Estimation, Prediction, and Extremes of Multivariate Random Fields

如今，从天文、环境、生态科学到图像处理等许多科学领域中，在空间位置获得的多变量测量的数据集是普遍存在的。近年来，构建和理解多变量随机场作为多变量空间或时空数据模型的需求急剧增加。然而，多元随机场的统计理论和方法的发展仍处于发展阶段，并提出了实质性的挑战。本研究项目致力于在多元随机场的参数估计、预测和极值理论方面建立新的结果和开辟新的研究方向。预计这项工作的结果将进一步促进多变量随机场模型在统计学和其他科学领域的适用性。此外，参与这项研究还将培养研究生并发展他们在数学科学方面的职业生涯。这项研究解决了多元随机场的统计推断和极值理论中的重要问题。在定域渐近性框架下，重点研究了多元高斯场和相关随机场的光滑性/分形性指数和交叉相依结构对其参数估计和预报的影响，以及多元极值理论。所研究的许多问题都与多元随机场的几何和拓扑性质有着内在的联系。该项目设想的主要理论成果包括多个参数的同时估计及其联合性能的描述，固定域渐近下的预测，以及由欧几里得空间或球面索引的多变量高斯和相关随机场的漂移概率的精确渐近。在以前的工作中，研究者发展了研究多元随机场的概率和统计方法，并解决了关于游程概率、随机分形、高斯随机场、L过程和分数阶Lévy随机场的几个未决问题。这个项目旨在对多变量随机场的理解产生新的见解，并量化它们的交叉相关结构对估计器、克立格法和协克立格法以及极值理论性能的影响。

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.