Conventional geostatistical methodology solves the problem of predicting the realised value of a linear functional of a Gaussian spatial stochastic process, S(x), based on observations Y i = S(x i ) + Z i at sampling locations x i , where the Z i are mutually independent, zero-mean Gaussian random variables. We describe two spatial applications for which Gaussian distributional assumptions are clearly inappropriate. The first concerns the assessment of residual contamination from nuclear weapons testing on a South Pacific island, in which the sampling method generates spatially indexed Poisson counts conditional on an unobserved spatially varying intensity of radioactivity; we conclude that a coventional geostatistical analysis oversmooths the data and underestimates the spatial extremes of the intensity. The second application provides a description of spatial variation in the risk of campylobacter infections relative to other enteric infections in part of North Lancashire and South C...

Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its two-parameter extension, the Pitman-Yor process and finite dimensional Dirichlet priors. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, two-parameter Poisson-Dirichlet process, prediction rule, random probability measure, species sampling sequence.

. A continuous spatial model can be constructed by convolving a very simple, perhaps independent, process with a kernel or point spread function. This approach for constructing a spatial process o#ers a number of advantages over specification through a spatial covariogram. In particular, this process convolution specification leads to compuational simplifications and easily extends beyond simple stationary models. This paper uses process convolution models to build space and space-time models that are flexible and able to accomodate large amounts of data. Data from environmental monitoring is considered. 1 Introduction Modeling spatial data with Gaussian processes is the common thread of all geostatistical analyses. Some notable references in this area include Matheron (1963), Journel and Huijbregts (1978), Ripley (1981), Cressie (1991), Wackernagel (1995), and Stein (1999). A common approach is to model spatial dependence through the covariogram c(), so that covariance between any t...

This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailor-made to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12

We develop computational procedures for a class of Bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures. A key feature of our approach is that explicit expressions for posterior distributions of these models share many common structural features with the posterior distributions of Bayesian hierarchical models using the Dirichlet process. Using this fact, along with an approximation for the weighted gamma process, we show that with some care, one can adapt efficient algorithms used for the Dirichlet process to this setting. We discuss blocked Gibbs sampling procedures and Pólya urn Gibbs samplers. We illustrate our methods with applications to proportional hazard models, Poisson spatial regression models, recurrent events, and panel count data.

Abstract: We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial non-homogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mix-ture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clus-tering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.

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A class of random hazard rates, that is defined as a mixture of an indicator kernel convoluted with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of S-paths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over S-paths. The path characterization or the estimator is proved to be a Rao-Blackwellization of an existing partition characterization or partition-sum estimator. This accentuates the importance of S-path in Bayesian modeling of monotone hazard rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to approximate this class of estimates. It is shown that S-path characterization also exists in modeling with covariates by a proportional hazard model, and the proposed algorithm again applies. Numerical results of the method are given to demonstrate its practicality and effectiveness.

Information Systems (GIS) provide new opportunities for forest inventory. These technologies allow representation of forest variables using rasters with cell sizes on the order of 25 m. Such rasters can be estimated from remotely sensed data using models of the relationship between the image’s digital number and the forest variables. This thesis investigates the possibility of using estimation methods incorporating remotely sensed data as well as spatial similarity of neighbouring field measurements, to improve prediction accuracy compared to using only remotely sensed data. Two new spatial prediction methods are presented and evaluated: ordinary kriging using information about edges detected in remotely sensed images, and prediction using Markov Chain Monte Carlo (MCMC) simulation of a new Bayesian state-space model. In addition, ordinary kriging, stratified ordinary kriging, ordinary cokriging, collocated ordinary cokriging, simple kriging with varying local means, and spatial regression using the autoregressive response model, are also evaluated. The methods are applied to predict forest stem volume per hectare in boreal forest in northern Sweden (Lat. 64°14’N, Long. 19°40’E) using Landsat TM data and a large field sampled dataset. Prediction accuracy, as well as

We formulate the inverse problem of solving Fredholm integral equations of the first kind as a nonparametric Bayesian inference problem, using Lévy random fields (and their mixtures) as prior distributions. Posterior distributions for all features of interest are computed employing novel Markov chain Monte Carlo numerical methods in infinite-dimensional spaces, based on generalizations and extensions of the authors ’ Inverse Lévy Measure (ILM) algorithm. The method is also well suited for deconvolution problems, for inverting Laplace and Fourier transforms, and for other linear and nonlinear problems in which the unknown feature is high- (or even infinite-) dimensional and where the corresponding forward problem may be solved rapidly. The methods are illustrated for an application to an important problem in rheology: that of inferring the molecular weight distribution of polymers from conventional rheological measurements, in which we achieve not just a point estimate but a posterior probability density plot representing all uncertainty about the weight.