Spatial statistics for large spatial datasets is challenging. The size of the dataset, n, causes problems in computing optimal spatial predictors such as kriging, since its computational complexity is on the order of the cube of n. In addition, a large dataset is often defined on a large spatial domain, so that the spatial process of interest typically exhibits nonstationary behavior over that domain. In this research, a family of nonstationary covariance functions is defined using a set of basis functions that is fixed in number, which is motivated by a spatial random effects (SRE) model. This leads to a spatial prediction method we call Fixed Rank Kriging (FRK). FRK relies on computational simplifications when n is large, for obtaining the spatial best linear unbiased predictor (BLUP) and its mean squared prediction error for a hidden spatial process. A weighted-least-squares method is derived to estimate the covariance-function parameters, and these are substituted into the FRK equations. The manuscript, "Fixed rank kriging for very large spatial datasets" by Noel Cressie and Gardar Johannesson is to appear in the Journal of the Royal Statistical Society, Series B. A follow-up manuscript, "Global statistical analysis of MISR aerosol data: A massive data product from NASA's Terra satellite" by Tao Shi and Noel Cressie, is to appear in Environmetrics.

In the study of stationary processes on the real line, the spectral density function is a parameter of considerable interest. In joint research between Chunfeng Huang, Tailen Hsing, and Noel Cressie, a new estimator of the spectral density function is obtained by a regularized inversion of estimated covariances. In particular, the data are not required to be observed on a grid and the estimator is not based on the periodogram. For data that are observed on a grid, the estimator is derived in closed form, and the mean squared error of the estimator can be computed.

• Office of Naval Research, Probability and Statistics Program