Victor De Oliveira

Objective Bayesian Analysis of Spatially Correlated Data

Spatially varying phenomena are often modeled using Gaussian random fields, specified by their mean function and covariance function. The spatial correlation structure of these models is commonly specified to be of a certain form (e.g., spherical, power exponential, rational quadratic, or Matérn) with a small number of unknown parameters. We consider objective Bayesian analysis of such spatial models, when the mean function of the Gaussian random field is specified as in a linear model. It is thus necessary to determine an objective (or default) prior distribution for the unknown mean and covariance parameters of the random field. We first show that common choices of default prior distributions, such as the constant prior and the independent Jeffreys prior, typically result in improper posterior distributions for this model. Next, the reference prior for the model is developed and is shown to yield a proper posterior distribution. A further attractive property of the reference prior is that it can be used directly for computation of Bayes factors or posterior probabilities of hypotheses to compare different correlation functions, even though the reference prior is improper. An illustration is given using a spatial dataset of topographic elevations.

Bayesian Prediction of Transformed Gaussian Random Fields

Abstract A model for prediction in some types of non-Gaussian random fields is presented. It extends the work of Handcock and Stein to prediction in transformed Gaussian random fields, where the transformation is known to belong to a parametric family of monotone transformations. The Bayesian transformed Gaussian model (BTG) provides an alternative to trans-Gaussian kriging taking into account the major sources of uncertainty, including uncertainty about the “normalizing transformation” itself, in the computation of the predictive density function. Unlike trans-Gaussian kriging, this approach mitigates the consequences of a misspecified transformation, giving in this sense a more robust predictive inference. Because the mean of the predictive distribution does not exist for some commonly used families of transformations, the median is used as the optimal predictor. The BTG model is applied in the spatial prediction of weekly rainfall amounts. Cross-validation shows the predicting performance of the BTG mo...

Bayesian prediction of clipped Gaussian random fields

This work provides a framework to perform prediction in some types of binary random elds. It is assumed the binary random eld is obtained by clipping a Gaussian random eld at a xed level. The model, following a Bayesian approach, is used to map a binary outcome over a bounded region D of the plane: For each location s0 2 D, we compute the optimal predictor of Z(s0), 0 or 1, given the binary data from a realization of the random eld, and provide measures of prediction uncertainty amenable for binary outcomes. The optimal predictor and the measure of prediction uncertainty are computed through data augmentation using Markov Chain Monte Carlo methods; a less computationally demanding plug-in approach is also described. A brief description of a geostatistical method called indicator kriging is given as well as some of its shortcomings. The prediction ability of the model is illustrated with two simulated binary maps, obtaining satisfactory results, and comparisons between the Bayesian, plug-in, and indicator kriging approaches are given. c 2000 Elsevier Science B.V. All rights reserved.

Bayesian Inference and Prediction of Gaussian Random Fields Based on Censored Data

This work develops a Bayesian approach to perform inference and prediction in Gaussian random fields based on spatial censored data. These type of data occur often in the earth sciences due either to limitations of the measuring device or particular features of the sampling process used to collect the data. Inference and prediction on the underlying Gaussian random field is performed, through data augmentation, by using Markov chain Monte Carlo methods. Previous approaches to deal with spatial censored data are reviewed, and their limitations pointed out. The proposed Bayesian approach is applied to a spatial dataset of depths of a geologic horizon that contains both left- and right-censored data, and comparisons are made between inferences based on the censored data and inferences based on “complete data” obtained by two imputation methods. It is seen that the differences in inference between the two approaches can be substantial.This work develops a Bayesian approach to perform inference and prediction in Gaussian random fields based on spatial censored data. These type of data occur often in the earth sciences due either to limitations of the measuring device or particular features of the sampling process used to collect the data. Inference and prediction on the underlying Gaussian random field is performed, through data augmentation, by using Markov chain Monte Carlo methods. Previous approaches to deal with spatial censored data are reviewed, and their limitations pointed out. The proposed Bayesian approach is applied to a spatial dataset of depths of a geologic horizon that contains both left- and right-censored data, and comparisons are made between inferences based on the censored data and inferences based on “complete data” obtained by two imputation methods. It is seen that the differences in inference between the two approaches can be substantial.

A note on the correlation structure of transformed Gaussian random fields

Summary Transformed Gaussian random fields can be used to model continuous time series and spatial data when the Gaussian assumption is not appropriate. The main features of these random fields are specified in a transformed scale, while for modelling and parameter interpretation it is useful to establish connections between these features and those of the random field in the original scale. This paper provides evidence that for many ‘normalizing’ transformations the correlation function of a transformed Gaussian random field is not very dependent on the transformation that is used. Hence many commonly used transformations of correlated data have little effect on the original correlation structure. The property is shown to hold for some kinds of transformed Gaussian random fields, and a statistical explanation based on the concept of parameter orthogonality is provided. The property is also illustrated using two spatial datasets and several ‘normalizing’ transformations. Some consequences of this property for modelling and inference are also discussed.

