Spatial and Spatio-temporal Processes: Asymptotics, Misspecification and Multivariate Extension

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The investigator develops appropriate infill or fixed domain asymptotic results for the evaluation of approximation methods for spatial data. The infill asymptotic framework is generally preferred for spatial data. However, infill asymptotic results for estimation are in general difficult to derive, and there exist only a few explicit infill asymptotic results pertaining to specific models. In this proposal, a general approach to establishing the infill asymptotic properties is outlined and followed to establish asymptotic distributions of estimators that maximize some approximated likelihood functions such as Vecchia's approximation and covariance tapering. These infill asymptotic results assure that the approximation methods may yield asymptotically efficient estimators. In addition, for spatio-temporal data, the investigator considers a new asymptotic framework in which the temporal domain is increasing while the spatial sampling domain is fixed. Under this asymptotic framework, the investigator shows that an incorrect covariance function (such as covariance tapering) generally results in biased estimators. The investigator establishes asymptotic results that allow for the correction of biases. The adjusted estimators are expected to be asymptotically normal and unbiased. These asymptotic results allow one to employ incorrect but simpler spatio-temporal covariance functions and then adjust for the bias. Finally, the investigator extends the results from the univariate case to the multivariate case when multiple spatial variables are observed across space and/or over time.Data across space and time are routinely observed in many scientific studies that are very important to the society such as those on global warming, environmental monitoring, precision agriculture, epidemiology, hot spot detection in homeland security, etc. The immense amount of data and the correlation across space and time have raised new challenges to the modeling and analysis of such space-time data. The primary goal of this project is tackle these challenges by studying computationally feasible and efficient approaches to the analysis of vast space-time data, which also bear no or little loss of statistical efficiency. It is expected that this project will make more accessible and feasible the analysis of huge spatial and spatial-temporal data to scientists in broader disciplines, and thus enables scientists to retrieve significant and reliable information from the vast amount of data.