Statistical Inference for Spatial Data

9971127本文重点研究了空间统计中的两个主要领域：点过程的推理和高斯随机场的预测与推理。点状过程的工作在很大程度上是由对类星体和地球之间的重元素吸收体的重要宇宙学目录的分析中产生的问题所激发的。这个目录提供了一种方法来评估宇宙中物质在非常大的空间尺度上的聚集，并描述随着宇宙的演化，这种聚集的变化。吸收器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。这个过程是在许多区域而不是一个大的连续区域上观察到的，这一事实为估计点过程的性质和获得有效的估计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，尽管在吸收器目录中，如果区域大小不同，问题会相当困难。提出的高斯随机场的工作解决了随机场的协方差结构部分未知时与预测有关的几个问题。在一个固定和有界的区域中，随着观测值的增加，在自然渐近状态下，目前只有在一些非常简单的情况下，才有可能证明基于估计协方差结构的预测是渐近最优的。Putter和Young最近的研究为证明这些结果提供了一种很有希望的方法。本文还讨论了在协方差结构未知的情况下，用于预测的观测网络的设计。对于预测，随机场的局部行为是至关重要的，实践经验表明，当试图推断这种局部行为时，均匀间隔的观测是糟糕的设计。本文旨在为这一实证发现提供理论支持。例如，初步的工作表明，通过在两个间距非常不同的网格上进行观测，而不是在一个均匀间隔的网格上进行观测，有巨大的潜力来改进高斯过程的分形维数的估计。空间统计学是一个快速发展的研究领域，对自然科学和社会科学具有广泛的适用性。尽管目前广泛使用，但空间统计中许多常用程序的理论性质尚不清楚。本提案的第一个主要主题是分析由空间位置组成的数据。这项工作的动机是一个宇宙学数据集，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几乎均匀的早期状态演变到现在高度团块状态这一基本宇宙学问题的关键组成部分。除了提出的工作直接应用于宇宙学之外，所解决的问题也自然地发生在显微镜中。本提案的第二个主要主题是对温度、污染浓度和土壤特征等数量的预测，这些数量的水平在空间上随机变化。两个具体问题包括：当这个量的值在空间中如何波动存在不确定性时，研究这种预测的性质；以及选择观测的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.