Random Field Research Articles

Shift-invariant homogeneous classes of random fields

Given an Rd-valued random field (rf) Z(t),t∈T and an α-homogeneous mapping κ we define the corresponding equivalent class of rf's (denoted by Cκ[Z]) which include representers of the same tail measure νZ. When T is an additive group, tractable equivalent classes of interest are the shift-invariant ones, which contain in particular all independent random shifts of Z. This contribution is mainly concerned with the investigation of the probabilistic properties of shift-invariant Cκ[Z]'s. Important objects introduced in our setting are tail and spectral tail rf's. Further, the class of universal maps U acting on elements of Cκ[Z] turns out to be crucial for properties of functionals of Z. Applications of our findings concern max-stable and symmetric α-stable rf's, their maximal indices as well as their random shift-representations.

Gaussian random field approximation via Stein's method with applications to wide random neural networks

We derive upper bounds on the Wasserstein distance (W1), with respect to sup-norm, between any continuous Rd valued random field indexed by the n-sphere and the Gaussian, based on Stein's method. We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the W1 distance. The smoothing is based on covariance functions constructed using powers of Laplacian operators, designed so that the associated Gaussian process has a tractable Cameron-Martin or Reproducing Kernel Hilbert Space. This feature enables us to move beyond one dimensional interval-based index sets that were previously considered in the literature. Specializing our general result, we obtain the first bounds on the Gaussian random field approximation of wide random neural networks of any depth and Lipschitz activation functions at the random field level. Our bounds are explicitly expressed in terms of the widths of the network and moments of the random weights. We also obtain tighter bounds when the activation function has three bounded derivatives.

Statistical Modeling of Right-Censored Spatial Data Using Gaussian Random Fields

Consider a fixed number of clustered areas identified by their geographical coordinates that are monitored for the occurrences of an event such as a pandemic, epidemic, or migration. Data collected on units at all areas include covariates and environmental factors. We apply a probit transformation to the time to event and embed an isotropic spatial correlation function into our models for better modeling as compared to existing methodologies that use frailty or copula. Composite likelihood technique is employed for the construction of a multivariate Gaussian random field that preserves the spatial correlation function. The data are analyzed using counting process and geostatistical formulation that led to a class of weighted pairwise semiparametric estimating functions. The estimators of model parameters are shown to be consistent and asymptotically normally distributed under infill-type asymptotic spatial statistics. Detailed small sample numerical studies that are in agreement with theoretical results are provided. The foregoing procedures are applied to the leukemia survival data in Northeast England. A comparison to existing methodologies provides improvement.

Threshold network GARCH model

Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model and its variations have been widely adopted in the study of financial volatilities, while the extension of GARCH‐type models to high‐dimensional data is always difficult because of over‐parameterization and computational complexity. In this article, we propose a multi‐variate GARCH‐type model that can simplify the parameterization by utilizing the network structure that can be appropriately specified for certain types of high‐dimensional data. The asymmetry in the dynamics of volatilities is also considered as our model adopts a threshold structure. To enable our model to handle data with extremely high dimension, we investigate the near‐epoch dependence (NED) of our model, and the asymptotic properties of our quasi‐maximum‐likelihood‐estimator (QMLE) are derived from the limit theorems for NED random fields. Simulations are conducted to test our theoretical results. At last we fit our model to log‐returns of four groups of stocks and the results indicate that bad news is not necessarily more influential on volatility if the network effects are considered.

