Collaborative Research: Spectral Functional Principal Components on Abelian Groups with Applications to Spatial Functional Data

Colorado State University, Fort Collins CO

Investigators

Abstract

Massive data sets on gridded 2D and 3D domains have recently become available through computer climate model outputs, records from satellite remote sensing and brain scans, among others. These data sets have both temporal and spatial dimension. For example, a state of vegetation is observed on a grid covering an agricultural area at regular time intervals, every day or every week. Such data can be viewed as functions of time, one function per spatial grid unit. Their chief characteristic is the spatial dependence of curves observed at the grid nodes. There is an increasing need to develop statistical tools, which will allow researchers to extract useful information from such data. The PIs will develop such tools. The data and problems that motivate this research arise in several science fields, which have important impacts on society. For example, conclusions drawn from future climate models help the government and corporations plan for the allocation of various assets. Brain research on trauma experienced by military veterans and on Alzheimer's disease are recognized as important societal goals. The statistical research the PIs will conduct will provide useful quantitative tools to help scientists in these fields. Mathematical foundations of the new approach will be created, together with domain-specific approaches. The new methods will be implemented in R packages and made available to research community, government agencies and commercial enterprises. In the course of the proposed research, two Ph.D. students will be trained. The PIs will create a new framework for inference for functional data defined on domains with an additive group structure. The new dimension reduction approach will have characteristics of a multi-scale, data-driven representation, which takes into account the dependence of the functions defined on group elements, for example spatial grid nodes. The PIs will use methods of Fourier analysis on Abelian groups, spectral theory for functional data, invariance principles in Hilbert spaces, computationally efficient spatio-temporal spline representations, routines for downloading and manipulating massive data sets. The PIs will develop several inferential procedures, including bootstrap-based inference, tests for the spatial and distributional structure, and applications to the evaluation of the accuracy of computer climate models. The PIs will also develop corresponding computational techniques, which will lead to the computationally fast representation of various data structures of large to massive size. 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.