need to collapse neural activity over time or trials, which may cause the loss of information pertinent to understanding the function of a neuron or circuit. We introduce a new method that can determine not only the trial-to-trial dynamics that accompany the learning of a contingency by a neuron, but also the latency of this learning with respect to the onset of a conditioned stimulus. The backbone of the method is a separable two-dimensional (2D) random field (RF) model of neural spike rasters, in which the joint conditional intensity function of a neuron over time and trials depends on two latent Markovian state sequences that evolve separately but in parallel. Classical tools to estimate state-space models cannot be applied readily to our 2D separable RF model. We develop efficient statistical and computational tools to estimate the parameters of the separable 2D RF model. We apply these to data collected from neurons in the prefrontal cortex in an experiment designed to characterize the neural underpinnings of the associative learning of fear in mice. Overall, the separable 2D RF model provides a detailed, interpretable characterization of the dynamics of neural spiking that accompany the learning of a contingency.

On the Decay of Correlations in the Random Field Ising Model

In a celebrated 1990 paper, Aizenman and Wehr proved that the two-dimensional random field Ising model has a unique infinite volume Gibbs state at any temperature. The proof is ergodic-theoretic in nature and does not provide any quantitative information. This article proves the first quantitative version of the Aizenman-Wehr theorem. The proof introduces a new method for proving decay of correlations that may be interesting in its own right. A fairly detailed sketch of the main ideas behind the proof is also included.

Anomalous transport in heterogeneous media

The diffusion dynamics of particles in heterogeneous media is studied using particle-based simulation techniques. A special focus is placed on systems where the transport of particles at long times exhibits anomalies such as subdiffusive or superdiffusive behavior. First, a two-dimensional model system is considered containing gas particles (tracers) that diffuse through a random arrangement of pinned, disk-shaped particles. This system is similar to a classical Lorentz gas. However, different from the original Lorentz model, soft instead of hard interactions are considered and we also discuss the case where the tracer particles interact with each other. We show that the modification from hard to soft interactions strongly affects anomalous-diffusive transport at high obstacle densities. Second, non-linear active micro-rheology in a glass-forming binary Yukawa mixture is investigated, pulling single particles through a deeply supercooled state by applying a constant force. Here, we observe superdiffusion in force direction and analyze its origin. Finally, we consider the Brownian dynamics of a particle which is pulled through a two-dimensional random force field. We discuss the similarities of this model with the Lorentz gas as well as active micro-rheology in glass-forming systems.

Critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model on a triangular lattice.

We present a numerical study of the critical behavior of the nonequilibrium zero-temperature random field Ising model in two dimensions on a triangular lattice. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations of systems with linear sizes up to L=65536, show that the model is in a different universality class than the same model on a quadratic lattice, which is relevant for a better understanding of model universality and the analysis of experimental data.

Modeling and statistical analysis of non-Gaussian random fields with heavy-tailed distributions.

In this paper, we investigate and develop an alternative approach to the numerical analysis and characterization of random fluctuations with the heavy-tailed probability distribution function (PDF), such as turbulent heat flow and solar flare fluctuations. We identify the heavy-tailed random fluctuations based on the scaling properties of the tail exponent of the PDF, power-law growth of qth order correlation function, and the self-similar properties of the contour lines in two-dimensional random fields. Moreover, this work leads to a substitution for the fractional Edwards-Wilkinson (EW) equation that works in the presence of μ-stable Lévy noise. Our proposed model explains the configuration dynamics of the systems with heavy-tailed correlated random fluctuations. We also present an alternative solution to the fractional EW equation in the presence of μ-stable Lévy noise in the steady state, which is implemented numerically, using the μ-stable fractional Lévy motion. Based on the analysis of the self-similar properties of contour loops, we numerically show that the scaling properties of contour loop ensembles can qualitatively and quantitatively distinguish non-Gaussian random fields from Gaussian random fluctuations.

