Gaussian Markov Random Field Research Articles

Scaling priors for intrinsic Gaussian Markov random fields applied to blood pressure data

An Intrinsic Gaussian Markov Random Field (IGMRF) can be used to induce conditional dependence in Bayesian hierarchical models. IGMRFs have both a precision matrix, which defines the neighborhood structure of the model, and a precision, or scaling, parameter. Previous studies have shown the importance of selecting the prior for this scaling parameter appropriately for different types of IGMRF, as it can have a substantial impact on posterior estimates. Here, we focus on cases in one and two dimensions, where tuning of the prior is achieved by mapping it to the marginal SD of an IGMRF of corresponding dimensionality. We compare the effects of scaling various IGMRFs, including an application to real two‐dimensional blood pressure data using MCMC methods.

Time-Varying Gaussian Markov Random Fields Learning for Multivariate Time Series Clustering

Time-Varying Gaussian Markov Random Fields Learning for Multivariate Time Series Clustering

Penalty parameter selection and asymmetry corrections to Laplace approximations in Bayesian P-splines models

Laplace P-splines (LPS) combine the P-splines smoother and the Laplace approximation in a unifying framework for fast and flexible inference under the Bayesian paradigm. The Gaussian Markov random field prior assumed for penalized parameters and the Bernstein-von Mises theorem typically ensure a razor-sharp accuracy of the Laplace approximation to the posterior distribution of these quantities. This accuracy can be seriously compromised for some unpenalized parameters, especially when the information synthesized by the prior and the likelihood is sparse. Therefore, we propose a refined version of the LPS methodology by splitting the parameter space in two subsets. The first set involves parameters for which the joint posterior distribution is approached from a non-Gaussian perspective with an approximation scheme tailored to capture asymmetric patterns, while the posterior distribution for the penalized parameters in the complementary set undergoes the LPS treatment with Laplace approximations. As such, the dichotomization of the parameter space provides the necessary structure for a separate treatment of model parameters, yielding improved estimation accuracy as compared to a setting where posterior quantities are uniformly handled with Laplace. In addition, the proposed enriched version of LPS remains entirely sampling-free, so that it operates at a computing speed that is far from reach to any existing Markov chain Monte Carlo approach. The methodology is illustrated on the additive proportional odds model with an application on ordinal survey data.

Efficient Inference of Spatially-Varying Gaussian Markov Random Fields With Applications in Gene Regulatory Networks.

In this paper, we study the problem of inferring spatially-varying Gaussian Markov random fields (SV-GMRF) where the goal is to learn a network of sparse, context-specific GMRFs representing network relationships between genes. An important application of SV-GMRFs is in inference of gene regulatory networks from spatially-resolved transcriptomics datasets. The current work on inference of SV-GMRFs are based on the regularized maximum likelihood estimation (MLE) and suffer from overwhelmingly high computational cost due to their highly nonlinear nature. To alleviate this challenge, we propose a simple and efficient optimization problem in lieu of MLE that comes equipped with strong statistical and computational guarantees. Our proposed optimization problem is extremely efficient in practice: we can solve instances of SV-GMRFs with more than 2 million variables in less than 2 minutes. We apply the developed framework to study how gene regulatory networks in Glioblastoma are spatially rewired within tissue, and identify prominent activity of the transcription factor HES4 and ribosomal proteins as characterizing the gene expression network in the tumor peri-vascular niche that is known to harbor treatment resistant stem cells.

Covariance–Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference

ABSTRACT The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field u on R d as the solution of an elliptic SPDE L β u = W where L is a second-order differential operator, 2 β ∈ N is a positive parameter that controls the smoothness of u and W is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying β > d / 4 . Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator L − 2 β of the Gaussian field u by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of the method is performed and the accuracy of the method is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference. Supplementary materials for this article are available online.

