Venezuelan Rainfall Data Analysed by Using a Bayesian Space–time Model

We consider a set of data from 80 stations in the Venezuelan state of Guárico consisting of accumulated monthly rainfall in a time span of 16 years. The problem of modelling rainfall accumulated over fixed periods of time and recorded at meteorological stations at different sites is studied by using a model based on the assumption that the data follow a truncated and transformed multivariate normal distribution. The spatial correlation is modelled by using an exponentially decreasing correlation function and an interpolating surface for the means. Missing data and dry periods are handled within a Markov chain Monte Carlo framework using latent variables. We estimate the amount of rainfall as well as the probability of a dry period by using the predictive density of the data. We considered a model based on a full second-degree polynomial over the spatial co-ordinates as well as the first two Fourier harmonics to describe the variability during the year. Predictive inferences on the data show very realistic results, capturing the typical rainfall variability in time and space for that region. Important extensions of the model are also discussed.     

A close look at the spatial structure implied by the CAR and SAR models

Modeling spatial interactions that arise in spatially referenced data is commonly done by incorporating the spatial dependence into the covariance structure either explicitly or implicitly via an autoregressive model. In the case of lattice (regional summary) data, two common autoregressive models used are the conditional autoregressive model (CAR) and the simultaneously autoregressive model (SAR). Both of these models produce spatial dependence in the covariance structure as a function of a neighbor matrix W and often a fixed unknown spatial correlation parameter. This paper examines in detail the correlation structures implied by these models as applied to an irregular lattice in an attempt to demonstrate their many counterintuitive or impractical results. A data example is used for illustration where US statewide average SAT verbal scores are modeled and examined for spatial structure using different spatial models.     

Fitting Gaussian Markov Random Fields to Gaussian Fields

This paper discusses the following task often encountered in building Bayesian spatial models: construct a homogeneous Gaussian Markov random field (GMRF) on a lattice with correlation properties either as present in some observed data, or consistent with prior knowledge. The Markov property is essential in designing computationally efficient Markov chain Monte Carlo algorithms to analyse such models. We argue that we can restate both tasks as that of fitting a GMRF to a prescribed stationary Gaussian field on a lattice when both local and global properties are important. We demonstrate that using the KullbackLeibler discrepancy often fails for this task, giving severely undesirable behaviour of the correlation function for lags outside the neighbourhood. We propose a new criterion that resolves this difficulty, and demonstrate that GMRFs with small neighbourhoods can approximate Gaussian fields surprisingly well even with long correlation lengths. Finally, we discuss implications of our findings for likelihood based inference for general Markov random fields when global properties are also important.     

A latent Gaussian Markov random-field model for spatiotemporal rainfall disaggregation

Summary. Rainfall data are often collected at coarser spatial scales than required for input into hydrology and agricultural models. We therefore describe a spatiotemporal model which allows multiple imputation of rainfall at fine spatial resolutions, with a realistic dependence structure in both space and time and with the total rainfall at the coarse scale consistent with that observed. The method involves the transformation of the fine scale rainfall to a thresholded Gaussian process which we model as a Gaussian Markov random field. Gibbs sampling is then used to generate realizations of rainfall efficiently at the fine scale. Results compare favourably with previous, less elegant methods.     

A Spatial-temporal Model for Temperature with Seasonal Variance

We propose a spatial-temporal stochastic model for daily average surface temperature data. First, we build a model for a single spatial location, independently on the spatial information. The model includes trend, seasonality, and mean reversion, together with a seasonally dependent variance of the residuals. The spatial dependency is modelled by a Gaussian random field. Empirical fitting to data collected in 16 measurement stations in Lithuania over more than 40 years shows that our model captures the seasonality in the autocorrelation of the squared residuals, a property of temperature data already observed by other authors. We demonstrate through examples that our spatial-temporal model is applicable for prediction and classification.     

Maximum likelihood estimation for directional conditionally autoregressive models

A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed using only the inter-distances or boundaries between the regions. To accommodate directional spatial variation, a new class of spatial models is proposed using different weights given to neighbors in different directions. The proposed model generalizes the usual CAR model by accounting for spatial anisotropy. Maximum likelihood estimators are derived and shown to be consistent under some regularity conditions. Simulation studies are presented to evaluate the finite sample performance of the new model as compared to the CAR model. Finally, the method is illustrated using a data set on the crime rates of Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

A hierarchical model for space–time surveillance data on meningococcal disease incidence

Summary. We describe a model-based approach to analyse space–time surveillance data on meningococcal disease. Such data typically comprise a number of time series of disease counts, each representing a specific geographical area. We propose a hierarchical formulation, where latent parameters capture temporal, seasonal and spatial trends in disease incidence. We then add—for each area—a hidden Markov model to describe potential additional (autoregressive) effects of the number of cases at the previous time point. Different specifications for the functional form of this autoregressive term are compared which involve the number of cases in the same or in neighbouring areas. The two states of the Markov chain can be interpreted as representing an 'endemic' and a 'hyperendemic' state. The methodology is applied to a data set of monthly counts of the incidence of meningococcal disease in the 94 départements of France from 1985 to 1997. Inference is carried out by using Markov chain Monte Carlo simulation techniques in a fully Bayesian framework. We emphasize that a central feature of our model is the possibility of calculating—for each region and each time point—the posterior probability of being in a hyperendemic state, adjusted for global spatial and temporal trends, which we believe is of particular public health interest.     