Non-parametric approximations for anisotropy estimation in two-dimensional differentiable Gaussian random fields

Spatially referenced data often have autocovariance functions with elliptical isolevel contours, a property known as geometric anisotropy. The anisotropy parameters include the tilt of the ellipse (orientation angle) with respect to a reference axis and the aspect ratio of the principal correlation lengths. Since these parameters are unknown a priori, sample estimates are needed to define suitable spatial models for the interpolation of incomplete data. The distribution of the anisotropy statistics is determined by a non-Gaussian sampling joint probability density. By means of analytical calculations, we derive an explicit expression for the joint probability density function of the anisotropy statistics for Gaussian, stationary and differentiable random fields. Based on this expression, we obtain an approximate joint density which we use to formulate a statistical test for isotropy. The approximate joint density is independent of the autocovariance function and provides conservative probability and confidence regions for the anisotropy parameters. We validate the theoretical analysis by means of simulations using synthetic data, and we illustrate the detection of anisotropy changes with a case study involving background radiation exposure data. The approximate joint density provides (i) a stand-alone approximate estimate of the anisotropy statistics distribution (ii) informed initial values for maximum likelihood estimation, and (iii) a useful prior for Bayesian anisotropy inference.

Polynomial chaos and scaling limits of disordered systems

Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d -dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O’Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant , including the disordered pinning model, the (long-range) directed polymer model in dimension 1 + 1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

Method of lossless image compression in the Earth remote sensing systems

A method of lossless digital image compression is presented in the paper. Its implementation demands minimum computing resources and it can be efficiently used in Earth remote sensing systems. The method makes possible maximum removal of statistical redundancy of images without spectral transformations. A digital image is resolved into bit planes which are processed independently of one another. That allows their parallel processing without any restriction on the pixel length of the image. Each bit plane is approximated by a two-dimensional Markov random field with discrete values. Prediction of elements of the bit plane is accomplished on the basis of the theory of conventional Markov processes. Only operations of logical comparison are used for the prediction, whereas computational operations such as multiplication and addition are not required, which makes it possible to increase the speed of image processing. The research conducted shows that the method is not inferior to known methods of lossless compression in its efficiency, while being considerably superior to them in the speed of processing. The method is applicable both to panchromatic and color images and can be used in any systems of digital image processing.

Data-worth analysis through probabilistic collocation-based Ensemble Kalman Filter

We propose a new and computationally efficient data-worth analysis and quantification framework keyed to the characterization of target state variables in groundwater systems. We focus on dynamically evolving plumes of dissolved chemicals migrating in randomly heterogeneous aquifers. An accurate prediction of the detailed features of solute plumes requires collecting a substantial amount of data. Otherwise, constraints dictated by the availability of financial resources and ease of access to the aquifer system suggest the importance of assessing the expected value of data before these are actually collected. Data-worth analysis is targeted to the quantification of the impact of new potential measurements on the expected reduction of predictive uncertainty based on a given process model. Integration of the Ensemble Kalman Filter method within a data-worth analysis framework enables us to assess data worth sequentially, which is a key desirable feature for monitoring scheme design in a contaminant transport scenario. However, it is remarkably challenging because of the (typically) high computational cost involved, considering that repeated solutions of the inverse problem are required. As a computationally efficient scheme, we embed in the data-worth analysis framework a modified version of the Probabilistic Collocation Method-based Ensemble Kalman Filter proposed by Zeng et al. (2011) so that we take advantage of the ability to assimilate data sequentially in time through a surrogate model constructed via the polynomial chaos expansion. We illustrate our approach on a set of synthetic scenarios involving solute migrating in a two-dimensional random permeability field. Our results demonstrate the computational efficiency of our approach and its ability to quantify the impact of the design of the monitoring network on the reduction of uncertainty associated with the characterization of a migrating contaminant plume.

Influence of the aspect ratio and boundary conditions on universal finite-size scaling functions in the athermal metastable two-dimensional random field Ising model.

