Intrinsic 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.

Control charts for monitoring two-dimensional spatial count data with spatial correlations

The quality characteristic of some manufacturing processes is represented by two-dimensional (2D) spatial count data in which spatial correlations are commonly observed. Statistical process control (SPC) is important and challenging for monitoring such kind of data. However, there is a scarcity of methods for monitoring spatial count data in SPC. In this paper, we integrate an intrinsic Gaussian Markov random field (IGMRF) with the hierarchical Bayesian model for modelling 2D spatial count data. Then, we adopt the Markov chain Monte Carlo (MCMC) method to tackle the issue of hierarchical model parameter estimation. Moreover, based on the estimates, we develop corresponding monitoring schemes utilizing the multivariate exponentially weighted moving average (MEWMA) procedure to detect shifts in the general trend and the spatial correlation. Extensive numerical results are presented with regard to the performance of the proposed schemes for detecting changes in the coefficient and the random spatial error. Finally, a real example from the wafer manufacturing process is used to illustrate the implementation and effectiveness of the proposed approach.

P-spline smoothing for spatial data collected worldwide

Spatial data collected worldwide from a huge number of locations is frequently used in environmental and climate studies. Spatial modelling for this type of data presents both methodological and computational challenges. In this work we illustrate a computationally efficient non-parametric framework in order to model and estimate the spatial field while accounting for geodesic distances between locations. The spatial field is modelled via penalized splines (P-splines) using intrinsic Gaussian Markov Random Field (GMRF) priors for the spline coefficients. The key idea is to use the sphere as a surrogate for the Globe, then build the basis of B-spline functions on a geodesic grid system. The basis matrix is sparse as is the precision matrix of the GMRF prior, thus computational efficiency is gained by construction. We illustrate the approach with a real climate study, where the goal is to identify the Intertropical Convergence Zone using high-resolution remote sensing data.

Approximate large-scale Bayesian spatial modeling with application to quantitative magnetic resonance imaging

We consider the Bayesian inference of nonlinear, large-scale regression problems in which the parameters model the spatial distribution of some property. A homoscedastic Gaussian sampling distribution is supposed as well as certain assumptions about the regression function. Propriety of the posterior and the existence of its moments are explored when using improper prior distributions expressing different levels of prior knowledge, ranging from a purely noninformative prior over intrinsic Gaussian Markov random field priors to a partition prior. The considered class of problems includes magnetic resonance fingerprinting (MRF). We apply an approximate Bayesian inference to this particular application and demonstrate its practicability in dimensions up to $$10^5$$ or larger. The benefit of incorporating substantial prior knowledge is illustrated. By analyzing simulated realistic MRF data, it is shown that MAP estimates can significantly improve the results achieved with maximum likelihood estimation.

Penalized complexity priors for degrees of freedom in Bayesian P-splines

Bayesian penalized splines (P-splines) assume an intrinsic Gaussian Markov random field prior on the spline coefficients, conditional on a precision hyper-parameter [Formula: see text]. Prior elicitation of [Formula: see text] is difficult. To overcome this issue, we aim to building priors on an interpretable property of the model, indicating the complexity of the smooth function to be estimated. Following this idea, we propose penalized complexity (PC) priors for the number of effective degrees of freedom. We present the general ideas behind the construction of these new PC priors, describe their properties and show how to implement them in P-splines for Gaussian data.

