Collaborative Research on Bayesian Nonparametric Methods for Spatial and Spatiotemporal Data

主要研究人员：Kottas，AThanasios和Gelfand，AlanProposal number：dms-0505085和dms-0504953建议标题：空间和时空数据的贝叶斯非参数方法协作研究机构：加州大学圣克鲁斯分校和杜克大学研究人员开发用于空间和时空数据分析的贝叶斯非参数方法。点参考空间数据产生于大气科学、生态学、环境科学和流行病学等多个领域。事实上，这类数据往往会随着时间的推移而被复制，比如通过在监测点进行采样。在某些情况下，经过适当的初步处理，副本可被视为独立的。更常见的情况是，时间相关性被保留，时间离散化，空间过程的时间序列出现。在任何一种情况下，几乎所有空间过程的建模都是参数指定的；事实上，它几乎总是一个高斯过程，最常见的假设是平稳的。研究人员研究了新类别的非参数空间模型，以消除这些假设。这些型号适用于上述任一复制设置。在其最简单的形式中，研究人员使用狄利克雷过程来创建随机空间过程，这些过程是非高斯、非平稳和具有非齐次方差的。这些过程是通过它们的有限维分布来定义的，并且被称为空间狄利克雷过程。然后，将空间狄里克莱特过程与纯误差过程卷积，以创建具有块块分量的说明性空间过程。这种模型是分层次的，可以通过马尔可夫链蒙特卡罗方法进行拟合。在应用中，研究人员使用空间Dirichlet过程将空间随机效应引入建模，或者直接通过独立的重复或嵌入动态模型来处理时间相关性。研究人员研究了与使用空间Dirichlet过程有关的一系列问题，包括它们的理论全局和局部性质；它们作为混合模型的使用；它们与半参数混合的使用；它们在动态模型中的实现；它们在给定时间点的内插和未来时间点的预测的使用；它们与数据的非高斯第一阶段规范的使用；它们在描述多变量分布和作为特例的扩展回归建模中的使用；它们在空间点过程数据建模中的应用；以及它们对更丰富的所谓广义空间Dirichlet过程的扩展。点参考空间数据出现在环境科学、气候学、生态学、流行病学和房地产市场等不同的应用领域。随着研究人员收集越来越多的空间和时空数据，对分析的需求越来越大，以增强他们对他们正在采样的复杂过程的理解。这激发了对足够丰富的模型的需求，以适应各种全球和局部行为。这项研究的主要动机是扩大这类科学家可用的时空建模工具的目录。这项研究工作提出了第一种非参数贝叶斯时空数据分析的方法。近年来，非参数贝叶斯方法由于在工程和生物医学领域的某些问题上的成功应用而得到了越来越多的应用，将这种方法应用到时空环境中也有望取得类似的成功。特别是，预计对于流行病学、环境污染和天气模拟等领域，研究人员将重视这项工作提供的灵活的模拟框架。而且，随着适应模型的计算技术的进步，这种方法的使用预计会增加。

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.