Block-Adaptive Point Cloud Attribute Coding With Region-Aware Optimized Transform

Block-based compression scheme shows remarkable success in image and video coding. However, existing tree-type block partition methods usually divide point clouds into clusters with few or disjoint points due to the irregular sampling, which is adverse for the subsequent transform to exploit the local region correlation. Moreover, the widely-used optimal transform, e.g., Discrete Cosine Transform (DCT), is deduced with the assumption of the specified probability model. Thus these transforms cannot adequately conform to diverse statistical characteristics of point cloud attributes. To address the above two problems, we propose a block-adaptive codec with the optimized transform for point cloud attribute compression, including Progressive Clustering (PC) for block partition and Region-Aware Signal Modeling (RASM) for transform. PC is designed via the split-and-merge strategy. In the splitting phase, over-segmented clusters are obtained by iteratively searching the nearest neighbors to maintain the spatial continuity of points in one cluster. In the merging step, blocks are determined by merging over-segmented clusters under the guidance of minimizing texture complexity, which adjusts block sizes to adapt to texture variation. RASM adaptively captures the color correlations to respond to diverse statistical characteristics by optimizing the overall Rate-Distortion (RD) cost. Then, the optimal transform bases are obtained via the eigen-decomposition of the color correlation representation with respect to the Gaussian Markov Random Field, where input colors are obtained considering the linear relationship between geometry and color variations. The linear coefficients also need to be encoded to combine geometry to reconstruct the transform bases at the decoder in the same way as the encoder. Particularly, we accelerate the RASM by constraining the independent relationship between signals in the local region, which hardly influences RD performance. Extensive experiments not only indicate the effectiveness of PC and RASM but also show the significant RD gains achieved by our approach compared with several state-of-the-art platforms.

Machine learning based priors for Bayesian inversion in MR imaging

The Bayesian approach allows the incorporation of informative prior knowledge to effectively enable and improve the solution of inverse problems. Obtaining prior information in probabilistic terms is, however, a challenging task. Recently, machine learning has been applied for the training of generative models to facilitate the translation of historically or otherwise available data to a prior distribution. In this work, we apply this methodology to undersampled magnetic resonance imaging. In particular, we employ an autoencoder as part of a generative model to statistically regularise and solve the high-dimensional inverse problem using Bayesian inversion. Comparison with a classical Gaussian Markov random field prior is performed and numerical examples highlight the possible advantages of data-driven priors.

High-dimensional modeling of spatial and spatio-temporal conditional extremes using INLA and Gaussian Markov random fields

The conditional extremes framework allows for event-based stochastic modeling of dependent extremes, and has recently been extended to spatial and spatio-temporal settings. After standardizing the marginal distributions and applying an appropriate linear normalization, certain non-stationary Gaussian processes can be used as asymptotically-motivated models for the process conditioned on threshold exceedances at a fixed reference location and time. In this work, we adapt existing conditional extremes models to allow for the handling of large spatial datasets. This involves specifying the model for spatial observations at d locations in terms of a latent mll d dimensional Gaussian model, whose structure is specified by a Gaussian Markov random field. We perform Bayesian inference for such models for datasets containing thousands of observation locations using the integrated nested Laplace approximation, or INLA. We explain how constraints on the spatial and spatio-temporal Gaussian processes, arising from the conditioning mechanism, can be implemented through the latent variable approach without losing the computationally convenient Markov property. We discuss tools for the comparison of models via their posterior distributions, and illustrate the flexibility of the approach with gridded Red Sea surface temperature data at over 6,000 observed locations. Posterior sampling is exploited to study the probability distribution of cluster functionals of spatial and spatio-temporal extreme episodes.

Structural Gaussian priors for Bayesian CT reconstruction of subsea pipes

A non-destructive testing (NDT) application of X-ray computed tomography (CT) is inspection of subsea pipes in operation via 2D cross-sectional scans. Data acquisition is time-consuming and costly due to the challenging subsea environment. While reducing the number of projections in a CT scan yields time and cost savings, this compromises the reconstruction quality when conventional reconstruction methods are used. To address this issue we take a Bayesian approach to CT reconstruction and focus on designing an effective prior that introduces additional information to the problem. We propose a new class of structural Gaussian priors to enforce expected material properties in different regions of the reconstructed image based on available a-priori information about the pipe's layered structure. The prior is composed from Gaussian distributions, and we utilize this to propose an efficient computational strategy to sample the resulting posterior distribution. This is essential in practical NDT applications for fast processing of the large-scale data we see in CT. Numerical experiments with synthetic and real data show that the proposed structural Gaussian prior reduces artifacts and enhances contrast in the reconstruction, compared to using only a conventional Gaussian Markov random field prior or no prior information at all. Furthermore, we obtain uncertainty estimates that indicate the certainty of the reconstructions increase when more prior information is added.

