Random Field Model Research Articles

A formal goodness-of-fit test for spatial binary Markov random field models.

Binary spatial observations arise in environmental and ecological studies, where Markov random field (MRF) models are often applied. Despite the prevalence and the long history of MRF models for spatial binary data, appropriate model diagnostics have remained an unresolved issue in practice. A complicating factor is that such models involve neighborhood specifications, which are difficult to assess for binary data. To address this, we propose a formal goodness-of-fit (GOF) test for diagnosing an MRF model for spatial binary values. The test statistic involves a type of conditional Moran's I based on the fitted conditional probabilities, which can detect departures in model form, including neighborhood structure. Numerical studies show that the GOF test can perform well in detecting deviations from a null model, with a focus on neighborhoods as a difficult issue. We illustrate the spatial test with an application to Besag's historical endive data as well as the breeding pattern of grasshopper sparrows across Iowa.

Statistical batch-aware embedded integration, dimension reduction, and alignment for spatial transcriptomics.

Spatial transcriptomics (ST) technologies provide richer insights into the molecular characteristics of cells by simultaneously measuring gene expression profiles and their relative locations. However, each slice can only contain limited biological variation, and since there are almost always non-negligible batch effects across different slices, integrating numerous slices to account for batch effects and locations is not straightforward. Performing multi-slice integration, dimensionality reduction, and other downstream analyses separately often results in suboptimal embeddings for technical artifacts and biological variations. Joint modeling integrating these steps can enhance our understanding of the complex interplay between technical artifacts and biological signals, leading to more accurate and insightful results. In this context, we propose a hierarchical hidden Markov random field model STADIA to reduce batch effects, extract common biological patterns across multiple ST slices, and simultaneously identify spatial domains. We demonstrate the effectiveness of STADIA using five datasets from different species (human and mouse), various organs (brain, skin, and liver), and diverse platforms (10x Visium, ST, and Slice-seqV2). STADIA can capture common tissue structures across multiple slices and preserve slice-specific biological signals. In addition, STADIA outperforms the other three competing methods (PRECAST, fastMNN, and Harmony) in terms of the balance between batch mixing and spatial domain identification, and it demonstrates the advantage of joint modeling when compared to STAGATE and GraphST. The source code implemented by R is available at https://github.com/zhanglabtools/STADIA and archived with version 1.01 on Zenodo https://zenodo.org/records/13637744.

A High-Resolution and Robust Microwave Correlation Imaging Method Based on URRF Using MC-AAMPE Algorithm

This manuscript presents a novel framework for high-resolution and robust microwave correlation imaging. In order to generate a more diverse random radiation field distribution, the unified random radiation field (URRF) model is proposed. The URRF model can accurately characterize the joint random modulation in the signals’ phase, amplitude, and frequency. Furthermore, we build a parametric imaging model based on URRF which clearly describes the relationship between the image to be reconstructed and the signals by the URRF model. By using this imaging model, the reconstruction of an image is converted into solving a multi-parameter optimization problem with multiple constraints. To solve this optimization problem with high efficiency and accuracy, the model-constrained adaptive alternating multiple parameter estimation (MC-AAMPE) algorithm is proposed. This algorithm decomposes the high-dimensional multi-parameter optimization problem into several sub-optimization problems. The renewing solutions to these sub-optimization problems make the multi-parameter optimization converge to the image of the target and the parameters of clutter and noise, which are all unknown before the solution. In comparison with the existing methods, the proposed scheme generates images with higher resolution and is more robust under noise conditions. Extensive simulation experiments confirmed the effectiveness and robustness of the proposed method.