Generalized estimating equations by considering additive terms for analyzing time-course gene sets data

Time-course gene sets are collections of predefined groups of genes in some patients gathered over time. The analysis of time-course gene sets for testing gene sets which vary significantly over time is an important context in genomic data analysis. In this paper, the method of generalized estimating equations (GEEs), which is a semi-parametric approach, is applied to time-course gene set data. We propose a special structure of working correlation matrix to handle the association among repeated measurements of each patient over time. Also, the proposed working correlation matrix permits estimation of the effects of the same gene among different patients. The proposed approach is applied to an HIV therapeutic vaccine trial (DALIA-1 trial). This data set has two phases: pre-ATI and post-ATI which depend on a vaccination period. Using multiple testing, the significant gene sets in the pre-ATI phase are detected and data on two randomly selected gene sets in the post-ATI phase are also analyzed. Some simulation studies are performed to illustrate the proposed approaches. The results of the simulation studies confirm the good performance of our proposed approach.     

Non-stationary partition modeling of geostatistical data for malaria risk mapping

The most common assumption in geostatistical modeling of malaria is stationarity, that is spatial correlation is a function of the separation vector between locations. However, local factors (environmental or human-related activities) may influence geographical dependence in malaria transmission differently at different locations, introducing non-stationarity. Ignoring this characteristic in malaria spatial modeling may lead to inaccurate estimates of the standard errors for both the covariate effects and the predictions. In this paper, a model based on random Voronoi tessellation that takes into account non-stationarity was developed. In particular, the spatial domain was partitioned into sub-regions (tiles), a stationary spatial process was assumed within each tile and between-tile correlation was taken into account. The number and configuration of the sub-regions are treated as random parameters in the model and inference is made using reversible jump Markov chain Monte Carlo simulation. This methodology was applied to analyze malaria survey data from Mali and to produce a country-level smooth map of malaria risk.     

Network-based genomic discovery: application and comparison of Markov random field models

Summary.  As biological knowledge accumulates rapidly, gene networks encoding genomewide gene–gene interactions have been constructed. As an improvement over the standard mixture model that tests all the genes identically and independently distributed a priori , Wei and co-workers have proposed modelling a gene network as a discrete or Gaussian Markov random field (MRF) in a mixture model to analyse genomic data. However, how these methods compare in practical applications is not well understood and this is the aim here. We also propose two novel constraints in prior specifications for the Gaussian MRF model and a fully Bayesian approach to the discrete MRF model. We assess the accuracy of estimating the false discovery rate by posterior probabilities in the context of MRF models. Applications to a chromatin immuno-precipitation–chip data set and simulated data show that the modified Gaussian MRF models have superior performance compared with other models, and both MRF-based mixture models, with reasonable robustness to misspecified gene networks, outperform the standard mixture model.     

Maximum likelihood estimation for generalized conditionally autoregressive models of spatial data

Conditionally autoregressive (CAR) models are often used to analyze a spatial process observed over a lattice or a set of irregular regions. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. To accommodate directional and inherent anisotropy variation, a new class of spatial models is proposed that adaptively determines neighbors based on a bivariate kernel using the distances and angles between the centroid of the regions. The newly proposed model generalizes the usual CAR model in a sense of accounting for adaptively determined weights. Maximum likelihood estimators are derived and simulation studies are presented for the sampling properties of the estimates on the new model, which is compared to the CAR model. Finally the method is illustrated using a data set on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

A joint modeling approach for spatial earthquake risk variations

Modeling spatial patterns and processes to assess the spatial variations of data over a study region is an important issue in many fields. In this paper, we focus on investigating the spatial variations of earthquake risks after a main shock. Although earthquake risks have been extensively studied in the literatures, to our knowledge, there does not exist a suitable spatial model for assessing the problem. Therefore, we propose a joint modeling approach based on spatial hierarchical Bayesian models and spatial conditional autoregressive models to describe the spatial variations in earthquake risks over the study region during two periods. A family of stochastic algorithms based on a Markov chain Monte Carlo technique is then performed for posterior computations. The probabilistic issue for the changes of earthquake risks after a main shock is also discussed. Finally, the proposed method is applied to the earthquake records for Taiwan before and after the Chi-Chi earthquake.     

Bayesian inference for generalized additive mixed models based on Markov random field priors

Most regression problems in practice require flexible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a unified approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with fixed effects, metrical covariates with non-linear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random field priors with different forms and degrees of smoothness. We applied the approach in several case-studies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.     

