Multivariate Gaussian Markov Random Fields Research Articles

On coregionalized multivariate Gaussian Markov random fields: construction, parameterization, and Bayesian estimation and inference

On coregionalized multivariate Gaussian Markov random fields: construction, parameterization, and Bayesian estimation and inference

Non-Separable Spatio-Temporal Models via Transformed Multivariate Gaussian Markov Random Fields

Abstract Models that capture spatial and temporal dynamics are applicable in many scientific fields. Non-separable spatio-temporal models were introduced in the literature to capture these dynamics. However, these models are generally complicated in construction and interpretation. We introduce a class of non-separable transformed multivariate Gaussian Markov random fields (TMGMRF) in which the dependence structure is flexible and facilitates simple interpretations concerning spatial, temporal and spatio-temporal parameters. Moreover, TMGMRF models have the advantage of allowing specialists to define any desired marginal distribution in model construction without suffering from spatio-temporal confounding. Consequently, the use of spatio-temporal models under the TMGMRF framework leads to a new class of general models, such as spatio-temporal Gamma random fields, that can be directly used to model Poisson intensity for space–time data. The proposed model was applied to identify important environmental characteristics that affect variation in the abundance of Nenia tridens, a dominant species of gastropod in a well-studied tropical ecosystem, and to characterize its spatial and temporal trends, which are particularly critical during the Anthropocene, an epoch of time characterized by human-induced environmental change associated with climate and land use.

Some results on the Gaussian Markov Random Field construction problem based on the use of invariant subgraphs

The study of Gaussian Markov Random Fields has attracted the attention of a large number of scientific areas due to its increasing usage in several fields of application. Here, we consider the construction of Gaussian Markov Random Fields from a graph and a positive-definite matrix, which is closely related to the problem of finding the Maximum Likelihood Estimator of the covariance matrix of the underlying distribution. In particular, it is simultaneously required that the variances and the covariances between variables associated with adjacent nodes in the graph are fixed by the positive-definite matrix and that pairs of variables associated with non-adjacent nodes in the graph are conditionally independent given all other variables. The solution to this construction problem exists and is unique up to the choice of a vector of means. In this paper, some results focusing on a certain type of subgraphs (invariant subgraphs) and a representation of the Gaussian Markov Random Field as a Multivariate Gaussian Markov Random Field are presented. These results ease the computation of the solution to the aforementioned construction problem.

Bayesian disease mapping: Past, present, and future

On the occasion of the Spatial Statistics’ 10th Anniversary, I reflect on the past and present of Bayesian disease mapping and look into its future. I focus on some key developments of models, and on recent evolution of multivariate and adaptive Gaussian Markov random fields and their impact and importance in disease mapping. I reflect on Bayesian disease mapping as a subject of spatial statistics that has advanced to date, and continues to grow, in scope and complexity alongside increasing needs of analytic tools for contemporary health science research, such as spatial epidemiology, population and public health, and medicine. I illustrate (potential) utility and impact of some of the disease mapping models and methods for analysing and monitoring communicable disease such as the COVID-19 infection risks during an ongoing pandemic.

Bayesian estimation of multivariate Gaussian Markov random fields with constraint.

This article concerns with conditionally formulated multivariate Gaussian Markov random fields (MGMRF) for modeling multivariate local dependencies with unknown dependence parameters subject to positivity constraint. In the context of Bayesian hierarchical modeling of lattice data in general and Bayesian disease mapping in particular, analytic and simulation studies provide new insights into various approaches to posterior estimation of dependence parameters under "hard" or "soft" positivity constraint, including the well-known strictly diagonal dominance criterion and options of hierarchical priors. Hierarchical centering is examined as a means to gain computational efficiency in Bayesian estimation of multivariate generalized linear mixed effects models in the presence of spatial confounding and weakly identified model parameters. Simulated data on irregular or regular lattice, and three datasets from the multivariate and spatiotemporal disease mapping literature, are used for illustration. The present investigation also sheds light on the use of deviance information criterion for model comparison, choice, and interpretation in the context of posterior risk predictions judged by borrowing-information and bias-precision tradeoff. The article concludes with a summary discussion and directions of future work. Potential applications of MGMRF in spatial information fusion and image analysis are briefly mentioned.

Joint estimation of gas and wind maps for fast-response applications

This work addresses 2D gas and wind distribution mapping with a mobile robot for real-time applications. Our proposal seeks to estimate how gases released in the environment are distributed from a set of sparse and uncertain gas-concentration and wind-flow measurements; such that by exploiting the high correlation between these two magnitudes we may extrapolate their value for unexplored areas. Furthermore, because the air currents are completely conditioned by the environment, we assume a priori knowledge of static elements such as walls and obstacles when estimating both distribution maps. In particular, this joint estimation problem is modeled as a multivariate Gaussian Markov random field (GMRF), combining gas and wind observations under a common maximum a posteriori estimation problem. It considers two lattices of cells (a scalar gas-concentration field and a wind vector field) which are correlated following the physical laws of gas dispersal and fluid dynamics. Finally, we report various experiments in which our proposal is compared to other stochastic gas and gas-wind modeling methods under simulation, to evaluate their performance against a computer fluid-dynamics generated ground-truth, as well as under real and uncontrolled conditions.