A Markov random field model for change points detection

Detecting Change Points (CPs) in data sequences is a challenging problem that arises in a variety of disciplines, including signal processing and time series analysis. While many methods exist for PieceWise Constant (PWC) signals, relatively fewer address PieceWise Linear (PWL) signals due to the challenge of preserving sharp transitions. This paper introduces a Markov Random Field (MRF) model for detecting changes in slope. The number of CPs and their locations are unknown. The proposed method incorporates PWL prior information using MRF framework with an additional boolean variable called Line Process (LP), describing the presence or absence of CPs. The solution is then estimated in the sense of maximum a posteriori. The LP allows us to define a non-convex non-smooth energy function that is algorithmically hard to minimize. To tackle the optimization challenge, we propose an extension of the combinatorial algorithm DPS, initially designed for CP detection in PWC signals. Also, we present a shared memory implementation to enhance computational efficiency. Numerical studies show that the proposed model produces competitive results compared to the state-of-the-art methods. We further evaluate the performance of our method on three real datasets, demonstrating superior and accurate estimates of the underlying trend compared to competing methods.

BGAT-CCRF: A novel end-to-end model for knowledge graph noise correction

Knowledge graph (KG) noise correction aims to select suitable candidates to correct the noises in KGs. Most of the existing studies have limited performance in repairing the noisy triple that contains more than one incorrect entity or relation, which significantly constrains their implementation in real-world KGs. To overcome this challenge, we propose a novel end-to-end model (BGAT-CCRF) that achieves better noise correction results. Specifically, we construct a balanced-based graph attention model (BGAT) to learn the features of nodes in triples’ neighborhoods and capture the correlation between nodes based on their position and frequency. Additionally, we design a constrained conditional random field model (CCRF) to select suitable candidates guided by three constraints for correcting one or more noises in the triple. In this way, BGAT-CCRF can select multiple candidates from a smaller domain to repair multiple noises in triples simultaneously, rather than selecting candidates from the whole KG to repair noisy triples as traditional methods do, which can only repair one noise in the triple at a time. The effectiveness of BGAT-CCRF is validated by the KG noise correction experiment. Compared with the state-of-the-art models, BGAT-CCRF improves the fMRR metric by 3.58% on the FB15K dataset. Hence, it has the potential to facilitate the implementation of KGs in the real world.

The impact of crystal grain size on the behavior of disordered ferromagnetic systems: from thin to bulk geometry

We report the findings of an extensive and systematic study on the effect of crystal grain size on the response of field-driven disordered ferromagnetic systems with thin, intermediate, and bulk geometry. For numerical modeling we used the athermal nonequilibrium variant of the random field Ising model simulating the systems with tightly packed and uniformly cubic-shaped, magnetically exchange-coupled crystal grains, conducted over a wide range of grain sizes. Together with the standard hysteresis loop characterizations, we offer an in-depth examination of the avalanching response of the system, estimating the effective grain-size-related exponents by analyses of the distributions of various avalanche parameters, average avalanche shape and size, and power spectra. Our results demonstrate that grain size plays an important role in the behavior of the system, outweighing the effect of its geometry. For sufficiently small grains, the characteristics of the system response are largely unaffected by grain size; however, for larger grains, the effects become more noticeable and show up as distinct asymmetry in the magnetization susceptibilities and average avalanche shapes, as well as characteristic kinks in the distributions of avalanche parameters, susceptibilities, and magnetizations for the largest grain sizes. Our insights, unveiling the sensitivity of the system’s response to the underlying structure in terms of crystal grain size, may prove beneficial in interpreting and analyzing experimental results obtained from driven disordered ferromagnetic samples of different geometries, as well as in extending the range of possible applications.

Jamming is a first-order transition with quenched disorder in amorphous materials sheared by cyclic quasistatic deformations

Jamming is an athermal transition between flowing and rigid states in amorphous systems such as granular matter, colloidal suspensions, complex fluids and cells. The jamming transition seems to display mixed aspects of a first-order transition, evidenced by a discontinuity in the coordination number, and a second-order transition, indicated by power-law scalings and diverging lengths. Here we demonstrate that jamming is a first-order transition with quenched disorder in cyclically sheared systems with quasistatic deformations, in two and three dimensions. Based on scaling analyses, we show that fluctuations of the jamming density in finite-sized systems have important consequences on the finite-size effects of various quantities, resulting in a square relationship between disconnected and connected susceptibilities, a key signature of the first-order transition with quenched disorder. This study puts the jamming transition into the category of a broad class of transitions in disordered systems where sample-to-sample fluctuations dominate over thermal fluctuations, suggesting that the nature and behavior of the jamming transition might be better understood within the developed theoretical framework of the athermally driven random-field Ising model.