Exact Goodness‐of‐Fit Testing for the Ising Model

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness‐of‐fit test for the finite‐lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness‐of‐fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness‐of‐fit testing for the Ising model that avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.     

Spatial modelling for binary data using␣a␣hidden conditional autoregressive Gaussian process: a multivariate extension of the probit model

A Bayesian approach to modelling binary data on a regular lattice is introduced. The method uses a hierarchical model where the observed data is the sign of a hidden conditional autoregressive Gaussian process. This approach essentially extends the familiar probit model to dependent data. Markov chain Monte Carlo simulations are used on real and simulated data to estimate the posterior distribution of the spatial dependency parameters and the method is shown to work well. The method can be straightforwardly extended to regression models.     

Efficient calculation of the normalizing constant of the autologistic and related models on the cylinder and lattice

Summary. Motivated by the autologistic model for the analysis of spatial binary data on the two-dimensional lattice, we develop efficient computational methods for calculating the normalizing constant for models for discrete data defined on the cylinder and lattice. Because the normalizing constant is generally unknown analytically, statisticians have developed various ad hoc methods to overcome this difficulty. Our aim is to provide computationally and statistically efficient methods for calculating the normalizing constant so that efficient likelihood-based statistical methods are then available for inference. We extend the so-called transition method to find a feasible computational method of obtaining the normalizing constant for the cylinder boundary condition. To extend the result to the free-boundary condition on the lattice we use an efficient path sampling Markov chain Monte Carlo scheme. The methods are generally applicable to association patterns other than spatial, such as clustered binary data, and to variables taking three or more values described by, for example, Potts models.     

On Block Updating in Markov Random Field Models for Disease Mapping

Gaussian Markov random field (GMRF) models are commonly used to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly single-site updating algorithms have been considered. However, convergence and mixing properties of such algorithms can be extremely poor due to strong dependencies of parameters in the posterior distribution. In this paper, we propose various block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for non-standard full conditionals, and can be applied in a modular fashion in a large number of different scenarios. For illustration we consider three different applications: two formulations for spatial modelling of a single disease (with and without additional unstructured parameters respectively), and one formulation for the joint analysis of two diseases. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. Implementation of such block algorithms is relatively easy using methods for fast sampling of Gaussian Markov random fields ( Rue, 2001 ). By comparison, Monte Carlo estimates based on single-site updating can be rather misleading, even for very long runs. Our results may have wider relevance for efficient MCMC simulation in hierarchical models with Markov random field components.     

A comparison of efficiencies between quasi-likelihood and pseudo-likelihood estimates in non-separable spatial–temporal binary data

The goal of this paper is to compare the performance of two estimation approaches, the quasi-likelihood estimating equation and the pseudo-likelihood equation, against model mis-specification for non-separable binary data. This comparison, to the authors’ knowledge, has not been done yet. In this paper, we first extend the quasi-likelihood work on spatial data to non-separable binary data. Some asymptotic properties of the quasi-likelihood estimate are also briefly discussed. We then use the techniques of a truncated Gaussian random field with a quasi-likelihood type model and a Gibbs sampler with a conditional model in the Markov random field to generate spatial–temporal binary data, respectively. For each simulated data set, both of the estimation methods are used to estimate parameters. Some discussion about the simulation results are also included.     

A markov regression model for the analysis of the postpartum lactational amenorrhea

Failure time data represent a particular case of binary longitudinal data. The corresponding analysis of the effect of explanatory covariates repeatedly collected over time on the failure rate has been largely facilitated by the Cox semi-parametric regression model. However, neither the interpretation of the estimated parameters associated with time-dependent covariates is straight-forward, nor does this model fully account for the dynamics of the effect of a covariate over time. Markovian regression models appear as complementary tools to address these specific issues from the predictive point of view. We illustrate these aspects using data from the WHO multicenter study, which was designed to analyze the relation between the duration of postpartum lactational amenorrhea and the breastfeeding pattern. One of the main advantage of this approach applied to the field of reproductive epidemiology was to provide a flexible tool, easily and directly understood by clinicians and fieldworkers, for simulating situations, which were still unobserved, and to predict their effects on the duration of amenorrhea.     

Parameter estimation and inference with spatial lags and cointegration

This article studies dynamic panel data models in which the long run outcome for a particular cross-section is affected by a weighted average of the outcomes in the other cross-sections. We show that imposing such a structure implies a model with several cointegrating relationships that, unlike in the standard case, are nonlinear in the coe?cients to be estimated. Assuming that the weights are exogenously given, we extend the dynamic ordinary least squares methodology and provide a dynamic two-stage least squares estimator. We derive the large sample properties of our proposed estimator under a set of low-level assumptions. Then our methodology is applied to US financial market data, which consist of credit default swap spreads, as well as firm-specific and industry data. We construct the economic space using a “closeness” measure for firms based on input–output matrices. Our estimates show that this particular form of spatial correlation of credit default swap spreads is substantial and highly significant.