Statistical Inference for Spatial Data

9971127该提案侧重于空间统计中的两个主要领域：点过程的推断和高斯随机场的预测和推断。 点过程的工作主要是由分析一个重要的宇宙学目录中出现的问题引起的，该目录是沿着类星体和地球之间的视线沿着的，被称为重元素吸收体。 这个星表提供了一种手段，用于评估宇宙中物质在非常大的空间尺度上的聚集，并描述宇宙演化过程中这种聚集的变化。 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。 事实上，该过程是在许多地区，而不是一个大的连续区域观察到的点过程的估计属性和获得有效的标准误差的估计产生了一些有趣的机会和挑战。 例如，人们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，尽管在吸收体目录中，如果区域具有不同的大小，则问题相当困难。 高斯随机场的拟议工作解决了几个问题的预测时，随机场的协方差结构是部分未知的。 在一个固定的和有界的域中的观测值的数量不断增加的自然渐近制度下，只有在一些非常简单的情况下，目前才有可能证明基于估计的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有希望的方法。 该建议还解决了协方差结构未知时预测观测网络的设计问题。 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，有巨大的潜力，以改善估计的分形维数的高斯过程中采取的意见，两个网格的非常不同的间距，而不是在一个单一的均匀间隔grail.Spatial统计是一个快速增长的调查领域，广泛适用于自然科学和社会科学。 尽管它目前的广泛使用，在空间统计中的许多常用程序的理论特性知之甚少。 该提案的第一个主要专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，提供了有关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目前高度聚集状态的基本宇宙学问题的关键组成部分。 除了直接应用所提出的工作有宇宙学，解决的问题也自然发生在显微镜。 该提案的第二个主要议题是预测温度、污染浓度和土壤特性等量，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时，有不确定性的价值如何在空间波动，并选择位置的观测，以产生准确的预测。 空间预测广泛应用于大气科学、污染监测、水文和采矿。

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.