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

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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.