Statistical Inference for Spatial Data

9971127这项建议集中在空间统计学的两个主要领域：点过程的推断和高斯随机场的预测和推断。点过程的工作在很大程度上是由分析类星体和地球之间视线沿线位置的重要宇宙学目录中产生的问题引起的，这些位置被称为重元素吸收体。这个星表提供了一种方法，用于评估宇宙中物质在非常大的空间尺度上的聚集性，并描述随着宇宙的演化而在这种聚集性中发生的变化。吸收体目录可以最简单地建模为在大量间隔上观察到的点过程的实现的集合。这一过程是在多个区域而不是一个大的连续区域观察到的，这一事实为估计点过程的性质和获得估计的有效标准误差带来了一些有趣的机会和挑战。例如，可以使用以区域为采样单位的重采样方法来获得置信度声明，尽管如果像在吸收目录中那样区域具有不同的大小，则该问题相当困难。所提出的关于高斯随机场的工作解决了当随机场的协方差结构部分未知时与预测有关的几个问题。在一个固定且有界的区域中的观测数量不断增加的自然渐近机制下，目前只有在一些非常简单的情况下才能证明基于估计协方差结构的预测是渐近最优的。Putter和Young最近的工作为证明这些结果提供了一种很有前途的方法。该建议还讨论了当协方差结构未知时用于预测的观测网络的设计。对于预测来说，随机场的局部行为是关键的，实践经验表明，当试图推断这种局部行为时，均匀分布的观测是糟糕的设计。这一建议旨在为这一实证研究结果提供理论支持。例如，初步工作表明，通过在两个间距非常不同的网格上进行观测，而不是在单个等间距网格上进行观测，改进高斯过程的分维估计有巨大的潜力。空间统计学是一个快速增长的研究领域，在自然科学和社会科学中具有广泛的适用性。尽管它目前被广泛使用，但人们对空间统计中许多常用程序的理论性质知之甚少。这项提议的第一个主要主题是分析由空间位置组成的数据。这项工作的动机是宇宙学数据集，它提供了关于宇宙大尺度结构的重要信息。估计这种大尺度结构是解决宇宙如何从其近乎均匀的早期状态演化到现在的高度聚集状态这一基本宇宙学问题的关键组成部分。除了被提议的工作直接应用于宇宙学之外，所解决的问题也在显微镜下自然发生。该提案中的第二个主要主题是对温度、污染浓度和土壤特性等量的预测，这些量的水平在空间上随机变化。两个具体的问题包括：研究这种预测的性质，当这个量的值在空间中如何波动存在不确定性时，以及选择观测位置以产生准确的预测。空间预测在大气科学、污染监测、水文和采矿等领域有着广泛的应用。

9971127This proposal focuses on two major areas in spatial statistics: inference for point processes and prediction and inference for Gaussian random fields. The work in point processes is largely motivated by questions arising in the analysis of an important cosmological catalog of locations along the lines-of-sight between quasars and the Earth of what are known as heavy-element absorbers. This catalog provides a means for assessing the clustering of matter in the universe over very large spatial scales and for describing changes in this clustering as the universe has evolved. The absorber catalog can be most simply modeled as a collection of realizations of a point process observed on a large number of intervals. The fact that the process is observed over many regions rather than one large contiguous region produces some interesting opportunities and challenges for estimating properties of the point process and obtaining valid standard errors for the estimates. For example, one can use resampling methods with the regions as sampling units to obtain confidence statements, although the problem is rather difficult if, as in the absorber catalog, the regions are of different sizes. The proposed work on Gaussian random fields addresses several problems related to prediction when the covariance structure of the random field is partially unknown. Under the natural asymptotic regime of an increasing number of observations in a fixed and bounded domain, only in some very simple cases is it presently possible to prove that predictions based on an estimated covariance structure are asymptotically optimal. Recent work by Putter and Young provide a promising approach for proving such results. This proposal also addresses the design of observation networks for prediction when the covariance structure is unknown. For prediction, the local behavior of the random field is critical and practical experience has demonstrated that evenly spaced observations are poor designs when trying to infer this local behavior. This proposal aims to provide theoretical support to this empirical finding. For example, preliminary work shows that there is tremendous potential for improving the estimation of the fractal dimension of a Gaussian process by taking observations on two grids of very different spacing rather than on a single evenly spaced grid.Spatial statistics is a rapidly growing area of inquiry with broad applicability to the natural and social sciences. Despite its present widespread usage, the theoretical properties of many commonly applied procedures in spatial statistics are poorly understood. The first major topic of this proposal is the analysis of data made up of locations in space. This work is motivated by a cosmological data set that provides important information about the large-scale structure of the universe. Estimating this large-scale structure is a critical component in resolving the fundamental cosmological problem of how the universe evolved from its nearly uniform early state to its present highly clumped state. In addition to the direct application the proposed work has to cosmology, the problems addressed also occur naturally in microscopy. The second major topic in this proposal is the prediction of quantities such as temperature, pollution concentrations and soil characteristics whose levels vary randomly across space. Two specific problems include studying the properties of such predictions when there is uncertainty about how the values of this quantity fluctuate in space and selecting the locations of observations to yield accurate predictions. Spatial prediction is widely used in the atmospheric sciences, pollution monitoring, hydrology and mining.