Study of the Influence of Random Fields on Phase Transitions Using the Example of the Exactly Solvable Model

Study of the Influence of Random Fields on Phase Transitions Using the Example of the Exactly Solvable Model

Simulating non-stationary and non-Gaussian cross-correlated fields using multivariate Karhunen–Loève expansion and L-moments-based Hermite polynomial model

In practical engineering, cross-correlated random fields are often used to model structural materials or random loads containing multiple correlations. Effective and accurate simulation of these cross-correlation fields is an important prerequisite for subsequent reliability analysis and uncertainty quantification of complex systems. Therefore, this paper proposes a new simulation method for non-stationary non-Gaussian cross-correlated random fields to satisfy realistic engineering requirements. In this new method, L-moments-based Hermite polynomials model (LHPM) is extended to non-stationary non-Gaussian cross-correlated fields. Then, a transformation model from non-stationary non-Gaussian correlation matrix function (CMF) to the underlying Gaussian CMF is explicitly given. Multivariate K-L expansion is further used to approximate the underlying Gaussian cross-correlated fields, where the eigenvalue and eigenfunctions are estimated via the Nyström method with uniform weights. The approximation permits the application of few random variables to characterize the entire Gaussian cross-correlated fields. Ultimately, the simulated underlying Gaussian cross-correlated fields are mapped back to the target non-stationary non-Gaussian cross-correlated fields based on LHPM. Three typical examples, including exponential kernel, Wiener process kernel and spatially varying non-Gaussian and nonstationary seismic ground motions, are used to validate the effectiveness of the proposed method. The source code is readily available at: https://github.com/zhaozhao23/Simulation-of-non-stationary-and-non-Gaussian-cross-correlated-fields.

Mesoscopic simulation of uniaxial compression fracture of concrete via the nonlocal macro–meso-scale consistent damage model

To capture the uniaxial compression fracture of concrete composite is still a great challenging problem. To this end, the newly proposed nonlocal macro–meso-scale consistent damage (NMMD) model is extended to simulate uniaxial compression fracture of concrete composites. In the NMMD model, a mesoscopic structure composed of material point pairs is attached to each material point. A decomposition scheme suitable for masonry-like materials is adopted to decompose the elongation quantity of point pairs. The topologic damage is defined as the weighted summation of mesoscopic damage in neighboring material point pairs driven by the decomposed elongation quantity. The physically-based energetic degradation function is employed to bridge the geometric discontinuity in solids and the energy dissipation, and thus the topologic damage is integrated into the framework of continuum damage mechanics. The Monte-Carlo method and two-phase random field model are employed to generate stochastic circular and irregular aggregates representing the meso structure of concrete, respectively. The interface transition zone is prescribed by reducing the critical value of interface point pairs. Numerical results indicate that both the crack patterns and stress–strain curves of the uniaxial compression experiments on concrete panels can be well captured by the proposed NMMD model without mesh size sensitivity. The influence of horizontal constraints on the top and bottom surfaces is investigated and shown to have an effect on the crack patterns, strength and deformation capability of the specimen. The necessity of decomposition is also illustrated.

Geotechnical correlation field-informed and data-driven prediction of spatially varying geotechnical properties

Geotechnical measurements are often limited, leading to the use of interpolation techniques for interpreting spatial variations in geotechnical properties from sparse geo-data. Traditional geostatistical methods suffer from significant computational complexity. On the other hand, data-driven approaches often lack integration with geotechnical domain knowledge, potentially oversimplifying or complicating predictions related to the spatial variability of geotechnical properties. This study introduces a novel framework that combines geotechnical knowledge with data-driven methods to model inherent soil spatial variability incorporating Geotechnical Correlation Field (GCF) that reflects domain knowledge. The GCF, influenced by Autocorrelation Function (ACF) types and Scale of Fluctuation (SOF), provides a flexible basis for accurately representing spatially varying geotechnical properties. Using a large synthetic database comprising known ACF types and SOFs, we constructed a series of specialized neural networks. These networks identify random field parameters at different sites based on sparse data, and the estimated parameters can be directly used to calculate GCFs for a given site. The performance of the proposed method is validated using a set of synthetic data and a real case history in New Zealand. The results demonstrate the model can accurately predict random field parameters for irregularly spaced geo-data, even with limited information. Significantly, the GCFs offer improved physical interpretations and enhance the performance of subsurface modeling. The computational complexity of this method is independent of the number of soil cells, making it highly efficient and scalable.