Polynomial chaos expansion vs. Monte Carlo simulation in a stochastic analysis of wave propagation

The paper deals with the propagation of a stochastic wave in an elastic medium using the example of a seismic wave in a ground medium. Identification of subsoil parameters is never exact or complete which justifies the use of random field models or random variable models for input data; thus, the response of the subsoil is also random. In this paper and in the context of random variables, the focus is on a sensitivity analysis addressing the question of how the uncertainty of the input data (subgrade parameters) influences the obtained results (displacements). Two different methods of stochastic analysis are presented—the intrusive polynomial chaos approach supported by the Galerkin projection and Monte Carlo simulation—and compared by using an example of wave propagation in the elastic half-plane. Consistency in the results of both approaches has been achieved; however, the calculation efficiencies differ. The advantages and disadvantages of both approaches are discussed. The upper subsoil layer influences the variances of the random solutions much more than does the lower layer.

Adaptive Gaussian Markov random fields for child mortality estimation.

The under-5 mortality rate (U5MR), a critical health indicator, is typically estimated from household surveys in lower and middle income countries. Spatio-temporal disaggregation of household survey data can lead to highly variable estimates of U5MR, necessitating the usage of smoothing models which borrow information across space and time. The assumptions of common smoothing models may be unrealistic when certain time periods or regions are expected to have shocks in mortality relative to their neighbors, which can lead to oversmoothing of U5MR estimates. In this paper, we develop a spatial and temporal smoothing approach based on Gaussian Markov random field models which incorporate knowledge of these expected shocks in mortality. We demonstrate the potential for these models to improve upon alternatives not incorporating knowledge of expected shocks in a simulation study. We apply these models to estimate U5MR in Rwanda at the national level from 1985 to 2019, a time period which includes the Rwandan civil war and genocide.

Reliability analysis of deep tunnels in spatially varying brittle rocks using interval and random field modelling

Rock properties are estimated using objective lab/in-situ testing and subjective judgements invoking different types of uncertainties, i.e., aleatory, and epistemic, along them, often indicated by varying information levels. This study presents a unified reliability method to integrate the spatial variability of inputs modelled via alternate uncertainty models (intervals and probability-boxes (p-boxes)) with those modelled as stochastic variables via probability distributions. The methodology employs advanced Karhunen–Loève decomposition to generate interval and random fields of inputs modelled via alternate and stochastic models, respectively. Input properties are allocated to the zones of the numerical model in Fast Lagrangian Analysis of Continua-2D (FLAC-2D) based on their spatial dependency and correlation functions through a developed MATLAB-FLAC coupled code. The methodology is demonstrated for a deep tunnel in Canada to be constructed along a massive rock prone to brittle failures. Intact rock properties are modelled as stochastic variables due to objective estimation, while Geological Strength Index (GSI) and deformation modulus are modelled using alternate models (interval and p-box, respectively) due to subjective and hybrid estimation (double uncertainty propagation algorithm). The results of the methodology are compared with those of traditional deterministic and random field methods. The methodology reduces the subjectivity invoked by including unavailable additional information (e.g., assuming probability distributions of inputs based on literature) and propagates the originally available information of inputs accurately. The final outputs are the p-boxes of response parameters (i.e., displacements and damage zone) instead of their fixed values (deterministic analysis) and probability distributions (traditional reliability analysis), indicating the propagation of both impreciseness and variability of inputs by the method. For this case study, the p-boxes of outputs were bounding their values/distributions estimated via traditional analyses, verifying the accuracy of the methodology. Further, the impreciseness in the outputs, highest in the damage zone extent, was due to imprecision in the estimated GSI.

