Estimation, Prediction, and Extremes of Multivariate Random Fields

如今，在空间位置获得的多变量测量数据集在许多科学领域中无处不在，从天文学、环境和生态科学到图像处理。近年来，构建和理解多元随机场作为多元空间或时空数据模型的需求急剧增加。然而，多元随机场统计理论和方法的发展仍处于进化阶段，并提出了巨大的挑战。该研究项目的重点是在多元随机场的参数估计、预测和极值理论方面建立新颖的成果并开辟新的研究方向。预计该工作成果将进一步促进多元随机场模型在统计和其他科学领域的适用性。此外，参与这项研究将培养研究生并发展他们在数学科学领域的职业生涯。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。特别重点研究多元高斯及相关随机场的平滑度/分形指数和交叉依赖结构对其在固定域渐进框架下的参数估计和预测以及多元极值理论的影响。许多正在研究的问题本质上与多元随机场的几何和拓扑性质有关。该项目设想的主要理论发现包括同时估计多个参数并描述其联合性能、固定域渐近下的预测以及偏移概率的精确渐近，对于由欧几里德空间或球体索引的多元高斯和相关随机场。在之前的工作中，研究人员开发了用于研究多元随机场的概率和统计方法，并解决了有关偏移概率、随机分形、高斯随机场、Lévy 过程和分数 Lévy 随机场的几个突出的开放问题。该项目旨在为理解多元随机场提供新颖的见解，并量化其相互依赖结构对估计量、克里金法和联合克里金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.