Modeling Multivariate and Space-Time Processes: Foundations and Innovations

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Geophysical processes for temperature and pressure are often highly correlated and are evolving in space over time with complex structures. For instance, many atmospheric processes such as turbulent processes can exhibit long-range dependence with correlation decays slowly as distance increases. While existing covariance models are successful in describing the smoothness behavior of these processes, the correlation in these models often decays exponentially fast and hence is inadequate. The data resulting from many geophysical processes are often continuously indexed and exhibit complicated dependence structures in many disciplines, including geophysics, ecology, environmental and climate sciences, engineering, public health, economics, political sciences, and business science. This project will develop new multivariate and space-time covariance functions with their theoretical properties to characterize complex behaviors such as long-range dependence and asymmetry and develop robust estimation procedures for estimating smoothness behaviors and long-range dependence. The project will also develop and distribute user-friendly open-source software, facilitate its broad adoption for complex data analytical problems, and provide training opportunities for next-generation statisticians and data scientists. This project is jointly funded by the Statistics Program and the Established Program to Stimulate Competitive Research (EPSCoR). This project will develop theoretical foundations and statistical models for inferring multivariate and space-time processes with long-range dependence using a model-based framework. This framework integrates and extends powerful techniques arising in the literature on scale-mixture modeling and objective Bayes. A scale-mixture technique is used to construct new multivariate and space-time covariance functions and offers flexible properties including arbitrary smoothness, long-range dependence, and asymmetry. Theoretical foundation will be provided to study the practical usefulness of the resultant covariances in a principled and unified manner in terms of several properties such as origin/tail behaviors and screening effect and offer theoretical insights on prediction accuracy in both interpolative and extrapolative settings. Objective Bayes inference is used to enable robust parameter estimation for Gaussian processes under the confluent hypergeometric covariance function with the reference prior in which the smoothness and tail-decay parameters are allowed to be estimated. The developed statistical theory and inferential tools will provide new foundations for modeling multivariate and space-time processes in spatial statistics and related areas that use covariance models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.