Estimation, Prediction, and Extremes of Multivariate Random Fields

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Data sets with multivariate measurements obtained at spatial locations are nowadays ubiquitous in many scientific areas, ranging from astronomy, environmental, and ecological sciences to image processing. The need to construct and understand multivariate random fields as models for multivariate spatial or spatio-temporal data has increased dramatically in recent years. However, the development of statistical theory and methodology for multivariate random fields is still in an evolutionary stage and poses substantial challenges. This research project is focused on establishing novel results and opening new research directions in parameter estimation, prediction, and extreme value theory of multivariate random fields. It is anticipated that the results of the work will further promote the applicability of multivariate random field models in statistics and other scientific areas. Moreover, involvement in the research will train graduate students and develop their careers in the mathematical sciences.The research addresses significant questions in statistical inference and extreme value theory of multivariate random fields. Special emphasis is placed on investigating the effects of smoothness/fractal indices and cross-dependence structures of multivariate Gaussian and related random fields on their parameter estimation and prediction under the framework of fixed-domain asymptotics, and on multivariate extreme value theory. Many of the problems under investigation are intrinsically connected with geometric and topological properties of the multivariate random fields. The principal theoretical findings envisaged by this project include simultaneous estimation of multiple parameters and description of their joint performance, prediction under fixed-domain asymptotics, and precise asymptotics for the excursion probabilities, for multivariate Gaussian and related random fields indexed by the Euclidean space or spheres. In previous work, the investigator developed probabilistic and statistical methods for studying multivariate random fields and resolved several outstanding open problems on excursion probabilities, random fractals, Gaussian random fields, Lévy processes, and fractional Lévy random fields. This project aims to yield novel insights into the understanding of multivariate random fields and quantify the influence of their cross dependence structures on the performance of estimators, kriging and co-kriging, and extreme value theory.