Bayesian Nonparametric Mixture Estimation for Time-Indexed Functional Data inR

We present growfunctions for R that offers Bayesian nonparametric estimation models for analysis of dependent, noisy time series data indexed by a collection of domains. This data structure arises from combining periodically published government survey statistics, such as are reported in the Current Population Study (CPS). The CPS publishes monthly, by-state estimates of employment levels, where each state expresses a noisy time series. Published state-level estimates from the CPS are composed from household survey responses in a model-free manner and express high levels of volatility due to insufficient sample sizes. Existing software solutions borrow information over a modeled time-based dependence to extract a de-noised time series for each domain. These solutions, however, ignore the dependence among the domains that may be additionally leveraged to improve estimation efficiency. The growfunctions package offers two fully nonparametric mixture models that simultaneously estimate both a time and domain-indexed dependence structure for a collection of time series: (1) A Gaussian process (GP) construction, which is parameterized through the covariance matrix, estimates a latent function for each domain. The covariance parameters of the latent functions are indexed by domain under a Dirichlet process prior that permits estimation of the dependence among functions across the domains: (2) An intrinsic Gaussian Markov random field prior construction provides an alternative to the GP that expresses different computation and estimation properties. In addition to performing denoised estimation of latent functions from published domain estimates, growfunctions allows estimation of collections of functions for observation units (e.g., households), rather than aggregated domains, by accounting for an informative sampling design under which the probabilities for inclusion of observation units are related to the response variable. growfunctions includes plot functions that allow visual assessments of the fit performance and dependence structure of the estimated functions. Computational efficiency is achieved by performing the sampling for estimation functions using compiled C++.

Smooth Scalar-on-Image Regression via Spatial Bayesian Variable Selection

We develop scalar-on-image regression models when images are registered multidimensional manifolds. We propose a fast and scalable Bayes’ inferential procedure to estimate the image coefficient. The central idea is the combination of an Ising prior distribution, which controls a latent binary indicator map, and an intrinsic Gaussian Markov random field, which controls the smoothness of the nonzero coefficients. The model is fit using a single-site Gibbs sampler, which allows fitting within minutes for hundreds of subjects with predictor images containing thousands of locations. The code is simple and is provided in the online Appendix (see the “Supplementary Materials” section). We apply this method to a neuroimaging study where cognitive outcomes are regressed on measures of white-matter microstructure at every voxel of the corpus callosum for hundreds of subjects.

A hierarchical model for characterising spatial wafer variations

Silicon wafers are commonly used materials in the semiconductor manufacturing industry. Their geometric quality directly affects the production cost and yield. Therefore, improvement in the quality of wafers is critical for meeting the current competitive market needs. Conventional summary metrics such as total thickness variation, bow and warp can neither fully reflect the local variability within each wafer nor provide useful insight for root cause diagnosis and quality improvement. The advancement of sensing technology enables two-dimensional (2D) data mapping to characterise the geometric shapes of wafers, which provides more information than summary metrics. The objective of this research is to develop a statistical model to characterise the thickness variation of wafers based on 2D data maps. Specifically, the thickness variation of wafers is decomposed into macro-scale and micro-scale variations, which are modelled as a cubic curve and a first-order intrinsic Gaussian Markov random field, respectively. The models can successfully capture both the macro-scale mean trend and the micro-scale local variation, with important engineering implications for process monitoring, fault diagnosis and run-to-run control. A practical case study from a wafer manufacturing process is performed to show the effectiveness of the proposed methodology.

The impact of spatial scales and spatial smoothing on the outcome of bayesian spatial model.

Discretization of a geographical region is quite common in spatial analysis. There have been few studies into the impact of different geographical scales on the outcome of spatial models for different spatial patterns. This study aims to investigate the impact of spatial scales and spatial smoothing on the outcomes of modelling spatial point-based data. Given a spatial point-based dataset (such as occurrence of a disease), we study the geographical variation of residual disease risk using regular grid cells. The individual disease risk is modelled using a logistic model with the inclusion of spatially unstructured and/or spatially structured random effects. Three spatial smoothness priors for the spatially structured component are employed in modelling, namely an intrinsic Gaussian Markov random field, a second-order random walk on a lattice, and a Gaussian field with Matérn correlation function. We investigate how changes in grid cell size affect model outcomes under different spatial structures and different smoothness priors for the spatial component. A realistic example (the Humberside data) is analyzed and a simulation study is described. Bayesian computation is carried out using an integrated nested Laplace approximation. The results suggest that the performance and predictive capacity of the spatial models improve as the grid cell size decreases for certain spatial structures. It also appears that different spatial smoothness priors should be applied for different patterns of point data.

