Most existing methodologies of functional data analysis are limited to data structured in one domain, such as time. Two-way structured functional data are indexed by two functional domains, such as space and time, and each domain has its own notion of regularity, such as smoothness or sparsity. Fully considering the two-way structure of the data will lead to more accurate analysis results. Recent work by the lead PI on two-way regularization has provided some preliminary results and a good starting point for investigating new methodology of dimension reduction, feature extraction, regression and classification for two-way functional data. This research team plans to further develop the methodology for two-way functional data in several important directions:(a) Develop a Reproducing Kernel Hilbert Space theory for two-way regularized singular value decomposition (SVD); (b) Develop novel dimension reduction methods for data that are indexed by general domains such as manifolds; (c) Develop dimension reduction methods that effectively analyze discrete two-way functional data; (d) Develop two-way regularization methods for solving the magnetoencephalography (MEG) inverse problem; (e) Develop new classifiers for diagnosis of mental disorders using dynamic MEG images as predictors; (f) Develop robust methods that are resistant to outliers. The success of the research will add a new dimension to functional data analysis and significantly enrich the field.
Two-way structured functional data arise in various disciplines, including medicine, social sciences, earth sciences, economics, and business. But few existing methodologies fully take into account the two-way structure of this type of functional data. The novel statistical methods developed in this research will provide valuable tools for efficient use of such data. They will provide better understanding of scientific, social and economic phenomena and make more accurate predictions. In particular, the two-way reguarlized SVD provides a new analysis of variance method for analyzing high throughput bioinformatics data and for discovering interactions among biomarkers and clinical variables that are associated with a disease phenotype. The new MEG inverse solvers will facilitate noninvasive presurgical mapping of functional areas of the brain. The new classifiers will help diagonosis and assessment of mental diseases using dynamic MEG images. The proposed activities involve training of Ph.D. students who participate in the proposed projects and mentoring of the female, junior statistician P.I. in a psychology department. Research results will be disseminated through collaborative work, academic presentations, and journal publications. Web pages will be created to enable quick access to user-friendly and accessible software implementations of new methods as well as technical reports and relevant references.
