Bayesian spatiotemporal stochastic partial differential equation for high-resolution earthquake magnitude mapping: Application to Sumatra Island, Indonesia.

This work focuses on the inverse problem of recovering time-dependent source terms within a multi-term time-fractional diffusion equation (MTFDE), utilizing supplementary measurement data. We establish the existence and uniqueness of a weak solution for the corresponding direct problem and provide uniqueness and a stability estimate for the inverse problem. To numerically tackle this ill-posed inverse problem, a Bayesian inference framework is adopted. This involves implementing a priori modelling via Markov random fields (MRFs) and exploring the posterior state space using the Metropolis-Hastings (MH) algorithm. The performance and uncertainty quantification capabilities of this Bayesian approach are then assessed through three numerical experiments.

Abstract In two dimensions, all isometrically invariant Markov random fields on binary assignments are induced by energy functions that can be represented as linear combinations of area, perimeter, and Euler characteristic. This class of model includes the Ising model, both ferro- and antiferromagnetic, with and without a field, as well as the Baxter-Wu model. On the hexagonal lattice, we determine the low-temperature behavior for this class of model, and construct a phase diagram of said behavior. In particular, we identify regions with three geometric phases, regions with a single unique phase, and coexistence curves between them. We also characterize the behavior along two non-Peierls lines, where entropy fails to vanish the as temperature goes to zero.

Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.

ABSTRACT In the synthetic aperture radar (SAR) image change detection problem, the inherent speckle noise significantly degrades the discriminability of changes in the difference image. To address this issue, this paper proposes a local/nonlocal structural differences and Markov random field (MRF) co-segmentation-based method. First, this method constructs fully connected graphs within image patches to capture local structure, and constructs K-nearest neighbour (KNN) graphs among patches to capture the nonlocal structure information of the image. Then, two local and four nonlocal structural difference metrics are computed through graph cross-mapping, enhancing robustness to noise and exploiting spatio-temporal correlations. Finally, the MRF co-segmentation model integrates the change information in different images and the context information in original images for joint segmentation and fusion. The experiments with fourteen comparison methods under three data sets have demonstrated the effectiveness of the proposed method, achieving an average improvement of 6.43% in kappa coefficient.

This article presents a Markov Random Field (MRF) coupled energy function designed for non rigid image registration. The proposed function addresses the difficulties associated with accurate image registration while maintaining spatial coherence. It combines intensity based and feature based terms. The intensity based term measures the similarity between pixels in fixed and moved images for registration, while the feature based term uses the Scale-Invariant Feature Transform (SIFT) method, ensuring robust alignment of pixel intensities and structural features between image pairs. Two graph cut optimization algorithms, α-expansion, and αβ-swap, are evaluated to optimize the coupled energy function proposed. Experimental results calculated for all pairs in the FIRE retinal dataset demonstrate that the α-expansion algorithm outperforms the αβ-swap algorithm. The α-expansion algorithm exhibits a lower mean squared error (MSE) indicating that it creates a registered image more similar to the fixed in terms of similarity, and a higher structural similarity index value (SSIM), indicating that the algorithm preserves the important visual features of the image. Furthermore, results confirm that the proposed coupled MRF based registration approach significantly outperforms the traditional intensity based registration method.

The low-frequency variability of the mid-latitude atmosphere involves complex nonlinear and chaotic dynamical processes posing predictability challenges. It is characterized by sporadically recurring, often long-lived patterns of atmospheric circulation of hemispheric scale known as weather regimes. The evolution of these circulation regimes in addition to their link to large-scale teleconnections can help to extend the limits of atmospheric predictability. They also play a key role in sub- and inter-seasonal weather forecasting. Their identification and modeling remains an issue, however, due to their intricacy, including a clear conceptual picture. In recent years, the concept of metastability has been developed to explain regimes formation. This suggests an interpretation of circulation regimes as communities of states in the neighborhood of which the atmospheric system remains abnormally longer than typical baroclinic timescales. Here we develop a new and effective method to identify such communities by constructing and analyzing an operator of the system's evolution via hidden Markov model (HMM). The method makes use of graph theory and is based on probabilistic approach to partition the HMM transition matrix into weakly interacting blocks - communities of hidden states - associated with regimes. The approach involves nonlinear kernel principal component mapping to consistently embed the system state space for HMM building. Application to northern winter hemisphere using geopotential heights from reanalysis yields four persistent and recurrent circulation regimes. Statistical and dynamical characteristics of these circulation regimes and surface impacts are discussed. In particular, unexpected high correlations are obtained with EL-Niño Southern Oscillation and Pacific decadal oscillation with lead times of up to one year.

