Statistical Analysis of Complex, Highly-structured Functional Data

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Functional data, where the observational units represent curves, surfaces, or images, are becoming increasingly prevalent in modern measurement systems. While most statistical methods focus on simplifying assumptions such as independence, real-world functional data often contain far more complex structures. For example, in neuroimaging, the dependence network across brain regions needs to be characterized using graphs based on EEG or fMRI measurements; in proteomic and spectral analysis, researchers frequently encounter functional data with featured regions. These structures represent generic complexities in data that cannot be handled by existing analytical tools. This project focuses on the development of novel statistical theory and methods to model complex structures. The proposed tools will be applied to datasets from neuroimaging, mass spectrometry, genomics, and bioinformatics. The research will be integrated with various educational and outreach activities that will impact teaching and learning both within and beyond the university. The project has the following four interrelated objectives: (1) Develop a functional graphical model framework to characterize conditional independent relationships between random functions, and apply the methods to estimate large-scale brain networks; (2) Develop novel frequentist regularization strategies and Bayesian priors to select regions of functional data; (3) Use Markov random fields to characterize spatial dependency between functions, and apply them to functional areal and functional point-reference data; and (4) Educate and engage students in the field of high-dimensional data analysis, develop K-12 outreach programs, and mentor minority students.