Estimation, Prediction, and Extremes of Multivariate Random Fields

在空间位置获得的多变量测量数据集如今在许多科学领域无处不在，从天文学、环境、生态科学到图像处理。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。本课题旨在建立多元随机场参数估计、预测和极值理论的新成果，开辟新的研究方向。预计本研究结果将进一步促进多元随机场模型在统计学和其他科学领域的适用性。此外，参与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。该研究解决了多元随机场统计推断和极值理论中的重要问题。重点研究了多变量高斯随机场及其相关随机场的光滑/分形指标和交叉相关结构在定域渐近框架下对其参数估计和预测的影响，以及多变量极值理论。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。本项目设想的主要理论发现包括多个参数的同时估计及其联合性能的描述，固定域渐近预测，以及偏移概率的精确渐近，以及由欧几里得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，研究者开发了研究多元随机场的概率和统计方法，并解决了偏移概率、随机分形、高斯随机场、lsamvy过程和分数lsamvy随机场等几个突出的开放问题。该项目旨在对多元随机场的理解产生新的见解，并量化它们的交叉依赖结构对估计器、克里格和共克里格以及极值理论的性能的影响。

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.