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.
