Collaborative Research on Bayesian Nonparametric Methods for Spatial and Spatiotemporal Data

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ABSTRACT Principal Investigators: Kottas, Athanasios and Gelfand, AlanProposal Number: DMS - 0505085 and DMS - 0504953Proposal Title: Collaborative Research on Bayesian Nonparametric Methods for Spatial and Spatiotemporal DataInstitution: University of California Santa Cruz and Duke UniversityThe investigators develop Bayesian nonparametric methodology forspatial and spatio-temporal data analysis. Point-referenced spatialdata arises in several fields, including atmospheric science, ecology, environmental science, and epidemiology. In fact, often such data is replicated across time say through sampling at monitoring sites. In certain cases, with appropriate preliminary manipulation, thereplicates may be viewed as independent. More often, the temporal dependence is retained and, discretizing time, a time series ofspatial processes emerges. In either case, virtually all of themodeling for the spatial processes is specified parametrically; in fact, it is almost always a Gaussian process which is most frequentlyassumed to be stationary. The investigators study new classes of nonparametric spatial models to remove these assumptions. These models are applicable to either of the above replicated settings. In its simplest form, the investigators use Dirichlet processes to create random spatial processes, which are non-Gaussian, nonstationary, and have non-homogeneous variance. These processes are defined throughtheir finite dimensional distributions, and are referred to as spatial Dirichlet processes. A spatial Dirichlet process is then convolvedwith a pure error process to create an illustrative spatial processwith a nugget component. Such models are hierarchical and can befitted through Markov chain Monte Carlo methods. In application, the investigators use spatial Dirichlet processes to introduce spatialrandom effects into the modeling, either directly with independent replicates or embedded within a dynamic model to handle temporal dependence. The investigators study an assortment of problemsassociated with the use of spatial Dirichlet processes, includingtheir theoretical global and local properties; their use as mixingmodels; their use with semiparametric mixing; their implementation in dynamic models; their utilization for interpolation at given timepoints and for forecasting at future time points; their use with non-Gaussian first stage specifications for the data; their use in describing multivariate distributions and, as a special case, for extended regression modeling; their use in modeling spatial point process data; and their extension to richer classes of so-called generalized spatial Dirichlet processes. Point-referenced spatial data arises in application areas as diverse as environmental science, climatology, ecology, epidemiology, andreal estate markets. As researchers collect more and more space and space-time data, the need for analyses to enhance their understanding of the complex processes they are sampling grows. This inspires the need for sufficiently rich models to accommodate a variety of globaland local behaviors. The primary motivation for this research is to expand the catalog of space-time modeling tools available to such scientists. This research work suggests the first approach to nonparametric Bayesian spatial and spatio-temporal data analysis. Nonparametric Bayesian approaches have witnessed increased utilization in recent years as a result of their successful application to certain problems in, for example, engineering and biomedical fields.Similar success is anticipated by bringing this methodology to space-time settings. In particular, it is anticipated that, for fields such as epidemiology, environmental contamination and weathermodeling, researchers will value the flexible modeling framework the work offers. And, an increase in usage of the methodology is expected as the computational techniques to fit the models are advanced.