Bayesian multiscale analysis of images modeled as Gaussian Markov random fields

A Bayesian multiscale technique for the detection of statistically significant features in noisy images is proposed. The prior is defined as a stationary intrinsic Gaussian Markov random field on a toroidal graph, which enables efficient computation of the relevant posterior marginals. Hence the method is applicable to large images produced by modern digital cameras. The technique is demonstrated in two examples from medical imaging.

Physics-driven Bayesian hierarchical modeling of the nanowire growth process at each scale

Despite significant advances in nanoscience, current physical models are unable to predict nanomanufacturing processes under uncertainties. This research work aims to model the nanowire (NW) growth process at any scale of interest. The main idea is to integrate available data and physical knowledge through a Bayesian hierarchical framework with consideration of scale effects. At each scale the NW growth model describes the time–space evolution of NWs at different sites on a substrate. The model consists of two major components: NW morphology and local variability. The morphology component represents the overall trend characterized by growth kinetics. The area-specific variability is less understood in nanophysics due to complex interactions among neighboring NWs. The local variability is therefore modeled by an intrinsic Gaussian Markov random field to separate it from the growth kinetics in the morphology component. Case studies are provided to illustrate the NW growth process model at coarse and fine scales, respectively.

Linear Neighborhood Propagation and Its Applications

In this paper, a novel graph-based transductive classification approach, called Linear Neighborhood Propagation, is proposed. The basic idea is to predict the label of a data point according to its neighbors in a linear way. This method can be cast into a second-order intrinsic Gaussian Markov random field framework. Its result corresponds to a solution to an approximate inhomogeneous biharmonic equation with Dirichlet boundary conditions. Different from existing approaches, our approach provides a novel graph structure construction method by introducing multiple-wise edges instead of pairwise edges, and presents an effective scheme to estimate the weights for such multiple-wise edges. To the best of our knowledge, these two contributions are novel for semi-supervised classification. The experimental results on image segmentation and transductive classification demonstrate the effectiveness and efficiency of the proposed approach.

Fast estimation of spatially dependent temporal vegetation trends using Gaussian Markov random fields

There is a need for efficient methods for estimating trends in spatio-temporal Earth Observation data. A suitable model for such data is a space-varying regression model, where the regression coefficients for the spatial locations are dependent. A second order intrinsic Gaussian Markov Random Field prior is used to specify the spatial covariance structure. Model parameters are estimated using the Expectation Maximisation (EM) algorithm, which allows for feasible computation times for relatively large data sets. Results are illustrated with simulated data sets and real vegetation data from the Sahel area in northern Africa. The results indicate a substantial gain in accuracy compared with methods based on independent ordinary least squares regressions for the individual pixels in the data set. Use of the EM algorithm also gives a substantial performance gain over Markov Chain Monte Carlo-based estimation approaches.

Spatial stochastic volatility for lattice data

Spatial heteroscedasticity may arise jointly with spatial autocorrelation in lattice data collected from agricultural trials and environmental studies. This leads to spatial clustering not only in the level but also in the variation of the data, the latter of which may be very important, for example, in constructing prediction intervals. This article introduces a spatial stochastic volatility (SSV) component into the widely used conditional autoregressive (CAR) model to capture the spatial clustering in heteroscedasticity. The SSV component is a mean zero, conditionally independent Gaussian process given a latent spatial process of the variances. The logarithm of the latent variance process is specified by an intrinsic Gaussian Markov random field. The SSV model relaxes the traditional homoscedasticity assumption for spatial heterogeneity and brings greater flexibility to the popular spatial statistical models. The Bayesian method is used for inference. The full conditional distribution of the heteroscedasticity components can be shown to be log-concave, which facilitates an adaptive rejection sampling algorithm. Application to the well-known wheat yield data illustrates that incorporating spatial stochastic volatility may reveal the spatial heteroscedasticity hidden from existing analyses.