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

Maximum of the Gaussian Interface Model in Random External Fields

Maximum of the Gaussian Interface Model in Random External Fields

Real-time Denoising Algorithm for STEM Imaging Using Markov Random Field Model

Real-time Denoising Algorithm for STEM Imaging Using Markov Random Field Model

Hysteresis-loop phenomena in disordered ferromagnets with demagnetizing field and finite temperature.

Using numerical simulations, we investigate the impact of the demagnetization field and finite temperature on the hysteresis phenomena in disordered ferromagnetics systems. We model the behavior of thin systems employing the thermal nonequilibrium random field Ising model driven by a finite-driving rate protocol to study the shape of the hysteresis loop and demagnetization line and the magnetization fluctuations for varied parameters. Our results reveal a significant interplay of the disorder, the demagnetizing fields, and thermal fluctuations. In particular, at a fixed disorder and temperature, the increasing demagnetization coefficient gradually prologues the magnetization reversal process, altering the multifractal nature of the magnetization fluctuations. The process accompanies the appearance of extended linear segments in the hysteresis loop and changes in the demagnetization line while practically preserving the value of the coercive field and slightly changing the remanent magnetization. On the other hand, increasing temperature expands the system's response fluctuations and narrows the loop, affecting the coercive field and remanent magnetization. The interplay of thermal fluctuations and demagnetizing fields fully manifests in limiting the large-scale magnetization fluctuations, revealed by the multifractal spectra and the scaling functions of the avalanches. Our research offers new modeling perspectives with a more realistic scenario and provides insight into new hysteresis loop phenomena relevant to theoretical analysis and applications.

A multiobjective optimization framework for site investigation program based on Bayesian approach and NSGA‐II

AbstractSite investigation provides essential geotechnical parameter information for analysis and design. However, three conflicting objectives, namely exploration effort, robustness and parameter uncertainty, pose a challenge to the development of an optimal site investigation program. In this study, a three objective optimization framework for the site investigation program is proposed based on the Bayesian approach and the non‐dominated sorting genetic algorithm (NSGA‐II). The only inputs required by the proposed framework are prior distribution of geotechnical parameters and error information. The prior distribution of geotechnical parameters is derived from integrating engineering experience and measurements from basic exploration boreholes. The error information is obtained based on literature and expert judgment related to the specific project. Firstly, a design pool of candidate investigation programs is generated using Bayesian approach to determine the locations and number of exploration boreholes. The NSGA‐II is then applied to identify the optimal program that balances lower cost, higher robustness, and lower uncertainty. The proposed multiobjective optimization framework is illustrated and validated through a real site investigation case in Chongqing, China, aimed at determining the ultimate bearing capacity of the rock foundation. The spatial correlation of parameters within the study area is also considered. The optimal program is represented by the location and number of exploration boreholes. By comparing measurements with predictions from different site investigation programs, the efficiency of the proposed multiobjective framework is demonstrated. Additionally, the influence of engineering experience and random field modeling on the investigation program is discussed.

Spin cones in random-field XY models.

We determine the arrangement of spins in the ground state of the XY model with quenched, random fields, on a fully connected graph. Two types of disordered fields are considered, namely, randomly oriented magnetic fields and randomly oriented crystal fields. Orientations are chosen from a uniformly isotropic distribution, but disorder fluctuations in each realization of a finite system lead to a breaking of rotational symmetry. The result is an interesting pattern of spin orientations found by solving a system of coupled, nonlinear equationswithin perturbation theory and also by exact numerical continuation. All spins lie within a cone for small enough ratio of field to coupling strength, with an interesting distribution of spin orientations, with peaks at the cone edges. The orientation of the cone depends strongly on the realization of disorder, but the opening angle does not. In the case of random magnetic fields, the cone angle widens as the ratio increases till a critical value at which there is a first-order phase transition and the cone disappears. With random crystal fields, there is no phase transition and the cone angle approaches 180^{∘} for large values of the ratio. At finite low temperatures, Monte Carlo simulations show that the formation of a cone and its subsequent alignment along the equilibrium direction occur on two different timescales.

Random Field Ising Model Criticality in a Complex Binary Liquid System.