Information-Driven Gas Distribution Mapping for Autonomous Mobile Robots

The ability to sense airborne pollutants with mobile robots provides a valuable asset for domains such as industrial safety and environmental monitoring. Oftentimes, this involves detecting how certain gases are spread out in the environment, commonly referred to as a gas distribution map, to subsequently take actions that depend on the collected information. Since the majority of gas transducers require physical contact with the analyte to sense it, the generation of such a map usually involves slow and laborious data collection from all key locations. In this regard, this paper proposes an efficient exploration algorithm for 2D gas distribution mapping with an autonomous mobile robot. Our proposal combines a Gaussian Markov random field estimator based on gas and wind flow measurements, devised for very sparse sample sizes and indoor environments, with a partially observable Markov decision process to close the robot’s control loop. The advantage of this approach is that the gas map is not only continuously updated, but can also be leveraged to choose the next location based on how much information it provides. The exploration consequently adapts to how the gas is distributed during run time, leading to an efficient sampling path and, in turn, a complete gas map with a relatively low number of measurements. Furthermore, it also accounts for wind currents in the environment, which improves the reliability of the final gas map even in the presence of obstacles or when the gas distribution diverges from an ideal gas plume. Finally, we report various simulation experiments to evaluate our proposal against a computer-generated fluid dynamics ground truth, as well as physical experiments in a wind tunnel.

Spatially varying anisotropy for Gaussian random fields in three-dimensional space

Isotropic covariance structures can be unreasonable for phenomena in three-dimensional spaces. In the ocean, the variability of a response may vary with depth, and ocean currents may lead to spatially varying anisotropy. We construct a class of non-stationary anisotropic Gaussian random fields (GRFs) in three dimensions through stochastic partial differential equations (SPDEs), where computations are done efficiently using Gaussian Markov random field approximations. A key novelty is the parametrization of the spatially varying anisotropy through vector fields.In a simulation study, we find that simple stationary models obtain reasonable parameter estimates with a moderate number of observations and a single realization, whereas the most complex non-stationary anisotropic model requires dense observations and multiple realizations. Further, we construct a stationary and a non-stationary GRF prior for salinity in an ocean mass outside Trondheim, Norway, based on simulations from the complex numerical ocean model SINMOD. These GRF priors are then evaluated using in-situ measurements collected with an autonomous underwater vehicle. We find that the new model outperforms the stationary anisotropic GRF prior for real-time prediction of unobserved locations both in terms of root mean square error and continuous rank probability score.

Statistical Assessment of Some Interpolation Methods for Building Grid Format Digital Bathymetric Models

As far as the knowledge of the seabed is concerned, both for safe navigation and for scientific research, 3D models, particularly digital bathymetric models (DBMs), are nowadays of fundamental importance. This work aimed to evaluate the quality of DBMs according to the interpolation methods applied to obtain grid format 3D surfaces from scattered sample points. Other complementary factors affecting DBM vertical accuracy, such as seabed morphological complexity and surveyed points sampling density, were also analyzed by using a factorial ANOVA experimental design. The experiments were performed on a multibeam dataset provided by the Italian Navy Hydrographic Institute (IIM) with an original resolution of 1 m × 1 m grid spacing, covering a surface of 0.24 km2. Six different sectors comprising different seabed morphologies were investigated. Eight sampling densities were randomly extracted from every sector, each with four repetitions. Finally, four different interpolation methods were tested, including: radial basis multiquadric function (RBMF), ordinary kriging (OK), universal kriging (UK) and Gaussian Markov random fields (GMRF). The results demonstrated that both RBMF and OK produced very accurate DBM in areas characterized by low levels of seabed ruggedness at sampling densities of only 0.0128 points/m2 (equivalent grid spacing of 8.84 m). In contrast, a higher density of 0.1024 points/m2 (3.13 m grid spacing) was required to produce accurate DBM in areas with more complex seabed topography. On the other hand, UK and GMRF were strongly influenced by morphology and sampling density, yielding higher vertical random errors and more prone to slightly overestimate seabed depths. In addition, sampling density and morphology were the factors that most influenced the vertical accuracy of the interpolated DBM. In this sense, the highly statistically significant influence of the interaction between sampling density and morphology on the vertical accuracy of the interpolated DBM confirms the need to perform a preliminary analysis of seabed morphological complexity in order to increase, if necessary, the number of surveyed points in cases of complex morphologies.

