Collaborative Research: Multi-Scale Modeling of Non-Gaussian Random Fields

在各种环境、地球物理和气象过程中收集的数据往往表现出不同的变化模式，特别是在不同的尺度上。准确地描述这些数据中的起伏特征可以提高对物理现象的科学理解。开发用于模拟这类地球物理过程的新统计工具还可以加强监测和预测其波动对通信系统和感觉网络的影响的能力。尽管这种数据无处不在，但目前很少有统计方法可以在球域上全局地描述这种时空标量和矢量随机场。该项目的一个关键目标是提出一种在球面上构建非高斯随机场的多尺度方法，一方面为建模提供一个灵活的数学框架，另一方面使人们能够使用现代计算工具来拟合这些模型。另一个目标是扩展方法论，以处理在图形和网络上观察到的数据。该项目还旨在通过分析地面和卫星对地球磁场的测量，证明拟议方法在加强对地球物理过程的科学了解方面的有效性。所提出的球面过程统计框架是基于球面上的多分辨分析的思想。在这个应用中，单位球面上的一类针状框架被用作构建块，以在单位球面上构造满足自然物理约束的时空标量场和矢量场，从而使得能够灵活地逼近物理过程。提出了参数统计模型来模拟单位球和球壳上的随机矢量场。这些随机场以矢量针的形式表示，并且可以表现出非高斯特征。在这一建模范式下，提出了一套分析和预测地球物理中出现的大尺度时空标量和矢量过程的方法，例如基于地球主磁场或电离层电磁场的地面和卫星测量。对所提出的矢量针的结构和性质及其所表示的随机矢量场的理论问题也进行了研究。进一步利用多分辨率分析建模随机场的灵活框架，利用图谱小波构造图上的非高斯过程。这个合作项目需要汇集来自不同领域的技能和知识，如多分辨率分析、空间统计学、谱图理论、贝叶斯和大规模计算、空间物理和地球物理学。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Data collected on various environmental, geophysical and meteorological processes often exhibit different modes of variability, especially at different scales. An accurate description of the features of the fluctuations in these data can improve scientific understanding of the physical phenomena. Development of new statistical tools for modeling such geophysical processes can also enhance the ability to monitor and predict the impact of their fluctuations on communication systems and sensory networks. Despite the ubiquity of such data, few statistical methodologies are currently available to describe such spatiotemporal scalar and vector random fields globally on a spherical domain. One key objective of this project is to propose a multiscale approach for constructing non-Gaussian random fields on a sphere, that on the one hand provides a flexible mathematical framework for modeling, and on the other hand, enables one to fit these models by using modern computational tools. A further objective is to extend the methodologies to deal with data observed on graphs and networks. The project also aims to demonstrate the effectiveness of the proposed methodologies in enhancing scientific understanding of geophysical processes by analyzing ground-based and satellite-based measurements of the earth's magnetic fields. The proposed statistical framework for spherical processes is based on the idea of multiresolution analysis on a sphere. In this application, a class of needlet frames on the unit sphere is utilized as a building block to construct spatio-temporal scalar and vector fields on the unit sphere that satisfy natural physical constraints such as being curl-free or divergence-free, thereby enabling a flexible approach to approximating physical processes. Parametric statistical models are proposed to model random vector fields on the unit sphere and spherical shells. These random fields are represented in terms of vectorial needlets and can exhibit non-Gaussian features. A suite of methodologies is proposed under this modeling paradigm to analyze and predict large-scale spatiotemporal scalar and vector processes arising in geophysics, such as ground and satellite based measurements on the earth's main magnetic field or on ionospheric electro-magnetic fields. Theoretical questions related to the structure and properties of the proposed vectorial needlets and the random vector fields represented by them are also investigated. The flexible framework of modeling random fields through multiresolution analysis is further exploited to construct non-Gaussian processes on graphs by means of graph spectral wavelets. This collaborative project requires bringing together skills and knowledge from disparate areas such as multiresolution analysis, spatial statistics, spectral graph theory, Bayesian and large-scale computation, space physics, and geophysics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.