Random Field Research Articles

On fractional spherically restricted hyperbolic diffusion random field

The paper investigates solutions of the fractional hyperbolic diffusion equation in its most general form with two fractional derivatives of distinct orders. The solutions are given as spatial–temporal homogeneous and isotropic random fields and their spherical restrictions are studied. The spectral representations of these fields are derived and the associated angular spectrum is analysed. The obtained mathematical results are illustrated by numerical examples. In addition, the numerical investigations assess the dependence of the covariance structure and other properties of these fields on the orders of fractional derivatives.

Graph signal recovery using variational Bayes in Fourier pairs with Cramér–Rao bounds

In this paper, the graph signal recovery problem is addressed by employing an aggregation of samples in the vertex domain and the Fourier graph transform domain. The statistical graph signal is modeled using a Gaussian Markov Random Field (GMRF). The reconstruction process involves employing a variational Bayes (VB) algorithm, which is a fully Bayesian method that iteratively estimates all unknown parameters by computing the posteriors in a closed-form. Furthermore, the closed-form of the Cramér–Rao lower bound (CRLB) for graph signal estimation is also derived. Simulation results demonstrate the superiority of the proposed algorithm over some of state-of-the-art algorithms in the literature.

Necessary and sufficient conditions for continuity of hypercontractive processes and fields

Sample path properties of random processes are an interesting and extensively studied topic, especially in the case of Gaussian processes. In this article, we study the continuity properties of hypercontractive fields, providing natural extensions for some known Gaussian results beyond Gaussianity. Our results apply to both random processes and random fields alike.

Influence of slab depth spatial variability on skier-triggering probability and avalanche size

Abstract Spatial variability of snowpack properties adds uncertainty in the evaluation of avalanche hazard. We propose a combined mechanical–statistical approach to study how spatial variation of slab depth affects the skier-triggering probability and possible release size. First, we generate multiple slab depth maps on a plane fictional slope based on Gaussian Random Fields (GRF) for a specific set of mean, variance and correlation length. For each GRF, we derive analytically the Skier Propagation Index (SPI). We then simulate multiple skier tracks and computed the probability based on the number of skier hits where SPI is below 1. Finally, we use a depth-averaged material point method to evaluate the possible avalanche size for given slab depth variations. The results of this analysis show that large correlation lengths and small variances lead to a lower probability of skier-triggering as it reduces the size and the number of areas with low slab depth. Then, we show the effect of skiing style and skier group size on skier-triggering probability. Spatial variability also affects the possible avalanche size by adding stress fluctuation causing early or late tensile failure. Finally, we demonstrate with our models the well-known relationship between the probability and the size in avalanche forecasting.

Efficient 3D real-time adaptive AUV sampling of a river plume front

The coastal environment faces multiple challenges due to climate change and human activities. Sustainable marine resource management necessitates knowledge, and development of efficient ocean sampling approaches is increasingly important for understanding the ocean processes. Currents, winds, and freshwater runoff make ocean variables such as salinity very heterogeneous, and standard statistical models can be unreasonable for describing such complex environments. We employ a class of Gaussian Markov random fields that learns complex spatial dependencies and variability from numerical ocean model data. The suggested model further benefits from fast computations using sparse matrices, and this facilitates real-time model updating and adaptive sampling routines on an autonomous underwater vehicle. To justify our approach, we compare its performance in a simulation experiment with a similar approach using a more standard statistical model. We show that our suggested modeling framework outperforms the current state of the art for modeling such spatial fields. Then, the approach is tested in a field experiment using two autonomous underwater vehicles for characterizing the three-dimensional fresh-/saltwater front in the sea outside Trondheim, Norway. One vehicle is running an adaptive path planning algorithm while the other runs a preprogrammed path. The objective of adaptive sampling is to reduce the variance of the excursion set to classify freshwater and more saline fjord water masses. Results show that the adaptive strategy conducts effective sampling of the frontal region of the river plume.

