Spectrum Estimation for Spatial Processes

提出了一种新的、统一的空间过程谱密度估计方法。本文将深入探讨该方法的理论性质和实际实施问题。该项目的完成将为空间数据分析中某些情况下的kriging或最佳预测提供强大的新工具。基于广义协方差与谱密度的关系，提出了一种用正则化逆问题估计谱密度的新方法。然后可以通过估计的谱密度估计广义协方差，这为克里格算法铺平了道路。正则化逆问题本质上是一个约束优化问题，在再现核希尔伯特空间中求解。由此产生了一些关键问题。基于无偏风险和广义交叉验证思想的候选程序将被研究，以确定最优的平滑参数从数据。理论性质，包括均方误差界限和渐近性质，将进行研究，以评估该方法的性能。将寻求有效的计算算法来克服由数据的高维性质带来的困难。本课题的研究为空间数据分析提供了一个新的视角。调查人员所考虑的数据是在多个空间位置，也可能是在多个时间点观察到的数据。总体目标是确定数据生成过程，并在数据可用的时空区域之外进行预测。这类问题的关键之一是理解数据的各个部分之间的依赖关系。本项目中的方法针对的是一大类模型的这个问题。新理论和方法的潜在应用存在于许多背景中，包括环境、地理和传感器网络。

A new, unified approach for estimating spectral densities of spatial processes is proposed. The theoretical properties as well as practical implementation issues of this approach will be thoroughly explored. The completion of this project will provide powerful new tools for kriging, or optimal prediction in certain situations in spatial data analysis. Based on the relationship between the generalized covariance and the spectral density, a new approach is formulated for estimating the spectral density in terms of solving a regularized inverse problem. The generalized covariance can then be estimated thorough the estimated spectral density, which paves the way for kriging. The regularized inverse problem is solved in a reproducing kernel Hilbert space essentially as a constrained optimization problem. A number of crucial issues arise from that. Candidate procedures based on the ideas of unbiased-risk and generalized cross-validation will be studied for the determination of the optimal smoothing parameters from data. Theoretical properties, including mean squared error bounds and asymptotic properties, will be investigated to assess the performance of the approach. Efficient computational algorithms will be sought to overcome the difficulties brought by the high-dimensional nature of the data.The research in this project offers a new perspective on the analysis of spatial data. The kind of data that the investigator has in mind are data observed at multiple spatial locations and possibly also at multiple time points. The general goals are to identify the data generation process and to make predictions beyond the spatial-temporal region where data are available. One of the keys in such problems is to understand the dependence relationship between the various pieces of the data. The approach in this project targets this problem for a broad class of models. Potential applications of the new theory and methodology exist in numerous contexts, including the environment, geography, and sensor networks.