Some links between conditional and coregionalized multivariate Gaussian Markov random fields

Multivariate disease mapping models are attracting considerable attention. Many modeling proposals have been made in this area, which could be grouped into three large sets: coregionalization, multivariate conditional and univariate conditional models. In this work we establish some links between these three groups of proposals. Specifically, we explore the equivalence between the two conditional approaches and show that an important class of coregionalization models can be seen as a large subclass of the conditional approaches. Additionally, we propose an extension to the current set of coregionalization models with some new unexplored proposals. This extension is able to reproduce asymmetric cross-spatial covariances for different diseases. This shows that the previously accepted belief that coregionalization was not able to reproduce models with asymmetric cross-covariances was wrong.

Some recent work on multivariate Gaussian Markov random fields

Some recent work on conditional formulation of multivariate Gaussian Markov random fields is presented. The focus is on model constructions by compatible conditionals and coregionalization. Special attention is given to multivariate generalizations of univariate models. Beginning with univariate model constructions, a survey of key approaches to formulating multivariate extensions is presented. Two challenges in the formulation and implementation of multivariate models are highlighted: (1) entanglement of spatial and non-spatial components, and (2) enforcement for positivity condition. Managing the two challenges by decomposition, separation, and constrained parameterization is discussed. Also highlighted is the challenge of flexible modeling of (conditional) cross-spatial dependencies and, in particular, asymmetric cross-spatial dependencies. Interpretation of asymmetric cross-spatial dependencies is also discussed. A coregionalization framework which connects and unifies the various lines of model development is presented. The framework enables a systematic development of a broad range of models via linear and spatially varying coregionalization, respectively, with extensions to locally adaptive models. Formulation of multivariate models over variable-specific lattices is discussed. Selected models are illustrated with examples of Bayesian multivariate and spatiotemporal disease mapping. Potential applications of coregionalization models in imaging analysis, covariance modeling, dimension reduction, and latent variable analysis are briefly mentioned.

Comments on: Some recent work on multivariate Gaussian Markov random fields

I provide comments on the article ‘Some recent work on multivariate Gaussian Markov random fields’ by Ying MacNab.

Comments on: Some recent work on multivariate Gaussian Markov random fields

Comments on: Some recent work on multivariate Gaussian Markov random fields

Comments on: Some recent work on multivariate Gaussian Markov random fields

Comments on: Some recent work on multivariate Gaussian Markov random fields

Efficient likelihood computations for some multivariate Gaussian Markov random fields

Data collected from spatial locations are often multivariate. Gaussian conditional autoregressive (CAR) models, also known as Gaussian Markov random fields, are frequently used to analyze such continuous data, or as models for the parameters of discrete distributions. Two difficulties in Gaussian maximum likelihood estimation are ensuring that the parameter estimates are allowable values, and computing the likelihood efficiently. It is shown here that, for some commonly-used multivariate CAR models, checking for allowable parameter values can be facilitated, and the likelihood can be computed very quickly.

Statistical classification of multivariate conditionally autoregressive Gaussian random field observations

The problem of supervised classifying of a multivariate Gaussian Markov random field (GMRF) observation into one of two populations specified by different regression mean models is considered. We focus on a multivariate conditionally autoregressive model, a subclass of GMRF with parametrical structure proposed by Pettitt et al. (2002) and generalized by Jin et al. (2005). For complete parametric certainty and with fixed training sample locations, the formula of the Bayes error rate is derived. Plug-in Bayes discriminant function obtained by replacing the regression and scale parameters with their ML estimators in the Bayes discriminant function is investigated. The novel formulas for the actual error rate and the approximation of the expected error rate (AER) associated with plug-in Bayes discriminant function are derived. These results are multivariate generalizations of the univariate ones. A numerical analysis of the derived formulas is implemented for the bivariate conditionally autoregressive model sampling on a regular 2-dimensional unit spacing lattice with the neighborhood structure based on the Euclidean distance. A comparison of the proposed classifier performance for different parametric structures of populations is done by studying the accuracy of the derived AER. Finally, the influence of the neighborhood order on the accuracy of the derived AER is examined.

A Gaussian Markov random field approach to convergence analysis

Spatial models have been widely applied in the context of growth regressions with spatial spillovers usually modelled by simultaneous autoregressions (SAR). Although largely used, such a class of models present some logical difficulties connected with the error behaviour, the lack of identifiability of the model parameters and their substantive interpretation. To overcome these logical pitfalls, in this paper we introduce a new specification of regional growth regressions by applying multivariate Gaussian Markov random fields (GMRFs). We discuss the theoretical properties of the proposed model and show some empirical results on the economic growth pattern of 254 NUTS-2 European regions in the period 1992–2006. We show that the proposed GMRF model is able to capture the complexity of the phenomenon including the possibility of estimating site-specific convergence parameters which may highlight clustering of regions and spatial heterogeneities in the speed of convergence.