Person group detection with global trajectory extraction in a disjoint camera network

Person group detection refers to the grouping of people with similar spatio-temporal trajectories. In this work, we address the problem of automatically detecting groups of people from disjoint camera views, which has an essential application in public safety but has not been seriously studied. The main challenge of this task is that the sparse distribution of cameras in a large surveillance area makes it difficult to infer and match people’s trajectories across cameras. To address this challenge, we propose a CCRF (Cyclic Conditional Random Fields) based model for cross-camera trajectory extraction, which takes both visual appearance and heterogeneous spatio-temporal information (including camera locations, video capture times, and the map of the surveillance area) as input and infers multiple candidate cross-camera trajectories for each person. Then, for each pair of people, we propose to use a dynamic trajectory warping (DTW) method to measure the similarity of their trajectories. DTW uses visual features to optimize the selection of trajectories and addresses the problem of trajectory length matching. Since there is no existing dataset that can directly support our research, we enrich our previously built Person Trajectory Dataset by adding the person group annotation and then verify the effectiveness of the proposed method on this dataset. The dataset and code are released at https://github.com/zhangxin1995/PTD_GROUP.

Lightweight Attentive Graph Neural Network with Conditional Random Field for Diagnosis of Anterior Cruciate Ligament Tear.

Anterior cruciate ligament (ACL) tears are prevalent orthopedic sports injuries and are difficult to precisely classify. Previous works have demonstrated the ability of deep learning (DL) to provide support for clinicians in ACL tear classification scenarios, but it requires a large quantity of labeled samples and incurs a high computational expense. This study aims to overcome the challenges brought by small and imbalanced data and achieve fast and accurate ACL tear classification based on magnetic resonance imaging (MRI) of the knee. We propose a lightweight attentive graph neural network (GNN) with a conditional random field (CRF), named the ACGNN, to classify ACL ruptures in knee MR images. A metric-based meta-learning strategy is introduced to conduct independent testing through multiple node classification tasks. We design a lightweight feature embedding network using a feature-based knowledge distillation method to extract features from the given images. Then, GNN layers are used to find the dependencies between samples and complete the classification process. The CRF is incorporated into each GNN layer to refine the affinities. To mitigate oversmoothing and overfitting issues, we apply self-boosting attention, node attention, and memory attention for graph initialization, node updating, and correlation across graph layers, respectively. Experiments demonstrated that our model provided excellent performance on both oblique coronal data and sagittal data with accuracies of 92.94% and 91.92%, respectively. Notably, our proposed method exhibited comparable performance to that of orthopedic surgeons during an internal clinical validation. This work shows the potential of our method to advance ACL diagnosis and facilitates the development of computer-aided diagnosis methods for use in clinical practice.

A New Construction of Covariance Functions for Gaussian Random Fields

A New Construction of Covariance Functions for Gaussian Random Fields

Reentrant ferromagnetic ordering of the random-field Heisenberg model in d>2 dimensions: Fourier-Legendre renormalization-group theory.

The random-magnetic-field classical Heisenberg model is solved in spatial dimensions d≥2 using the recently developed Fourier-Legendre renormalization-group theory for 4π steradians continuously orientable spins, with renormalization-group flows of 12500 variables. The random-magnetic-field Heisenberg model is exactly solved in 10 hierarchical models, for d=2,2.26,2.46,2.58,2.63,2.77,2.89,3. For nonzero random fields, ferromagnetic order is seen for d>2. This ordering, at d=2.46,2.58,2.63,2.77,2.89,3, shows reentrance as a function of temperature.

