Geostatistical Modeling of Spatial Discrete Data

与分析连续数据的方法相比，用于分析空间离散数据的统计方法相对不发达。这是一个值得注意的方法上的差距，因为前者通常是在地球科学和社会科学中收集的。例如，美国各地的政府机构定期收集不同原因导致的死亡人数，并根据年龄、性别和种族等不同的人口统计变量进行分类。本项目旨在通过发展对地质统计离散数据模型的全面研究来填补这一空白。该项目由三部分组成。首先，开发了一类层次空间模型，旨在改善目前使用模型的研究者发现的一些局限性。其中一些与这些模型所表示的空间关联结构有关的限制，在数据主要由小计数组成时尤其严重，正是在最需要描述数据离散性的模型的情况下。研究了这些新模型的性质和基于似然的拟合方法。其次，开发了一类非分层空间模型，旨在表示广泛的空间离散数据，而不仅仅是计数，具有与分层空间模型类中的空间关联结构相补充的空间关联结构。本课程中的模型是通过使用类似于copula的方法分别对边缘和空间关联结构进行建模来构建的。本文还研究了这些模型的性质和基于似然的拟合方法。第三，研究了最近提出的贝叶斯方法来评估统计模型的拟合优度，并探讨了其在上述模型类别中使用的合理性。该方法基于关键量在不同参数值下的分布恒等式，适用于分层模型和非分层模型。开发这样的方法是迫切需要的，因为众所周知，缺乏正式的方法来评估空间模型的模型充分性。如今，许多地球科学和社会科学（如生态学、流行病学、人口学和地理学）都经常收集空间数据，但分析离散数据（如死亡人数）的方法远不如分析连续数据（如温度）的相应方法发达。研究者建议通过构建新的模型类来填补这一空白，这些模型类一方面改善了研究者发现的当前使用模型的一些局限性，另一方面增加了模型所代表的数据模式。该项目还将开发方法来评估新提出的模型的模型充分性，这是科学中普遍存在的任务，因为任何模型都不能完美地代表所研究的现象。在这个项目过程中发展的统计方法将对地球科学和社会科学产生直接的方法和实际影响，在这些领域，空间离散数据是常规收集的，但用于分析这些数据的模型和方法很少。所提出的模型类别将大大增加空间数据分析人员可用的工具库，以及表示空间离散数据的广泛行为的可能性。研究生将参与该项目，这将有助于他们在贝叶斯方法和空间统计方面的统计训练，以及对德克萨斯大学圣安东尼奥分校应用统计学博士课程未来的预测。

Statistical methods for the analysis of spatial discrete data are relatively underdeveloped when compared to methods for continuous data. This is a notable methodological gap since the former are routinely collected in the earth and social sciences. For instance, death counts due to different causes are collected on a regular basis by government agencies throughout the entire U.S. and classified according to different demographic variables, such as age, gender and race. This project aims at filling this gap by developing a comprehensive study of models for geostatistical discrete data. The project consists of three parts. First, a class of hierarchical spatial models is developed that seeks to ameliorate some limitations identified by the investigator of currently used models. Some of these limitations, relating to the spatial association structures representable by these models, are especially severe when the data consist mostly of small counts, precisely the case when models describing the discreteness of the data are most needed. The properties of these new models and likelihood based methods to fit them are studied. Second, a class of non-hierarchical spatial models is developed that seeks to represent a wide range of spatial discrete data, not just counts, having spatial association structures that are complementary to those in the class of hierarchical spatial models. The models in this class are constructed by separately modeling the marginal and spatial association structures, using an approach akin to copulas. The properties of these models and likelihood based methods to fit them are also studied. Third, a recently proposed Bayesian method to assess goodness-of-fit of statistical models is studied and its soundness for use in the aforementioned classes of models explored. The method, based on a distributional identity between pivotal quantities evaluated at different parameter values, is applicable to both hierarchical and non-hierarchical models. Developing such methods is a pressing need since formal methods to assess model adequacy of spatial models are notoriously lacking.Spatial data are nowadays routinely collected in many earth and social sciences, such as ecology, epidemiology, demography and geography, but methodology for the analysis of discrete data (say death counts) is much less developed than the corresponding methodology for the analysis of continuous data (say temperature). The investigator proposes to fill this gap by constructing new classes of models that on the one hand ameliorate some limitations identified by the investigator of currently used models, and on the other hand increase the data patterns represented by the models. The project will also develop methodology to assess model adequacy for the newly proposed models, a ubiquitous task in science since any model is an imperfect representation of the phenomenon under study. The statistical methodology developed in the course of this project would have immediate methodological and practical impacts on the earth and social sciences, where spatial discrete data are routinely collected but models and methods for their analysis are scarce. The proposed classes of models will substantially increase the arsenal of tools available to spatial data analysts and the possibility of representing a wide range of behaviors for spatial discrete data. Graduate students will be engaged in the project which will contribute to their statistical training in Bayesian methods and Spatial Statistics, as well as the projection into the future of the Ph.D. program in Applied Statistics at the University of Texas at San Antonio.