Estimation and Prediction in Spatial Statistics

摘要：张浩提议：DMS 0405782本研究涉及两个主要目标的地质统计学：估计空间相关性和预测值在未采样的网站。 具体的研究问题可以分为三组：（i）填充高斯过程的渐近性。 空间统计学的最新进展表明，渐近结果对空间数据的分析是有用的。 研究了变差函数参数估计的填充渐近性质。 此外，通过理论和数值研究，给出了在有限样本情形下采用何种渐近方法的指导原则，因为存在两种不同的渐近方法：增域渐近方法和填充渐近方法。 在这两种渐近性下，结果有很大的不同。 (ii)基于单变量模型的地质统计学。 空间广义线性混合模型（GLMM）用于基于模型的地质统计学中，对空间非高斯变量进行建模和预测。 本计画研究广义似然矩中参数极大似然估计的相合性与渐近分布。 (iii)基于多元模型的地质统计学中多元协方差函数的估计与推断。 这个项目开发并实现了估计多元协方差函数的显式算法。 它还开发了多变量空间GLMM，这是一个强大的模型，当一个或多个空间变量是非高斯，如二项式计数。 在S-Plus和R中研究和实施了推理方法。地质统计数据出现在许多领域，包括水文学、生态学、农业、自然资源评估、环境科学和健康研究。 在广泛的应用中，有一个真实的需要理论工作来理解估计和预测的性质。 本文对一元goestatistics的研究结果部分地满足了需要，并易于应用于地质统计数据的分析。 还非常需要为多个空间变量建立适当的统计模型。 例如，在环境和健康研究中，经常在不同地点观察到多个变量。 由于空间相关性，对这些空间变量进行建模是一个具有挑战性的问题，并且尚未得到充分研究。 本研究开发的方法和算法建模这样的多个空间变量。 因此，研究成果对各种科学学科产生广泛影响。

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.