This work studies universal finite size scaling functions for the number of one-dimensional spanning avalanches in a two-dimensional (2D) disordered system with boundary conditions of different nature and different aspect ratios. To this end, we will consider the 2D random field Ising model at T=0 driven by the external field H with athermal dynamics implemented with periodic and forced boundary conditions. We have chosen a convenient scaling variable z that accounts for the deformation of the distance to the critical point caused by the aspect ratio. In addition, assuming that the dependence of the finite size scaling functions on the aspect ratio can be accounted for by an additional multiplicative factor, we have been able to collapse data for different system sizes, different aspect ratios, and different types of the boundary conditions into a single scaling function Q̂.

Дифузія частинок у двовимірному випадковому полі швидкостi

Дифузія частинок у двовимірному випадковому полі швидкостi

Spatial time-dependent reliability analysis of reinforced concrete slab bridges subject to realistic traffic loading

Resistance and loads are often correlated in time and space. The paper assesses the influence of these correlations on structural reliability/probability of failure for a typical two-lane reinforced concrete (RC) slab bridge under realistic traffic loading. Spatial variables for structural resistance are cover and concrete compressive strength, which in turn affect the strength and chloride-induced corrosion of RC elements. Random variables include pit depth and model error. Correlation of weights between trucks in adjacent lanes and inter-vehicle gaps are also included and are calibrated against weigh-in-motion data. Reliability analysis of deteriorating bridges needs to incorporate uncertainties associated with parameters governing the deterioration process and loading. One of the major unanswered questions in the work carried out to date is the influence of spatial variability of load and resistance on failure probability. Spatial variability research carried out to date has been mainly focused on predicting the remaining lifetime of a corroding structure and spatial variability of material, dimensional and environmental properties. A major shortcoming in the work carried out to date is the lack of an allowance for the spatial variability of applied traffic loads. In this article, a two-dimensional (2D) random field is developed where load effects and time-dependent structural resistance are calculated for each segment in the field. The 2D spatial time-dependent reliability analysis of an RC slab bridge found that a spatially correlated resistance results in only a small increase in probability of failure. Despite the fact that load effect at points along the length of a bridge is strongly correlated, the combined influence of correlation in load and resistance on probability of failure is small.

Time-Dependent System Reliability Analysis Using Random Field Discretization

This paper proposes a novel and efficient methodology for time-dependent system reliability analysis of systems with multiple limit-state functions of random variables, stochastic processes, and time. Since there are correlations and variations between components and over time, the overall system is formulated as a random field with two dimensions: component index and time. To overcome the difficulties in modeling the two-dimensional random field, an equivalent Gaussian random field is constructed based on the probability equivalency between the two random fields. The first-order reliability method (FORM) is employed to obtain important features of the equivalent random field. By generating samples from the equivalent random field, the time-dependent system reliability is estimated from Boolean functions defined according to the system topology. Using one system reliability analysis, the proposed method can get not only the entire time-dependent system probability of failure curve up to a time interval of interest but also two other important outputs, namely, the time-dependent probability of failure of individual components and dominant failure sequences. Three examples featuring series, parallel, and combined systems are used to demonstrate the effectiveness of the proposed method.

Metal–insulator transition in iron doped 2H-TaSe2: Suggestion of chiral unitary localization

We investigated the conduction of electrons in a quasi two-dimensional random magnetic field (RMF) system, namely iron-doped transition metal dichalcogenide 2H-FexTa1−xSe2. We observed a clear metal–insulator transition by increasing the Fe doping rate. The temperature dependence of the resistances is consistent with the unitary class of Anderson localization. However, the observed critical sheet resistance is between 2 and 19kΩ, which is inconsistent with the unitary class (2h/e2=51.6kΩ). We propose that chiral symmetry can be present in 2H-FexTa1−xSe2 because the magnetoresistance of the crystals is similar to that of dirty graphites.