Estimation, Prediction, and Extremes of Multivariate Random Fields

在空间位置获得的具有多变量测量的数据集如今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一个渐进的阶段，并提出了实质性的挑战。本研究计画致力于在多元随机场之参数估计、预测及极值理论等方面建立新的研究成果，并开拓新的研究方向。预计这项工作的成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，并发展他们的职业生涯在mathematicalscience.The研究解决重大问题的统计推断和多元随机场的极值理论。特别强调的是放在调查的光滑性/分形指数和交叉依赖结构的多元高斯和相关的随机场的参数估计和预测的框架下的固定域渐近的影响，和多元极值理论。研究中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研究成果包括多个参数的同时估计和描述其联合性能，固定域渐近下的预测，和精确的渐近的漂移概率，多变量高斯和相关的随机场索引的欧几里得空间或领域。在以前的工作中，研究者开发了研究多元随机场的概率和统计方法，并解决了几个突出的开放问题，如漂移概率，随机分形，高斯随机场，Lévy过程和分数Lévy随机场。该项目旨在对多元随机场的理解产生新的见解，并量化其交叉依赖结构对估计器，克里金和协克里金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.