Diseases prevalent in Kharif crops grown in Barmer district of Rajasthan, India

In the Kharif season, pearl millet (Pennisetum glaucum) is the predominant crop of the Barmer district followed by cluster bean (Cyamopsis tetragonoloba) and moth bean (Vigna aconitifolia). Sesame (Sesamum indicum) and green gram (Vigna radiata) are other important Kharif oilseed and pulse crops, respectively. A survey was conducted to determine the status of major diseases of Kharif crops grown in the Barmer district of the Marwar region of Rajasthan, India. Five blocks and three villages in each block were surveyed through a random field survey method during the collection of diseased samples every Sunday for the practical classes of agricultural students of the College of Agriculture, Baytu; in August and September 2023. Per cent disease incidence was recorded on randomly selected plants in a particular field of selected location. The incidence of diseases was observed based on typical field symptoms and later the association was confirmed through microscopic examinations in the laboratory. In survey, yellow mosaic (virus), leaf crinkle/ curl (virus), bacterial leaf spot/ blight (Xanthomonas phaseoli) of green gram;green ear/ downy mildew (Sclerospora graminicola), rust (Puccinia substriata) and blast (Pyricularia grisea) of pearl millet; Phytophthora stem blight (Phytophthora parasitica var. sesame) and phyllody (Phytoplasma) of sesame; yellow mosaic (virus)and bacterial leaf spot/ blight (Xanthomonas axonopodis pv. cyamopsidis) of cluster beanwere recorded with >50% incidence. However; charcoal rot (Macrophomina phaseolina), web blight (Rhizoctonia solani), Cercospora leaf spot (Cercospora canescens) and anthracnose (Colletotrichum lindemuthianum) of green gram; Alternaria blight/ leaf spot (Alternaria sesame) of sesame,sorghum rust (Puccinia purpurea) and Alternaria blight/ leaf spot (Alternaria cyamopsidis) of cluster beanwere noticed with 10 to 50% incidence.

Quantitative heterogeneous source apportionment of toxic metals through a hybrid method in spatial random fields

Toxic metals in soils pose hazards to food security and human health. Accurate source apportionment provides foundation for pollution prevention. In this study, a novel hybrid method that combines positive matrix factorization, Bayesian maximum entropy and integrative predictability criterion is proposed to provide a new perspective for exploring the heterogeneity of pollution sources in spatial random fields. The results suggest that Cd, As and Cu are the predominant pollutants, with exceedance rates of 27%, 12% and 11%, respectively. The new method demonstrates superiority in predicting toxic metals when combined major and all sources as auxiliary information., with the improvements of 44% and 46%, respectively, Although the major sources identified with the hybrid method are the primary contributors to the accumulation of toxic metals (e.g. coal combustion for Hg, traffic emission for Pb and Zn, industrial activities for As, agricultural activities for Cd and Cu and natural sources for Cr and Ni), the impact of nonmajor sources on toxic metal sin specific regions should not be ignored (e.g. industrial activities on Ni, Pb and Zn in the north and natural sources on Cd, Cu, As, Pb and Zn in the south). For better pollution control, specific local sources should be considered.

Analysis of the stability of cracked soil slopes using a numerical model incorporating stochastic theory

Desiccation crack is a prevalent natural phenomenon that plays a significant role in the stability of soil slopes. In this study, a hydromechanical coupling model incorporating a layer of stochastic cracks is developed for analyzing cracked soil slopes. To properly consider the anisotropy and spatial variability of desiccation cracks, three crack indices are generated through cross-correlated random fields via Cholesky decomposition. The seepage and mechanical behavior of a cracked slope are analyzed by adjusting stochastic parameters and rainfall conditions. Applied to the Ningzhen Mountains area in China, the model investigates the stability of slopes under various annual meteorological conditions. The results indicate that neglecting the spatial variability of cracked layer properties can lead to inaccurate assessments of instability risks at the base and water accumulation at the top of slopes. During heavy rainfall, slopes with deeper (up to 5 m) and weaker cracked layers often show a roughly planar sliding morphology. Moreover, the uncertainties in crack depth have the most pronounced influence on the uncertainties of the slope stability, more than horizontal permeability or crack aperture. The average crack aperture’s influence on slope stability depends on the relationship between crack infiltration rate and rainfall intensity.