Hierarchical Poisson models for spatial count data

This work proposes a class of hierarchical models for geostatistical count data that includes the model proposed by Diggle et al. (1998) [13] as a particular case. For this class of models the main second-order properties of the count variables are derived, and three models within this class are studied in some detail. It is shown that for this class of models there is a close connection between the correlation structure of the counts and their overdispersions, and this property can be used to explore the flexibility of the correlation structures of these models. It is suggested that the models in this class may not be adequate to represent data consisting mostly of small counts with substantial spatial correlation. Three geostatistical count datasets are used to illustrate these issues and suggest how the results might be used to guide the selection of a model within this class.

Interpolation performance of a spatio-temporal model with spatially varying coefficients: application to PM10 concentrations in Rio de Janeiro

In this work we present a Bayesian analysis in linear regression models with spatially varying coefficients for modeling and inference in spatio-temporal processes. This kind of model is particularly appealing in situations where the effect of one or more explanatory processes on the response present substantial spatial heterogeneity. We describe for this model how to make inference about the regression coefficients and response processes under two scenarios: when the explanatory processes are known throughout the study region, and when they are known only at the sampling locations. Using a simulation experiment we investigate how parameter inference and interpolation performance are affected by some features of the data and prior distribution that is used. The proposed methodology is used to model the dataset on PM10 levels in the metropolitan region of Rio de Janeiro presented in Paez and Gamerman (2003).

On shortest prediction intervals in log-Gaussian random fields

This work considers the problem of constructing prediction intervals in log-Gaussian random fields. New prediction intervals are derived that are shorter than the standard prediction intervals of common use, where the reductions in length can be substantial in some situations. We consider both the case when the covariance parameters are known and unknown. For the latter case we propose a bootstrap calibration method to obtain prediction intervals with better coverage properties than the plug-in (estimative) prediction intervals. The methodology is illustrated using a spatial dataset consisting of cadmium concentrations from a potentially contaminated region in Switzerland.

The Effect of Uncertainty in Vascular Wall Material Properties on Abdominal Aortic Aneurysm Wall Mechanics

Clinical management of abdominal aortic aneurysms (AAA) can benefit from patient-specific computational biomechanics-based assessment of the disease. Individual variations in shape and aortic material properties are expected to influence the assessment of AAA wall mechanics. While patient-specific geometry can be reproduced using medical images, the accurate individual and regionally varying tissue material property estimation is currently not feasible. This work addresses the relative uncertainties arising from variations in AAA material properties and its effect on the ensuing wall mechanics. Computational simulations were performed with five different isotropic material models based on an ex-vivo AAA wall material characterization and a subject population sample of 28 individuals. Care was taken to exclude the compounding effects of variations in all other geometric and biomechanical factors. To this end, the spatial maxima of the principal stress (σ max), principal strain (e max), strain-energy density (ψ max), and displacement (δ max) were calculated for the diameter-matched cohort of 28 geometries for each of the five different constitutive materials. This led to 140 quasi-static simulations, the results of which were assessed on the basis of intra-patient (effect of material constants) and inter-patient (effect of individual AAA shape) differences using statistical averages, standard deviations, and Box and Whisker plots. Mean percentage variations for σ max, e max, ψ max, and δ max for the intra-patient analysis were 1.5, 7.1, 8.0, and 6.1, respectively, whereas for the inter-patient analysis these were 11.1, 4.5, 15.3, and 12.9, respectively. Changes in the material constants of an isotropic constitutive model for the AAA wall have a negligible influence on peak wall stress. Hence, this study endorses the use of population-averaged material properties for the purpose of estimating peak wall stress, strain-energy density, and wall displacement. Conversely, strain is more dependent on the material constant variation than on the differences in AAA shape in a diameter-matched population cohort.

Bayesian Spatial Modeling of Housing Prices Subject to a Localized Externality

This work proposes a non stationary random field model to describe the spatial variability of housing prices that are affected by a localized externality. The model allows for the effect of the localized externality on house prices to be represented in the mean function and/or the covariance function of the random field. The correlation function of the proposed model is a mixture of an isotropic correlation function and a correlation function that depends on the distances between home sales and the localized externality. The model is fit using a Bayesian approach via a Markov chain Monte Carlo algorithm. A dataset of 437 single family home sales during 2001 in the city of Cedar Falls, Iowa, is used to illustrate the model.