Inverter overheating is a critical fault factor in rail transit systems. To address the challenges of sparse low-voltage data and high-dimensional input features, we propose a hybrid prediction framework for inverter temperature. The Random Masked Dual DCGAN (RTDG) model is introduced to enhance low-voltage data diversity, while a Gaussian Markov Random Field (GMRF) method performs dimensionality reduction by identifying key variables. To capture spatio-temporal dependencies, an enhanced Transformer architecture (STTr) is constructed, integrating state space modeling and temporal normalization. These components are fused using a weighted stacking strategy. The model is trained and validated on real-world rail transit datasets. Performance is evaluated using MSE, RMSE, and MAE metrics. Experimental results show that the proposed model outperforms conventional approaches, achieving a 4.93% improvement over single models and a 9.73% gain compared to non-augmented training. This framework supports intelligent fault prevention and contributes to the safe, efficient operation of modern rail systems.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-19004-8.

Retinal vessel segmentation plays a critical role in the early diagnosis of ocular and systemic diseases such as diabetic retinopathy and hypertension. This paper proposes an unsupervised segmentation method that integrates dual-feature fusion and spatially constrained co-clustering. The approach combines multi-scale and multi-directional vascular structural information captured by B-COSFIRE filters with local contrast enhancement from the Top-hat transformation, resulting in a complementary feature representation. Furthermore, a spatial constraint based on Markov Random Fields is incorporated into an information-theoretic co-clustering framework to improve topological consistency. Experimental results on the STARE database demonstrate that the proposed method achieves competitive performance, with an accuracy of 0.9511, sensitivity of 0.7439, and specificity of 0.9747, effectively enhancing vascular continuity and robustness.

The paper presents a novel probability-informed approach to improving the accuracy of small object semantic segmentation in high-resolution imagery datasets with imbalanced classes and a limited volume of samples. Small objects imply having a small pixel footprint on the input image, for example, ships in the ocean. Informing in this context means using mathematical models to represent data in the layers of deep neural networks. Thus, the ensemble Quadtree-informed Graph Self-Attention Networks (QiGSANs) are proposed. New architectural blocks, informed by types of Markov random fields such as quadtrees, have been introduced to capture the interconnections between features in images at different spatial resolutions during the graph convolution of superpixel subregions. It has been analytically proven that quadtree-informed graph convolutional neural networks, a part of QiGSAN, tend to achieve faster loss reduction compared to convolutional architectures. This justifies the effectiveness of probability-informed modifications based on quadtrees. To empirically demonstrate the processing of real small data with imbalanced object classes using QiGSAN, two open datasets of synthetic aperture radar (SAR) imagery (up to 0.5 m per pixel) are used: the High Resolution SAR Images Dataset (HRSID) and the SAR Ship Detection Dataset (SSDD). The results of QiGSAN are compared to those of the transformers SegFormer and LWGANet, which constitute a new state-of-the-art model for UAV (Unmanned Aerial Vehicles) and SAR image processing. They are also compared to convolutional neural networks and several ensemble implementations using other graph neural networks. QiGSAN significantly increases the F1-score values by up to 63.93%, 48.57%, and 9.84% compared to transformers, convolutional neural networks, and other ensemble architectures, respectively. QiGSAN outperformed the base segmentors with the mIOU (mean intersection-over-union) metric too: the highest increase was 35.79%. Therefore, our approach to knowledge extraction using mathematical models allows us to significantly improve modern computer vision techniques for imbalanced data.

