Generalized spatial Dirichlet process models (original) (raw)

Bayesian nonparametric spatial modeling with Dirichlet process mixing

Journal of the American Statistical …, 2005

Customary modeling for continuous point-referenced data assumes a Gaussian process which is often taken to be stationary. When such models are fitted within a Bayesian framework, the unknown parameters of the process are assumed to be random so a random Gaussian process results. Here, we propose a novel spatial Dirichlet process mixture model to produce a random spatial process which is neither Gaussian nor stationary. We first develop a spatial Dirichlet process model for spatial data and discuss its properties. Due to familiar limitations associated with direct use of Dirichlet process models, we introduce mixing through this process against a pure error process. We then examine properties of models created through such Dirichlet process mixing. In the Bayesian framework, posterior inference is implemented using Gibbs sampling as we detail. Spatial prediction raises interesting questions but can be handled. Finally, we illustrate the approach using simulated data as well as a dataset involving precipitation measurements over the Languedoc-Roussillon region in southern France.

Spatial regression for marked point processes. (With discussion)

In a wide range of applications, dependence on smoothly-varying covariates leads spatial point count intensities to feature positive correlation for nearby locations. In applications where the points are "marked" with individual attributes, the distributions for points with varying attributes may also differ. We introduce a class of hierarchical spatial regression models for relating marked point process intensities to location-specific covariates and to individual-specific attributes, and for modeling the remaining intensity variation that arises from dependence on unobserved or unreported covariates. The magnitude of residual intensity variation is a measure of how completely the covariates explain the observed variations in point intensities. The models, extending those recently introduced by Wolpert and Ickstadt, treat the point patterns as doubly-stochastic Poisson random fields with a random inhomogeneous Poisson intensity given by a spatial mixture of gamma or other infinitely-divisible independent-increment random fields. Bayesian prior distributions for a third level of hierarchy are elicited to reflect beliefs about the homogeneity, continuity, and similar features of the intensity. Inference is based on posterior and predictive distributions computed using Markov chain Monte Carlo methods featuring data augmentation and an efficient method for sampling from independent-increment random fields. The models are illustrated in an application to a four-dimensional spatial regression analysis of origin/destination trip data from the 1994/95 Metro survey of Portland, Oregon.

Spatial process modelling for univariate and multivariate dynamic spatial data

Environmetrics, 2005

There is a considerable literature in spatiotemporal modelling. The approach adopted here applies to the setting where space is viewed as continuous but time is taken to be discrete. We view the data as a time series of spatial processes and work in the setting of dynamic models, achieving a class of dynamic models for such data. We seek rich, flexible, easy-to-specify, easy-to-interpret, computationally tractable specifications which allow very general mean structures and also non-stationary association structures. Our modelling contributions are as follows. In the case where univariate data are collected at the spatial locations, we propose the use of a spatiotemporally varying coefficient form. In the case where multivariate data are collected at the locations, we need to capture associations among measurements at a given location and time as well as dependence across space and time. We propose the use of suitable multivariate spatial process models developed through coregionalization. We adopt a Bayesian inference framework. The resulting posterior and predictive inference enables summaries in the form of tables and maps, which help to reveal the nature of the spatiotemporal behaviour as well as the associated uncertainty. We illuminate various computational issues and then apply our models to the analysis of climate data obtained from the National Center for Atmospheric Research to analyze precipitation and temperature measurements obtained in Colorado in 1997.

Modeling disease incidence data with spatial and spatio temporal Dirichlet process mixtures

2008

Disease incidence or mortality data are typically available as rates or counts for specified regions, collected over time. We propose Bayesian nonparametric spatial modeling approaches to analyze such data. We develop a hierarchical specification using spatial random effects modeled with a Dirichlet process prior. The Dirichlet process is centered around a multivariate normal distribution. This latter distribution arises from a log-Gaussian process model that provides a latent incidence rate surface, followed by block averaging to the areal units determined by the regions in the study. With regard to the resulting posterior predictive inference, the modeling approach is shown to be equivalent to an approach based on block averaging of a spatial Dirichlet process to obtain a prior probability model for the finite dimensional distribution of the spatial random effects. We introduce a dynamic formulation for the spatial random effects to extend the model to spatio-temporal settings. Posterior inference is implemented through Gibbs sampling. We illustrate the methodology with simulated data as well as with a data set on lung cancer incidences for all 88 counties in the state of Ohio over an observation period of 21 years.