While Ising criticality in classical liquids has been firmly established both theoretically and experimentally, much less is known about criticality in liquids in which the growth of the correlation length is frustrated by finite-size effects. A theoretical approach for dealing with this issue is the random-field Ising model (RFIM). While experimental critical-exponent values have been reported for magnetic samples (here, we consider γ, ν and η), little experimental information is available for critical fluctuations in corresponding liquid systems. In this paper, we present a study on a binary liquid consisting of 3-methyl pyridine and heavy water in a very light-weight porous gel. We find that the experimental results are in agreement with the theoretical predictions from the RFIM.

Bethe M-layer construction on the Ising model

In statistical physics, one of the standard methods to study second order phase transitions is the renormalization group that usually leads to an expansion around the corresponding fully connected solution. Unfortunately, often in disordered models, some important finite dimensional second-order phase transitions are qualitatively different or absent in the corresponding fully connected model: in such cases the standard expansion fails. Recently, a new method, the M-layer one, has been introduced that performs an expansion around a different soluble mean field model: the Bethe lattice one. This new method has been already used to compute the upper critical dimension DU of different disordered systems such as the Random Field Ising model or the Spin glass model with field. If then one wants to go beyond and construct an expansion around DU to understand how critical quantities get renormalized, the actual computation of all the numerical factors is needed. This next step has still not been performed, being technically more involved. In this paper we perform this computation for the ferromagnetic Ising model without quenched disorder, in finite dimensions: we show that, at one-loop order inside the M-layer approach, we recover the continuum quartic field theory and we are able to identify the coupling constant g and the other parameters of the theory, as a function of macroscopic and microscopic details of the model such as the lattice spacing, the physical lattice dimension and the temperature. This is a fundamental step that will help in applying in the future the same techniques to more complicated systems, for which the standard field theoretical approach is impracticable.

Probabilistic analysis of homogenized elastic property for resin products fabricated by additive manufacturing based on three-dimensional random field modeling of microstructure

Additive manufacturing (AM) techniques have been used in several fields of science and industry, and fabrication techniques are being updated. For this fact, especially, for industrial use, mechanical property evaluation methodologies for AM products and standards for product quality assessment should also be well established. In this paper, a probabilistic evaluation of the homogenized elastic properties of a resin product fabricated by a material extrusion-based AM technique is attempted by considering the randomness of both material and microscopic geometrical quantities. This AM method fabricates a resin structure by piling up melted resin, and to decrease consumed material and influence of thermal deformation, the inner structure of the fabricated products will include many pores and its geometry is difficult to be well controlled. From this fact, the products will be regarded as a heterogeneous material with complex random microstructure. This will cause difficulty in the evaluation of its apparent material properties and therefore a probabilistic homogenization analysis is attempted for their quantitative estimation in this study. In particular, to investigate probabilistic properties of microscopic geometry, a random field modeling technique is employed for the evaluation of autocorrelation of the microscopic geometrical parameter, and the results of the autocorrelation identified by experimental observation are introduced to the probabilistic homogenization analysis. The two-dimensional or three-dimensional random field modeling is attempted, and the effectiveness of this approach is investigated by comparing it with the experimental result.

Evaluating the load-bearing capacity of corroded cables in long-span cable-stayed bridges: A stochastic corrosion field simulation approach

In cable-stayed bridges, cables are the primary load-bearing components. The corrosion of high-strength steel wires within these cables can degrade their bearing capacity, which in turn impacts the safety and durability of long-span cable-stayed bridges. To address this concern, a three-dimensional random field model was proposed in this study to evaluate the residual bearing capacity of corroded cables. This model considers the random distribution of the strength of high-strength wires along both axial and cross-sectional directions caused by corrosion. Subsequently, a three-dimensional spatial correlation model was developed based on field test results from two stay cables in service on an actual bridge. This model represents the strength distribution of corroded high-strength steel wires within the cable. Finally, the evaluation model developed in this study was employed to assess the bearing performance of two corroded cables with different degrees of corrosion. The results were then compared with test outcomes. The results show that the proposed evaluation model can accurately estimate the bearing performance of corroded cables.