ABSTRACT In tunnel engineering, the excavation response is significantly affected by stratigraphic uncertainty. Therefore, quantitative evaluation of stratigraphic uncertainty is crucial. In this paper, an automatic framework integrating stochastic stratigraphic modelling and numerical calculation is proposed to investigate the impact of stratigraphic uncertainty on ground surface settlement for tunnel projects. A three-dimensional stochastic stratigraphic modelling approach based on Markov random field and Bayesian machine learning is developed to construct a database of potential strata. Then, by integrating the non-intrusive analysis method, stratigraphic models are automatically mapped onto the numerical model, followed by numerical simulations in batches. A synthetic case study is used to determine the selection of model parameters and validate the reliability of the stochastic stratigraphic modelling approach. Finally, a subway tunnel is selected as a real-world case to demonstrate the effectiveness of the proposed framework. The local stratigraphic uncertainty of various stratigraphic layers is quantitatively measured based on the proposed local geological uncertainty index. A preliminary investigation is conducted on the impact of stratigraphic uncertainty on the distribution of tunnel surface settlement. The results show that a straightforward assessment of tunnel construction can be achieved with a small number of calculations, thus providing effective guidance for subsequent construction optimisation.

The Ising model is important in statistical modeling and inference in many applications, however its normalizing constant, mean number of active vertices and mean spin interaction – quantities needed in inference – are computationally intractable. We provide accurate approximations that make it possible to numerically calculate these quantities in the homogeneous case. Simulation studies indicate good performance of our approximation formulae that are scalable and unfazed by the size (number of nodes, degree of graph) of the Markov Random Field. The practical import of our approximation formulae is illustrated in performing Bayesian inference in a functional Magnetic Resonance Imaging activation detection experiment, in likelihood ratio testing, for anisotropy in the spatial patterns of yearly increases in pistachio tree yields, and for independence of the least significant bit in the three color channels of a gigapixel image.

ABSTRACT During tunnel excavation, the spatial heterogeneity of geological conditions and the limited boreholes inevitably introduce uncertainties in stratigraphic interfaces and geotechnical properties, significantly impacting the stability of a tunnel face. This study focuses on the comprehensive effect of stratigraphic and geotechnical uncertainty on the stability of a tunnel face. The Markov Random Field method was employed to construct a 2D geological model based on sparse borehole observations. The Karhunen - Loève (K-L) expansion method was used to generate random fields of soil properties of soil properties. The Upper-Bound Limit Analysis Method with the discretized three-dimensional rotational failure mechanism was employed to quantitatively assess the stability of a tunnel face in spatial heterogeneous soils. As a validation, the study utilised actual borehole data from the Zhuhai tunnel project to establish a geological model. The study introduced Monte Carlo Simulations to evaluate the probability distribution of the safety factor of the Zhuhai tunnel face under the influence of geological uncertainties. The study also explored the impact of borehole spacing and the autocorrelation distances on tunnel face stability. This research provides new theoretical insights into tunnel face stability, offering valuable practical tools for engineering decision-making and risk management.

Transmission line galloping—wind-induced conductor vibrations—threaten grid safety by damaging components and triggering outages. An AI-driven method using HD cameras captures galloping video, which is frame-segmented via DirectShow and clarified with adaptive gamma correction. A Markov random field model isolates conductors/spacers (foreground) from backgrounds, and DeepLabv3+ extracts spatial features for displacement/angle quantification and frequency analysis. Experimental results confirm robust preprocessing, accurate component segmentation, and reliable galloping pattern detection, enabling proactive maintenance and grid resilience. Keywords: Transmission line; Deep learning; Neural network model; Dance characteristics; Counting frequency; Streaming media; Gamma correction; Convolutional layer; Video image segmentation; MRF; Edge point; CBMA

Air pollution remains a critical environmental and public health challenge, demanding high-resolution spatial data to better understand its spatial distribution and impacts. This study addresses the challenges of conducting multivariate spatial analysis of air pollutants observed at aggregated levels, particularly when the goal is to model the underlying continuous processes and perform spatial predictions at varying resolutions. To address these issues, we propose a continuous multivariate spatial model based on Gaussian processes (GPs), naturally accommodating the support of aggregated sampling units. Computationally efficient inference is achieved using R-INLA, leveraging the connection between GPs and Gaussian Markov random fields (GMRFs). A custom projection matrix maps the GMRFs defined on the triangulation of the study region and the aggregated GPs at sampling units, ensuring accurate handling of changes in spatial support. This approach integrates shared information among pollutants and incorporates covariates, enhancing interpretability and explanatory power. This approach is used to downscale PM 2.5 , PM10 and ozone levels in Portugal and Italy, improving spatial resolution from 0.1 ° (10 km) to 0.02 ° (2 km), and revealing dependencies among pollutants. Our framework provides a robust foundation for analyzing complex pollutant interactions, offering valuable insights for decision-makers seeking to address air pollution and its impacts.

