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

收集到的关于各种环境、地球物理和气象过程的数据往往表现出不同的变异模式，特别是在不同的尺度上。准确描述这些数据中的波动特征可以提高对物理现象的科学理解。为模拟这种地球物理过程开发新的统计工具也可以提高监测和预测其波动对通信系统和传感网络的影响的能力。尽管这样的数据无处不在，目前很少有统计方法来描述这样的时空标量和矢量随机场全球上的球面域。该项目的一个关键目标是提出一种多尺度方法来构建球体上的非高斯随机场，一方面提供了一个灵活的建模数学框架，另一方面，使人们能够通过使用现代计算工具来拟合这些模型。另一个目标是扩大这些方法，以处理在图表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认识方面的有效性。球面过程的统计框架是基于球面多分辨率分析的思想。在该应用中，一类针状框架上的单位球被用作构建块，以构建时空标量和矢量场的单位球，满足自然的物理约束，如卷曲自由或发散自由，从而使一个灵活的方法来近似的物理过程。 提出了参数统计模型来模拟单位球和球壳上的随机向量场。这些随机场表示的矢量针状，可以表现出非高斯的功能。一套方法，提出了在这种模式下，分析和预测大规模的时空标量和矢量过程中出现的物理，如地面和卫星的测量地球的主磁场或电离层电磁场。有关的理论问题的结构和性能的建议矢量针和随机向量场所代表的他们也进行了调查。通过多分辨率分析建模随机场的灵活框架，进一步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，谱图理论，贝叶斯和大规模计算，空间物理学和空间物理学。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.