Estimation and Prediction in Spatial Statistics

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ABSTRACTPI: Hao Zhang proposal: DMS 0405782This research concerns two major objectives of geostatistics: estimation of spatial correlation and prediction of values at unsampled sites. Specific problems to be studied can be categorized into three groups: (i) Infill asymptotics for Gaussian processes. Recent advances in spatial statistics show that asymptotic results are useful for the analysis of spatial data. This research establishes infill asymptotic properties of estimators of variogram parameters. In addition, it also provides, through theoretical and numerical studies, some guidelines about which asymptotics to employ in a finite sample case because there are two distinct asymptotics: the increasing domain asymptotics and infill asymptotics. Results are quite different under the two asymptotics. (ii) Univariate model-based Geostatistics. Spatial generalized linear mixed models (GLMM) are used in model-based geostatistics to model and predict spatial non-Gaussian variables. This project studies consistency and asymptotic distributions of the maximum likelihood estimators of the parameters in the GLMM. (iii) Estimation of multivariate covariogram and inferences in multivariate model-based geostatistics. This project develops and implements explicit algorithms for estimating multivariate covariograms. It also develops the multivariate spatial GLMM that is a powerful model when one or more spatial variable is non-Gaussian such as binomial counts. Inferential methods are studied and implemented in S-Plus and R.Geostatistical data arise in many fields including hydrology, ecology, agriculture, natural resource evaluation, environmental sciences and health studies. In the midst of the wide applications there is a real need for theoretical work to understand the properties of estimation and prediction. The results of this research for univariate goestatistics partially meet the need and are readily applicable to the analysis of geostatistical data. There is also a great need to develop appropriate statistical models for multiple spatial variables. For example, in environmental and health studies, multiple variables are often observed at different locations. Due to spatial correlations, modeling these spatial variables is a challenging problem and is understudied. This research develops methods and algorithms for modeling such multiple spatial variables. Hence the research results have broad impacts on a variety of scientific disciplines.