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

收集到的各种环境、地球物理和气象过程的数据往往表现出不同的变率模式，特别是在不同尺度上。对这些数据波动特征的准确描述可以提高对物理现象的科学理解。开发用于模拟这种地球物理过程的新统计工具也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。尽管这些数据无处不在，但目前很少有统计方法可以在球形域上全局描述这些时空标量和矢量随机场。本项目的一个关键目标是提出一种多尺度方法来构建球面上的非高斯随机场，这一方面为建模提供了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。进一步的目标是扩展方法来处理在图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理解方面的有效性。提出的球面过程统计框架是基于球面多分辨率分析的思想。在此应用中，利用单位球上的一类针框作为构建块，在单位球上构造满足自然物理约束（如无旋度或无散度）的时空标量场和向量场，从而实现了一种灵活的近似物理过程的方法。提出了单位球和球壳上随机矢量场的参数统计模型。这些随机场用向量针表示，可以表现出非高斯特征。在这种建模范式下，提出了一套方法来分析和预测地球物理学中出现的大规模时空标量和矢量过程，例如基于地面和卫星的地球主磁场或电离层电磁场测量。本文还研究了所提出的向量针及其所表示的随机向量场的结构和性质的理论问题。利用多分辨率分析建模随机场的灵活框架，利用图谱小波在图上构造非高斯过程。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、谱图理论、贝叶斯和大规模计算、空间物理和地球物理学。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.