Asymptotic predictions on the velocity gradient statistics in low-Reynolds-number random flows: onset of skewness, intermittency and alignments

Stirring a fluid through a Gaussian forcing at a vanishingly small Reynolds number produces a Gaussian random field, while flows at higher Reynolds numbers exhibit non-Gaussianity, cascades, anomalous scaling and preferential alignments. Recent works (Yakhot & Donzis, Phys. Rev. Lett., vol. 119, 2017, 044501; Khurshid et al., Phys. Rev. E, vol. 107, 2023, 045102) investigated the onset of these turbulent hallmarks in low-Reynolds-number flows by focusing on the scaling of the velocity gradients and velocity increments. They showed that the onset of power-law scalings in the velocity gradient statistics occurs at low Reynolds numbers, with the scaling exponents being surprisingly similar to those in the inertial range of fully developed turbulence. In this work we address the onset of turbulent signatures in low-Reynolds-number flows from the viewpoint of the velocity gradient dynamics, giving insights into its rich statistical geometry. We combine a perturbation theory of the full Navier–Stokes equations with velocity gradient modelling. This procedure results in a stochastic model for the velocity gradient in which the model coefficients follow directly from the Navier–Stokes equations and statistical homogeneity constraints. The Fokker–Planck equation associated with our stochastic model admits an analytic solution that shows the onset of turbulent hallmarks at low Reynolds numbers: skewness, intermittency and preferential alignments arise in the velocity gradient statistics through a smooth transition as the Reynolds number increases. The model predictions are in excellent agreement with direct numerical simulations of low-Reynolds-number flows.

Part-of-speech Tagging for Low-resource Languages: Activation Function for Deep Learning Network to Work with Minimal Training Data

Numerous natural language processing (NLP) applications exist today, especially for the most commonly spoken languages such as English, Chinese, and Spanish. Popular traditional methods such as Rule based methods, Naive Bayes classifiers, Hidden Markov models, Conditional Random field-based classifiers, and other stochastic methods have contributed to this improvement in the past. Recently, deep learning has led to exciting breakthroughs in several areas of artificial intelligence, including image processing and natural language processing. It is important to label words as parts of speech to begin developing most of the NLP applications. A deep study in this area reveals that many popular approaches used for this purpose require massive training data. Therefore, these approaches have not been helpful for languages not rich in digital resources. Applying these methods with very little training data prompts the need for innovative problem-solving. This article describes our research, which examines the strengths and weaknesses of well-known approaches, such as conditional random fields and state-of-the-art deep learning models, when applied for part-of-speech tagging using minimal training data for Assamese and English. We also examine the factors affecting them. We discuss our deep learning architecture and the proposed activation function, which shows promise with little training data. The activation function categorizes words belonging to different classes with more confidence by using the outcomes of statistical methods with SMTaylor SoftMax in our deep learning model. With minimal training, our deep learning architecture using the proposed modification of SM-Taylor SoftMax improves accuracy upto 4%, for our small dataset. This technique is a combination of SMTaylor SoftMax and statistical probability distribution of words over tags.

Full inference for the anisotropic fractional Brownian field

The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired by the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.

Estimating major merger rates and spin parameters ab initio via the clustering of critical events

Abstract We build a model to predict from first principles the properties of major mergers. We predict these from the coalescence of peaks and saddle points in the vicinity of a given larger peak, as one increases the smoothing scale in the initial linear density field as a proxy for cosmic time. To refine our results, we also ensure, using a suite of ∼400 power-law Gaussian random fields smoothed at ∼30 different scales, that the relevant peaks and saddles are topologically connected: they should belong to a persistent pair before coalescence. Our model allows us to (a) compute the probability distribution function of the satellite-merger separation in Lagrangian space: they peak at three times the smoothing scale; (b) predict the distribution of the number of mergers as a function of peak rarity: haloes typically undergo two major mergers (>1:10) per decade of mass growth; (c) recover that the typical spin brought by mergers: it is of the order of a few tens of percent.