Variance modeling for nonstationary spatial processes with temporal replications

Journal of Geophysical Research: Atmospheres, 2003

1] We have previously formulated a Bayesian approach to the Sampson and Guttorp model for the nonstationary correlation function r(x, x 0 ) of a Gaussian spatial process . This model assumes that the nonstationarity can be encoded through a bijective space deformation, f, that defines a new coordinate system in which the spatial correlation function can be considered isotropic, namely r(x, x 0 ) = r(k f (x) À f (x 0 )k), where r belongs to a known parametric family. We extend this model to incorporate spatial heterogeneity in site-specific temporal variances. In our Bayesian framework the variances are considered (hidden) realizations of another spatial process, which we model as log-Gaussian, with correlation structure expressed in terms of the same spatial deformation function underlying that of the observed process. We demonstrate the method in simulations and in an application. Variance modeling for nonstationary spatial processes with temporal replications,

Bayesian Spatial Statistical Modeling

Handbook of Regional Science, 2013

Spatial statistics has in the last decade or two emerged as a major sub-specialism within statistics. Applications areas are diverse, and there is cross-fertilization with methodologies in other disciplines (econometrics, epidemiology, geography, geology, climatology, ecology, etc). This chapter reviews three major settings and techniques that have attracted attention from statisticians: spatial econometrics and simultaneous autoregressive models, spatial epidemiology and conditional autoregressive models, and geostatistical methods for point pattern data. The review is oriented to Bayesian inferences for such models, including discussion of choice of prior densities, questions of identification, outcomes of interest, and methods of estimation (using Markov chain Monte Carlo).

An autoregressive point source model for spatial processes

Environmetrics, 2009

We suggest a parametric modeling approach for nonstationary spatial processes driven by point sources. Baseline near-stationarity, which may be reasonable in the absence of a point source, is modeled using a conditional autoregressive (CAR) Markov random field. Variability due to the point source is captured by our proposed autoregressive point source (ARPS) model. Inference proceeds according to the Bayesian hierarchical paradigm, and is implemented using Markov chain Monte Carlo (MCMC) methods. The parametric approach allows a formal test of effectiveness of the point source. Application is made to a real dataset on electric potential measurements in a field containing a metal pole and the finding is that our approach captures the pole's impact on small-scale variability of the electric potential process.

Bayesian Wombling for Spatial Point Processes

Biometrics, 2009

In many applications involving geographically indexed data, interest focuses on identifying regions of rapid change in the spatial surface, or the related problem of the construction or testing of boundaries separating regions with markedly different observed values of the spatial variable. This process is often referred to in the literature as boundary analysis or wombling. Recent developments in hierarchical models for point-referenced (geostatistical) and areal (lattice) data have led to corresponding statistical wombling methods, but there does not appear to be any literature on the subject in the point process case, where the locations themselves are assumed to be random and likelihood evaluation is notoriously difficult. We extend existing point-level and areal wombling tools to this case, obtaining full posterior inference for multivariate spatial random effects that, when mapped, can help suggest spatial covariates still missing from the model. In the areal case we can also construct wombled maps showing significant boundaries in the fitted intensity surface, while the point-referenced formulation permits testing the significance of a postulated boundary. In the computationally demanding point-referenced case, our algorithm combines Monte Carlo approximants to the likelihood with a predictive process step to reduce the dimension of the problem to a manageable size. We apply these techniques to an analysis of colorectal and prostate cancer data from the northern half of Minnesota, where a key substantive concern is possible similarities in their spatial patterns, and whether they are affected by each patient's distance to facilities likely to offer helpful cancer screening options.

Spatial models for spatial statistics: some unification

Journal of Vegetation Science, 1993

AbstracL A general statistical framework is proposed for comparing linear models of spatial process and pattern. A spatial linear model for nested analysis of variance can be based on either fixed effects or random effects. originally used a fixed effects model, but there are also examples of random effects models in Lhe soil science literature. Assuming imrinsic stationarity for a linear model, the expectations of a spatial nested ANOVA lllld (wo teon local variance (lTLV, Hill 1973) are funclions of the variogram, and several examples are given. Paired quadrat variance (PQV. Ludwig & Goodall 1978) is a variogram estimator which can be used 10 approximate TIl..V. and we provide an example from ecological data. BOIh nested ANOVA and TILV can be seen as weighted lag-I variogram estimators that are functions of support, rather than distance. We show that there are two unbiased estimators for the variogram under aggregation, and computer simulation shows that the estimator with smaller variance depends on Ihe process autocorrelation.

Modelling spatially correlated data via mixtures: a Bayesian approach

Journal of The Royal Statistical Society Series B-statistical Methodology, 2000

The paper develops mixture models for spatially indexed data. We confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in the mixture that vary from one location to another. Our specific focus is on Poisson-distributed data, and applications in disease mapping. We work in a Bayesian framework, with the Poisson parameters drawn from gamma priors, and an unknown number of components. We propose two alternative models for spatially dependent weights, based on transformations of autoregressive Gaussian processes: in one (the logistic normal model), the mixture component labels are exchangeable; in the other (the grouped continuous model), they are ordered. Reversible jump Markov chain Monte Carlo algorithms for posterior inference are developed. Finally, the performances of both of these formulations are examined on synthetic data and real data on mortality from a rare disease.