Performance of the supervised generative classifiers of spatio-temporal areal data using various spatial autocorrelation indexes

This article is concerned with a generative approach to supervised classification of spatio-temporal data collected at fixed areal units and modeled by Gaussian Markov random field. We focused on the classifiers based on Bayes discriminant functions formed by the log-ratio of the class conditional likelihoods. As a novel modeling contribution, we propose to use decision threshold values induced by three popular spatial autocorrelation indexes, i.e., Moran’s I, Geary’s C and Getis–Ord G. The goal of this study is to extend the recent investigations in the context of geostatistical and hidden Markov Gaussian models to one in the context of areal Gaussian Markov models. The classifiers performance measures are chosen to be the average accuracy rate, which shows the percentage of correctly classified test data, balanced accuracy rate specified by the average of sensitivity and specificity and the geometric mean of sensitivity and specificity. The proposed methodology is illustrated using annual death rate data collected by the Institute of Hygiene of the Republic of Lithuania from the 60 unicipalities in the period from 2001 to 2019. Classification model selection procedure is illustrated on three data sets with class labels specified by the threshold to mortality index due to acute cardiovascular event, malignant neoplasms and diseases of the circulatory system. Presented critical comparison among proposed approach classifiers with various spatial autocorrelation indexes (decision threshold values) and classifier based hidden Markov model can aid in the selection of proper classification techniques for the spatio-temporal areal data.

Gaussian Markov Random Fields and totally positive matrices

The present paper focuses on the study of the conditions under which the covariance matrix of a multivariate Gaussian distribution is totally positive, paying particular attention to multivariate Gaussian distributions that are Gaussian Markov Random Fields. More specifically, it is proven that, if the graph over which the Gaussian Markov Random Field is defined consists of path graphs and the covariances between adjacent variables on the graph are non-negative, then there always exists a reordering of the variables that renders the resulting covariance matrix totally positive. Moreover, this reordering is identified and some cases for which the conditions for the covariance matrix of a multivariate Gaussian distribution to be totally positive are necessary and sufficient are provided.

Adaptive Gaussian Markov random field spatiotemporal models for infectious disease mapping and forecasting

Recent disease mapping literature presents adaptively parameterized spatiotemporal (ST) autoregressive (AR) or conditional autoregressive (CAR) models for Bayesian prediction of COVID-19 infection risks. These models were motivated to capture complex spatiotemporal dynamics and heterogeneities of infection risks. In the present paper, we synthesize, generalize, and unify the ST AR and CAR model constructions for models augmented by adaptive Gaussian Markov random fields, with an emphasis on disease forecasting. A general convolution construction is presented, with illustrative models motivated to (i) characterize local risk dependencies and influences over both spatial and temporal dimensions, (ii) model risk heterogeneities and discontinuities, and (iii) predict and forecast areal-level disease risks and occurrences. The broadened constructions allow rich options of intuitive parameterization for disease mapping and spatial regression. Illustrative parameterizations are presented for Bayesian hierarchical models of Poisson, zero-inflated Poisson, and Bernoulli data models, respectively. They are also discussed in the context of quantifying time-varying or time-invariant effects of (omitted) covariates, with application to prediction and forecasting areal-level COVID-19 infection occurrences and probabilities of zero-infection. The model constructions presented herein have much wider scope in offering a flexible framework for modelling complex spatiotemporal data and for estimation, learning, and forecasting purposes.