Unsupervised recognition of radar signals combining multi-block TFR with subspace clustering

In the realm of radar systems, the proliferation of new modulation techniques introduces an increased level of complexity in the identification of both existing and potentially novel modulation schemes. Conventional supervised recognition methodologies, which rely heavily on labeled datasets, exhibit limitations in distinguishing amongst a multitude of unlabeled signals. This paper introduces a novel framework that synergizes Subspace Clustering (SC) with Multi-Block Time-Frequency Representations (TFRs, denoted as mT), referred to as SC-mT. This framework leverages subspace clustering for the unsupervised categorization of radar modulations and employs an innovative multi-block strategy to augment classification precision. The process commences with the generation of TFR datasets for radar signals, followed by the construction of two distinct multi-block models, Model-A and Model-B. These models segment the radar TFRs into overlapping multi-block sets, utilizing the concept of random receptive fields. The subspace clustering algorithm is then applied to each block within the sub-TFR sets to procure an affinity matrix. The culmination of this process involves the aggregation of the affinity matrices from all blocks, facilitating the derivation of classification outcomes via spectral clustering. Empirical analyses affirm that the sequential integration of the SC-mT algorithm surpasses the classification efficacy of various conventional and cutting-edge algorithms. Notably, this algorithm attains a classification accuracy surpassing 90% for ten unlabeled signals, even in scenarios where the signal-to-noise ratio (SNR) is as low as -4 dB.

Lattice QCD Calculation of Electroweak Box Contributions to Superallowed Nuclear and Neutron Beta Decays.

We present the first lattice QCD calculation of the universal axial γW-box contribution □_{γW}^{VA} to both superallowed nuclear and neutron beta decays. This contribution emerges as a significant component within the theoretical uncertainties surrounding the extraction of |V_{ud}| from superallowed decays. Our calculation is conducted using two domain wall fermion ensembles at the physical pion mass. To construct the nucleon four-point correlation functions, we employ the random sparsening field technique. Furthermore, we incorporate long-distance contributions to the hadronic function using the infinite-volume reconstruction method. Upon performing the continuum extrapolation, we arrive at □_{γW}^{VA}=3.65(7)_{lat}(1)_{PT}×10^{-3}. Consequently, this yields a slightly higher value of |V_{ud}|=0.973 86(11)_{exp}(9)_{RC}(27)_{NS}, reducing the previous 2.1σ tension with the CKM unitarity to 1.8σ. Additionally, we calculate the vector γW-box contribution to the axial charge g_{A}, denoted as □_{γW}^{VV}, and explore its potential implications.

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

A physics and data co-driven surrogate modeling method for high-dimensional rare event simulation

This paper presents a physics and data co-driven surrogate modeling method for efficient rare event simulation of civil and mechanical systems with high-dimensional input uncertainties. The method fuses interpretable low-fidelity physical models with data-driven error corrections. The hypothesis is that a well-designed and well-trained simplified physical model can preserve salient features of the original model, while data-fitting techniques can fill the remaining gaps between the surrogate and original model predictions. The coupled physics-data-driven surrogate model is adaptively trained using active learning, aiming to achieve a high correlation and small bias between the surrogate and original model responses in the critical parametric region of a rare event. A final importance sampling step is introduced to correct the surrogate model-based probability estimations. Static and dynamic problems with input uncertainties modeled by random field and stochastic process are studied to demonstrate the proposed method.

Statistical theory of fluids confined in quenched disordered porous media.