Identifying the latent relationships between factors associated with traffic crashes through graphical models

Traffic safety field has been oriented toward finding the relationships between crash outcomes and predictor variables to understand crash phenomena and/or predict future crashes. In the literature, the main framework established for this purpose is based on constructing a modelling equation in which crash outcome (e.g., frequencies) is examined in relation to explanatory variables chosen based on the problem at hand. Despite the importance and success of this approach, there are two issues that are generally not discussed: 1) the latent relationships between factors associated with crashes are oftentimes not the focus of analysis or not observed; and 2) there are not many tools to make informed decisions on which variables might have an impact on the crash outcome and should be included in a safety model, particularly when observations are limited. To address these issues, this paper proposes the use of graphical models, namely a Markov random field (MRF) modelling, Bayesian network modelling, and a graphical XGBoost approach, to disclose relationship topologies of explanatory variables leading to fatal and incapacitating injury pedestrian crashes. The application of graph learning models in traffic safety has a high potential because they are not only useful to understand the mechanism behind the crash occurrence but also can assist in devising accurate and reliable prevention measures by identifying the true variable structure and essential factors jointly acting towards crash occurrence, similar to a pathological examination.

Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.

Estimation of the installation torque-capacity correlation of helical pile considering spatially variable clays

As the offshore industry moves into deeper water, helical piles are emerging as a potential foundation solution option. A large number of studies have been published to explore the installation torque-capacity correlation. However, in most previous studies, the inherent spatial variability of soil strength was neglected. The present research explores the installation and extraction behavior of helical piles through a large deformation random finite element method. A strain-softening soil constitutive model proposed in past literature is employed to model the soil strength remoulding. The validity of the numerical model used to simulate the installation process and the subsequent uplift process is verified by the installation torque and the uplift capacity, respectively. The spatial variation of soil strength is modeled using random field, and then a series of Monte Carlo simulations are performed to investigate the torque-capacity correlation of the helical piles under different random realizations. The analysis results show that for the helical piles with different penetration depths, the spatially random soil strength markedly affects the torque-capacity correlation. Moreover, a probabilistic analysis of the torque-capacity correlation is conducted, which may be of great interest to engineering practitioners in the design method of the helical pile.

LRD spectral analysis of multifractional functional time series on manifolds

AbstractThis paper addresses the estimation of the second-order structure of a manifold cross-time random field (RF) displaying spatially varying Long Range Dependence (LRD), adopting the functional time series framework introduced in Ruiz-Medina (Fract Calc Appl Anal 25:1426–1458, 2022). Conditions for the asymptotic unbiasedness of the integrated periodogram operator in the Hilbert–Schmidt operator norm are derived beyond structural assumptions. Weak-consistent estimation of the long-memory operator is achieved under a semiparametric functional spectral framework in the Gaussian context. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The performance of both estimation procedures is illustrated in the simulation study, in the context of multifractionally integrated spherical functional autoregressive–moving average (SPHARMA(p,q)) processes.

Influence of modeling choices and prior information on the Bayesian assessment of a reinforced concrete bridge

AbstractA lot of bridges are aging and reaching the anticipated service life. To gain insight in their remaining resistance, inspections and measurements can be performed. The resulting information and data can be used to update variables in the degradation models of these bridges. To account for spatial variation, degradation variables can be modeled with random fields. The random fields of the degradation variables are updated based on a Bayesian inference procedure, where different types of heterogenous and indirect data are accounted for. Nevertheless, in this procedure, various assumptions need to be made, such as the quantification of measurement uncertainty, the structural and degradation model to be used, the prior distributions of the variables of interest, and so forth. These assumptions can influence the results of the Bayesian inference. Hence, in this work, it will be investigated what the influence is of different assumptions and how they affect the localization and quantification of damage. This will be done by application to a reinforced concrete girder bridge. The main conclusions are that the model used in the Bayesian inference procedure should resemble the actual structure as good as possible and that simplifications that are allowed in the design can lead to posterior distributions deviating a lot from the actual situation. Moreover, it is illustrated how information from visual observations can be included in the definition of the prior distributions and how this has a beneficial effect on the posterior predictions.