Modeling US housing prices by spatial dynamic structural equation models

This article proposes a spatial dynamic structural equation model for the analysis of housing prices at the State level in the USA. The study contributes to the existing literature by extending the use of dynamic factor models to the econometric analysis of multivariate lattice data. One of the main advantages of our model formulation is that by modeling the spatial variation via spatially structured factor loadings, we entertain the possibility of identifying similarity “regions” that share common time series components. The factor loadings are modeled as conditionally independent multivariate Gaussian Markov Random Fields, while the common components are modeled by latent dynamic factors. The general model is proposed in a state-space formulation where both stationary and nonstationary autoregressive distributed-lag processes for the latent factors are considered. For the latent factors which exhibit a common trend, and hence are cointegrated, an error correction specification of the (vector) autoregressive distributed-lag process is proposed. Full probabilistic inference for the model parameters is facilitated by adapting standard Markov chain Monte Carlo (MCMC) algorithms for dynamic linear models to our model formulation. The fit of the model is discussed for a data set of 48 States for which we model the relationship between housing prices and the macroeconomy, using State level unemployment and per capita personal income.

A new paradigm to compare a subject to a statistical model. Application to the detection of skull abnormalities

This article deals with the construction of a statistical model which represents the location of clinically relevant regions of the skull. The landmark distribution which is estimated from 3-D CT scans is then modeled using a multivariate Gaussian Markov random field. The main contribution of this paper lies in a new way to characterize what constitutes an anomaly in a subject when it is compared to such a statistical model (which does not need to be a Gaussian Markov random field). Once global abnormality of the subject is detected, local anomalies are searched for by finding the smallest subset of landmarks whose well chosen displacement can render the subject normal according to the statistical model. A proof of concept of the idea is shown through preliminary experiments with a database of 20 subjects used for training and several subjects (including one real subject) used for testing.

Regression B‐spline smoothing in Bayesian disease mapping: with an application to patient safety surveillance

In the context of Bayesian disease mapping, recent literature presents generalized linear mixed models that engender spatial smoothing. The methods assume spatially varying random effects as a route to partially pooling data and 'borrowing strength' in small-area estimation. When spatiotemporal disease rates are available for sequential risk mapping of several time periods, the 'smoothing' issue may be explored by considering spatial smoothing, temporal smoothing and spatiotemporal interaction. In this paper, these considerations are motivated and explored through development of a Bayesian semiparametric disease mapping model framework which facilitates temporal smoothing of rates and relative risks via regression B-splines with mixed-effect representation of coefficients. Specifically, we develop spatial priors such as multivariate Gaussian Markov random fields and non-spatial priors such as unstructured multivariate Gaussian distributions and illustrate how time trends in small-area relative risks may be explored by splines which vary in either a spatially structured or unstructured manner. In particular, we show that with suitable prior specifications for the random effects ensemble, small-area relative risk trends may be fit by 'spatially varying' or randomly varying B-splines. A recently developed Bayesian hierarchical model selection criterion, the deviance information criterion, is used to assess the trade-off between goodness-of-fit and smoothness and to select the number of knots. The methodological development aims to provide reliable information about the patterns (both over space and time) of disease risks and to quantify uncertainty. The study offers a disease and health outcome surveillance methodology for flexible and efficient exploration and assessment of emerging risk trends and clustering. The methods are motivated and illustrated through a Bayesian analysis of adverse medical events (also known as iatrogenic injuries) among hospitalized elderly patients in British Columbia, Canada.

An automatic statistical segmentation algorithm for extraction of fire and smoke regions

Estimation of the extent and spread of wildland fires is an important application of high spatial resolution multispectral images. This work addresses a fuzzy segmentation algorithm to map fire extent, active fire front, hot burn scar, and smoke regions based on a statistical model. The fuzzy results are useful data sources for integrated fire behavior and propagation models built using Dynamic Data Driven Applications Systems (DDDAS) concepts that use data assimilation techniques which require error estimates or probabilities for the data parameters. The Hidden Markov Random Field (HMRF) model has been used widely in image segmentation, but it is assumed that each pixel has a particular class label belonging to a prescribed finite set. The mixed pixel problem can be addressed by modeling the fuzzy membership process as a continuous Multivariate Gaussian Markov Random Field. Techniques for estimating the class membership and model parameters are discussed. Experimental results obtained by applying this technique to two Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) images show that the proposed methodology is robust with regard to noise and variation in fire characteristics as well as background. The segmentation results of our algorithm are compared with the results of a K-means algorithm, an Expectation Maximization (EM) algorithm (which is very similar to the Fuzzy C-Means Clustering algorithm with entropy regularization), and an MRF-MAP algorithm. Our fuzzy algorithm achieves more consistent segmentation results than the comparison algorithms for these test images with the added advantage of simultaneously providing a proportion or error map needed for the data assimilation problem.

Multi-dimensional multivariate Gaussian Markov random fields with application to image processing

Recently, numerous practical applications of multivariate Gaussian Markov random fields (GMRF) on a lattice have emerged. However, the theory is not satisfactorily developed. We give various properties of multivariate GMRF for multi-dimensional lattice. In particular, some multivariate MRF are given. We discuss estimation procedures and give a numerical example from the area of image processing.