Estimation and Prediction in Spatial Statistics

摘要：张浩建议：DMS 0405782这项研究涉及到地统计学的两个主要目标：空间相关性的估计和未采样点值的预测。要研究的具体问题可以分为三类：(1)高斯过程的填充渐近性。空间统计学的最新进展表明，渐近结果对于空间数据的分析是有用的。本研究建立了变异函数参数估计量的填充渐近性质。此外，它还通过理论和数值研究提供了在有限样本情况下使用渐近性的指导原则，因为存在两种截然不同的渐近性：递增区域渐近性和填充渐近性。在这两个渐近条件下，结果截然不同。(2)基于单变量模型的地统计学。空间广义线性混合模型(GLMM)用于基于模型的地统计学，对空间非高斯变量进行建模和预测。本课题研究GLMM中参数的极大似然估计的相合性和渐近分布。(3)基于多变量模型的地质统计学中的多变量协方差函数估计和推断。这个项目开发并实现了用于估计多变量协方差函数的显式算法。它还发展了多变量空间GLMM，这是一个强大的模型，当一个或多个空间变量为非高斯变量时，如二项计数。在S+和R中研究并实现了推理方法。地统计学数据产生于许多领域，包括水文、生态、农业、自然资源评价、环境科学和健康研究。在广泛的应用中，确实需要理论工作来理解估计和预测的性质。单变量统计的研究结果部分满足需要，适用于地质统计学数据的分析。还非常需要为多个空间变量开发适当的统计模型。例如，在环境和健康研究中，经常在不同的地点观察到多个变量。由于空间相关性，对这些空间变量的建模是一个具有挑战性的问题，也是一个研究不足的问题。这项研究开发了对这种多空间变量进行建模的方法和算法。因此，研究成果对各种科学学科都有广泛的影响。

ABSTRACTPI: Hao Zhang proposal: DMS 0405782This research concerns two major objectives of geostatistics: estimation of spatial correlation and prediction of values at unsampled sites. Specific problems to be studied can be categorized into three groups: (i) Infill asymptotics for Gaussian processes. Recent advances in spatial statistics show that asymptotic results are useful for the analysis of spatial data. This research establishes infill asymptotic properties of estimators of variogram parameters. In addition, it also provides, through theoretical and numerical studies, some guidelines about which asymptotics to employ in a finite sample case because there are two distinct asymptotics: the increasing domain asymptotics and infill asymptotics. Results are quite different under the two asymptotics. (ii) Univariate model-based Geostatistics. Spatial generalized linear mixed models (GLMM) are used in model-based geostatistics to model and predict spatial non-Gaussian variables. This project studies consistency and asymptotic distributions of the maximum likelihood estimators of the parameters in the GLMM. (iii) Estimation of multivariate covariogram and inferences in multivariate model-based geostatistics. This project develops and implements explicit algorithms for estimating multivariate covariograms. It also develops the multivariate spatial GLMM that is a powerful model when one or more spatial variable is non-Gaussian such as binomial counts. Inferential methods are studied and implemented in S-Plus and R.Geostatistical data arise in many fields including hydrology, ecology, agriculture, natural resource evaluation, environmental sciences and health studies. In the midst of the wide applications there is a real need for theoretical work to understand the properties of estimation and prediction. The results of this research for univariate goestatistics partially meet the need and are readily applicable to the analysis of geostatistical data. There is also a great need to develop appropriate statistical models for multiple spatial variables. For example, in environmental and health studies, multiple variables are often observed at different locations. Due to spatial correlations, modeling these spatial variables is a challenging problem and is understudied. This research develops methods and algorithms for modeling such multiple spatial variables. Hence the research results have broad impacts on a variety of scientific disciplines.