Spectrum Estimation for Spatial Processes

提出了一种新的、统一的估计空间过程谱密度的方法。将深入探讨这一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的克里格法或在某些情况下的最佳预测提供强大的新工具。基于广义协方差与谱密度之间的关系，提出了一种通过求解正则化反问题来估计谱密度的新方法。然后，可以通过估计的谱密度来估计广义协方差，这为克里格法铺平了道路。正则化反问题本质上是一个约束优化问题，在再生核Hilbert空间中求解。由此产生了许多关键问题。基于无偏风险和广义交叉验证思想的候选程序将被用于从数据中确定最佳平滑参数。理论性质，包括均方误差界和渐近性质，将被研究以评估该方法的性能。人们将寻求高效的计算算法来克服数据的高维特性带来的困难。本项目的研究为空间数据的分析提供了一个新的视角。调查人员考虑的数据类型是在多个空间位置观察到的数据，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，并在可获得数据的时空区域之外进行预测。解决这类问题的关键之一是理解各种数据之间的依赖关系。本项目中的方法针对的是广泛类别的模型的这个问题。新理论和方法的潜在应用存在于许多背景下，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.