Correlation matrices of Gaussian Markov random fields over cycle graphs

Gaussian Markov Random Fields over graphs have been widely used in many fields of application. Here, we address the matrix construction problem that arises in the study of Gaussian Markov Random Fields with uniform correlation, i.e., those in which all correlations between adjacent nodes in the graph are equal. We provide a characterization of the correlation matrix of a Gaussian Markov Random Field with uniform correlation over a cycle graph, which is circulant and has a sparse inverse matrix, and study the relationship with the stationary Gaussian Markov Process on the circle. Two methods for computing the correlation matrix are also provided. Ultimately, asymptotic results for cycle graphs of large order point out the relation between Gaussian Markov Random Fields with uniform correlation over cycle and path graphs.

A New Approach for 4-D Exospheric Tomography Based on Optimal Interpolation and Gaussian Markov Random Fields

Exospheric tomography is a computational 3-D imaging technique that provides the estimates of the neutral density distributions of the terrestrial exosphere from space-based ultraviolet (UV) measurements. Variability of neutral densities during geomagnetically active conditions has been previously reported, motivating the development of time-dependent tomographic techniques that can characterize both the spatial and temporal scales of densities during these events. However, solving the dynamic exospheric tomography problem can be challenging owing to its ill-posedness. In this letter, we introduce a novel algorithm for 4-D exospheric tomography based on optimal interpolation (OI) and Gaussian Markov random field (GMRF) theory. The OI analysis enables iterative reconstructions of the exosphere when a statistical background field is provided. Its mean is selected from previous knowledge of the exosphere, and its covariance matrix is estimated using GMRF. To validate the performance, we apply our proposed methodology to six days of UV data acquired by National Aeronautics and Space Administration (NASA) two-wide angle imaging neutral-atom spectrometer (TWINS) mission during the geomagnetic storm that occurred on June 15, 2008.

Image-to-Height Domain Translation for Synthetic Aperture Sonar

Synthetic aperture sonar (SAS) intensity statistics are dependent upon the sensing geometry at the time of capture. Estimating bathymetry from acoustic surveys is challenging. While several methods have been proposed to estimate seabed relief via intensity, we develop the first large-scale study that relies on deep learning models. In this work, we pose bathymetric estimation from SAS surveys as a domain translation problem of translating intensity to height. Since no dataset of coregistered seabed relief maps and sonar imagery previously existed to learn this domain translation, we produce the first large simulated dataset containing coregistered pairs of seabed relief and intensity maps from two unique sonar data simulation techniques. We apply four types of models, with varying complexity, to translate intensity imagery to seabed relief: a shape-from-shading (SFS) approach, a Gaussian Markov random field (GMRF) approach, a conditional Generative Adversarial Network (cGAN), and UNet architectures. Each model is applied to datasets containing sand ripples, rocky, mixed, and flat sea bottoms. Methods are compared in reference to the coregistered simulated datasets using L1 error. Additionally, we provide results on simulated and real SAS imagery. Our results indicate that the proposed UNet architectures outperform an SFS, a GMRF, and a pix2pix cGAN model.

Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields

This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the ℓ1 norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the ℓ1 norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than ℓ1 for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable computation time.

Model‐Free Approach for Regional Ionospheric Multi‐Instrument Imaging

AbstractThe article proposes a straightforward Kalman filter‐based method for computationally efficient ionospheric electron density multi‐instrument imaging. The approach uses direct ionospheric measurements, such as ionosondes, and general physical assumptions to estimate the uncertainty associated with the previous reconstructed time step. Therefore the method does not require any electron density model of the ionosphere as a background. The uncertainty is represented by an inverse covariance matrix constructed with Gaussian Markov random fields, allowing the problem to be solved directly with relatively high resolution. The experiments utilize measurements from dense ground‐based GNSS and low Earth orbit beacon satellite receiver networks as well as ionosondes. A synthetic simulation verification and real data validation with a specific European Incoherent Scatter Scientific Association incoherent scatter radar measurement campaign is carried out over Northern European sector. The method can be controlled using parameters with probabilistic and physically realistic interpretations that can be applied to both simulated and real‐world data. The results show that the approach is feasible for near real‐time regional ionospheric imaging. Especially, the method can be seen as an expansion to local profile measurements field of view, but with sufficient measurement coverage, it also provides information further away from the specific instrument.