This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
To meet the rising need for analyzing functional data with complex structures, the investigators develop innovative statistical methods under three broad categories: (1) Multi-level functional data analysis. New methods for multi-level principal components analysis, multi-level clustering and classification will be developed. New methods will also be developed for fitting functional mixed effects models --- a very flexible class of models for functional data. (2) Correlated functional data. New methods will be developed for modeling time series of curves and spatially correlated curves. (3) Two-way functional data. A new principal components analysis is developed for two-way functional data, where both index domains of the data matrix are structured. In these projects, the structures of the functional data vary from case to case, but the common challenge is to deal with the covariance kernel of a random functional object. The main strategy is to reduce dimension through functional principal components. In addition, an alternative regularization strategy is also investigated based on shrinkage to simple structures. Penalized splines are used for estimating the principal components functions. By looking into functional data with complex structures, the research has significant potential to advance the knowledge of statistics.
Functional data are data that can be represented as a collection of curves or functions. Examples of functional data include, but not limited to, a patient's vital signs over time, a digitized image, geographical data and demographic data. As automated measuring systems make data collection easy, functional data become more prevalent with increasingly complex structure. Our research is motivated by analyzing functional data arising from studies on colon physiology and colon cancer, studies of US ethnic diversity dynamics and business operational management. The statistical methodology innovations proposed are widely applicable in various fields that involve functional data, such as environment and global change, health and medicine, etc. The success of the proposed research will benefit people with deeper understanding of functional data.