We develop a theory to calculate structural correlations and thermodynamic properties of a fluid confined in a random porous solid medium (matrix). We used density functional formalism to derive an annealed averaged expression for the density profile and excess free energy of fluid arising due to random fields of a particular realization of the matrix. After performing the second average over the quenched-disordered variables, the excess free energy is organized to give one- and two-body potentials for fluid particles. The average over disorder reduces the system to an effective one-component system of fluid in which particles feel one-body (external) potential and interact via effective pair potential. The effective pair potential is a sum of the bare (the one in the pure fluid) and the matrix-induced potential. The resulting partition function involves only fluid variables. Equations are derived for fluid-fluid and fluid-matrix correlation functions and for free energy, pressure, and chemical potential of the fluid. The theory is applied to a model system of hard spheres and results for the effective pair potential, correlation functions, and thermodynamic properties are reported. The effective pair potential is found to be attractive at the contact and develops a repulsive peak before decaying to zero. Results for pair correlation function and structure factor are compared with simulation results for several fluid densities at two matrix densities. In all the cases, a very good agreement has been found.

High Energy Storage Performance in BiFeO3-Based Lead-Free High-Entropy Ferroelectrics.

Dielectric capacitors are widely used in advanced electrical and electronic systems due to the rapid charge/discharge rates and high power density. High comprehensive energy storage properties are the ultimate ambition in the field of application achievements. Here, the high-entropy strategy is proposed to design and fabricate single-phase homogeneous (Bi0.5Ba0.1Sr0.1Ca0.2Na0.1)(Fe0.5Ti0.3Zr0.1Nb0.1)O3 ceramic, the hierarchical heterostructure including rhombohedral-tetragonal multiphase nanoclusters and locally disordered oxygen octahedral tilt can lead to the increased dielectric relaxation, diffused phase transition, diverse local polarization configurations, grain refinement, ultrasmall polar nanoregions, large random field, delayed polarization saturation and improved breakdown field. Accordingly, a giant Wrec ≈13.3 J cm-3 and a high η ≈78% at 66.4kV mm-1 can be simultaneously achieved in the lead-free high-entropy BiFeO3-based ceramic, showing an obvious advantage in overall energy-storage properties over BiFeO3-based lead-free ceramics. Moreover, an ultrafast discharge rate (t0.9 = 18ns) can be achieved at room temperature, concomitant with favorable temperature stability in the range of 20-160 °C, due to the enhanced diffuse phase transition and fast polarization response. This work provides a feasible pathway to design and generate dielectric materials exhibiting high comprehensive energy-storage performance.

Operator learning for homogenizing hyperelastic materials, without PDE data

In this work, we address operator learning for stochastic homogenization in nonlinear elasticity. A Fourier neural operator is employed to learn the map between the input field describing the material at fine scale and the deformation map. We propose a variationally-consistent loss function that does not involve solution field data. The methodology is tested on materials described either by piecewise constant fields at microscale, or by random fields at mesoscale. High prediction accuracy is obtained for both the solution field and the homogenized response. We show, in particular, that the accuracy achieved with the proposed strategy is comparable to that obtained with the conventional data-driven training method.

Random Responses of Shield Tunnel to New Tunnel Undercrossing Considering Spatial Variability of Soil Elastic Modulus

This paper investigates the effect of spatial variability of soil elastic modulus on the longitudinal responses of the existing shield tunnel to the new tunnel undercrossing using a random two-stage analysis method (RTSAM). The Timoshenko–Winkler-based deterministic method considering longitudinal variation in the subgrade reaction coefficient and the random field of the soil elastic modulus discretized by the Karhunen–Loeve expansion method are combined to establish the RTSAM. Then, the proposed RTSAM is applied to carry out a random analysis based on an actual engineering case. Results show that the increases in the scale of fluctuation and the coefficient of variation of the soil elastic modulus lead to higher variabilities of tunnel responses. A decreasing pillar depth and mean value of the soil elastic modulus and an increasing skew angle strengthen the effect of the spatial variability of the soil elastic modulus on tunnel responses. The variabilities of tunnel responses under the random field of the soil elastic modulus are overestimated by the Euler–Bernoulli beam model. The results of this study provide references for the uncertainty analysis of the new tunneling-induced responses of the existing tunnel under the random field of soil properties.