Disease mapping attempts to explain observed health event counts across areal units, typically using Markov random field models. These models rely on spatial priors to account for variation in raw relative risk or rate estimates. Spatial priors introduce some degree of smoothing, wherein, for any particular unit, empirical risk or incidence estimates are either adjusted towards a suitable mean or incorporate neighbor-based smoothing. While model explanation may be the primary focus, the literature lacks a comparison of the amount of smoothing introduced by different spatial priors. Additionally, there has been no investigation into how varying the parameters of these priors influences the resulting smoothing. This study examines seven commonly used spatial priors through both simulations and real data analyses. Using areal maps of peninsular Spain and England, we analyze smoothing effects using two datasets with associated populations at risk. We propose empirical metrics to quantify the smoothing achieved by each model and theoretical metrics to calibrate the expected extent of smoothing as a function of model parameters. We employ areal maps in order to quantitatively characterize the extent of smoothing within and across the models as well as to link the theoretical metrics to the empirical metrics.

Quality control is critical in cabinet panel manufacturing due to the complexity of the assembly process, which requires three-dimensional measurement methods for enhanced precision and efficiency compared to conventional two-dimensional techniques. Stereo vision offers an effective solution with high accuracy, efficiency, and cost-effectiveness, yet challenges like unclear edge disparities, occlusions, and weak textures persist. To overcome these, we propose a high-precision stereo reconstruction method combining guided image filtering with Markov random fields. Simulated and real-world experiments validate our approach, demonstrating significant improvements in challenging scenarios. This work aims to advance stereo vision's practical application in manufacturing.

This work proposes a new hybrid model for joint indoor localization and activity recognition by combining a Convolutional Neural Network-Gated Recurrent Unit (CNN-GRU) model with a Markov Random Field (MRF) for better classification. The CNN-GRU successfully captures spatial and temporal dependencies, while the MRF models the mutual relations of activities and locations by estimating their joint probability distribution. The new system was tested on a public smart home dataset with four activities (sitting, lying, walking, and standing) and four indoor locations (kitchen, bedroom, living room, and stairs). The hybrid framework obtained an accuracy of 95% for activity recognition and 93% for indoor localization with a combined activity-location classification accuracy of 81%. Such results confirm the ability of the system to provide robust predictions in real-world smart environments, make it highly suitable for healthcare and intelligent living applications, and is efficient and deployable in real-world scenarios, addressing the critical challenges of noisy and dynamic indoor environments.

Abstract. With the advancement of urbanization, building footprint data plays an important role in urban planning, 3D Real Scene and smart cities. Traditional manual contouring methods are time-consuming and laborious, while deep learning-based building extraction methods often require a large amount of labeled data and have limited generalization ability. In this paper, a zero-shot framework based on Segment Anything Model (SAM) is proposed for extracting and regularing building footprints from 3D mesh data. The method mainly consists of three steps: 1) Coarse Prompt Generation, irrelevant element’s masks such as ground and vegetation are eliminated by semi-global filtering and traditional classification method, and rough building mask is obtained as a boundary box prompt. 2) Fine mask generation: Using SAM's mask prompt capability, combined with logits map and grid elevation information with adaptive threshold to generate the fine mask prompt. Combine it with the updated bounding box to form hybrid prompt, and input SAM to generate a refined building mask. 3) Footprint regularization: Kinetic Partition, Markov random field, and Region Growth Algorithm are used to extract regularized building contours. Structural line segments from LSD guide the Kinetic Partitioning of the building. Markov random field matches building labels, while a region growth-based boundary reassignment refines the contours. The final regularized contour integrates the partitioned building zones. Our method achieved 78.31% AP50 on the Vaihingen dataset and obtained regular footprints that closely align with the true building contours on real Mesh data.

Estimation of the covariance matrix of a Gaussian Markov Random Field under a total positivity constraint