Система анализа тональности текста на телугу на основе нового пассивно-агрессивного классификатора с нечетким взвешиванием

Natural language processing (NLP) is a subset of artificial intelligence demonstrating how algorithms can interact with individuals in their unique languages. In addition, sentiment analysis in NLP is better in numerous programs, including evaluating sentiment in Telugu. Several unsupervised machine-learning algorithms, such as k-means clustering with cuckoo search, are used to detect Telugu text. However, these techniques struggle to cluster data with variable cluster sizes and densities, slow search speeds, and poor convergence accuracy. This study developed a unique ML-based sentiment analysis system for Telugu text to address the shortcomings. Initially, in the pre-processing stage, the proposed Linear Pursuit Algorithm (LPA) removes words in white spaces, punctuation, and stops. Then, for POS tagging, this research proposed a Conditional Random Field with Lexicon weighting; following that, a Contrived Passive Aggressive with Fuzzy Weighting Classifier (CPSC-FWC) is proposed to classify the sentiments in Telugu text. Consequently, the method we propose produces efficient outcomes in terms of accuracy, precision, recall, and f1-score.

Asymptotic covariances for functionals of weakly stationary random fields

Let (Ax)x∈Rd be a measurable, weakly stationary random field, i.e. E[Ax]=E[Ay], Cov(Ax,Ay)=K(x−y), ∀x,y∈Rd, with covariance function K:Rd→R.Assuming only that the integral covariance functionwt≔∫{|z|≤t}K(z)dz is regularly varying (which encompasses the classical assumptions found in the literature), we compute limt→∞Cov∫tDAxdxtd/2wt1/2,∫tLAydytd/2wt1/2 for D,L⊆Rd belonging to a certain class of compact sets.As an application, we combine this result with existing limit theorems to obtain multi-dimensional limit theorems for non-linear functionals of stationary Gaussian fields, in particular proving new results for the Berry’s random wave model. At the end of the paper, we also show how the problem for A with a general continuous covariance function K can be reduced to the same problem for a radial, continuous covariance function Kiso.The novel ideas of this work are mainly based on regularity conditions for (cross) covariograms of Euclidean sets and standard properties of regularly varying functions.

Automatic detection of myocardial ischaemia using generalisable spatio-temporal hierarchical Bayesian modelling of DCE-MRI

Dynamic Contrast Enhanced Magnetic Resonance Imaging (DCE-MRI) can be used as a non-invasive method for the assessment of myocardial perfusion. The acquired images can be utilised to analyse the spatial extent and severity of myocardial ischaemia (regions with impaired microvascular blood flow). In the present paper, we propose a novel generalisable spatio-temporal hierarchical Bayesian model (GST-HBM) to automate the detection of ischaemic lesions and improve the in silico prediction accuracy by systematically integrating spatio-temporal context information. We present a computational inference procedure with an adequate trade-off between accuracy and computational efficiency, whereby model parameters are sampled from the posterior distribution with Gibbs sampling, while lower-level hyperparameters are selected using model selection strategies based on the Watanabe Akaike information criterion (WAIC). We have assessed our method on both synthetic (in silico) data with known gold-standard and 12 sets of clinical first-pass myocardial perfusion DCE-MRI datasets. We have also carried out a comparative performance evaluation with four established alternative methods: Gaussian mixture model (GMM), opening and closing operations based on Gaussian mixture model (GMMC&Omax), Markov random field constrained Gaussian mixture model (GMM-MRF) and model-based hierarchical Bayesian model (M-HBM). Our results show that the proposed GST-HBM method achieves much higher in silico prediction accuracy than the established alternative methods. Furthermore, this method appears to provide a more robust delineation of ischaemic lesions in datasets affected by spatially variant noise.