Collaborative Research on Bayesian Nonparametric Methods for Spatial and Spatiotemporal Data

主要研究人员：kotta， Athanasios和Gelfand， alan提案号：DMS - 0505085和DMS - 0504953提案题目：空间和时空数据贝叶斯非参数方法的合作研究机构：加州大学圣克鲁斯分校和杜克大学研究人员开发了用于空间和时空数据分析的贝叶斯非参数方法。点参考空间数据出现在几个领域，包括大气科学、生态学、环境科学和流行病学。事实上，这样的数据通常是通过在监测点采样来复制的。在某些情况下，经过适当的初步处理，副本可以被视为独立的。更常见的是，时间依赖性被保留，离散时间，出现空间过程的时间序列。在任何一种情况下，几乎所有空间过程的建模都是参数化的；事实上，它几乎总是一个高斯过程，通常被认为是平稳的。研究者研究了新的非参数空间模型来消除这些假设。这些模型适用于上述任何一种复制设置。在最简单的形式中，研究者使用狄利克雷过程来创建随机空间过程，这些过程是非高斯的，非平稳的，并且具有非齐次方差。这些过程是通过它们的有限维分布来定义的，被称为空间狄利克雷过程。然后将空间狄利克雷过程与纯错误过程进行卷积，以创建具有块金组件的说明性空间过程。这种模型是分层的，可以通过马尔可夫链蒙特卡罗方法进行拟合。在应用中，研究人员使用空间狄利克雷过程将空间随机效应引入模型，直接与独立的重复或嵌入在动态模型中以处理时间依赖性。研究人员研究了与空间狄利克雷过程的使用相关的各种问题，包括它们的理论全局和局部性质；它们用作混合模型；它们在半参数混合中的应用；它们在动态模型中的实现；利用它们在给定时间点进行插值和在未来时间点进行预测；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扩展回归建模中的特殊应用；它们在空间点过程数据建模中的应用并将其扩展到更丰富的所谓广义空间狄利克雷过程。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多种应用领域。随着研究人员收集越来越多的空间和时空数据，需要进行分析，以提高他们对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，以适应各种全局和局部行为。这项研究的主要动机是扩展这些科学家可用的时空建模工具的目录。本研究提出了非参数贝叶斯时空数据分析的第一种方法。近年来，非参数贝叶斯方法成功地应用于工程和生物医学领域的某些问题，得到了越来越多的应用。通过将这种方法引入时空设置，预计也会取得类似的成功。特别是，预计在诸如流行病学、环境污染和天气建模等领域，研究人员将重视这项工作提供的灵活的建模框架。而且，随着拟合模型的计算技术的进步，预计该方法的使用将会增加。

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.