Modern data collection methods are now frequently returning observations that could be viewed as the results of digitized recording or sampling from random functions. This project investigates regression problems for which the response is scalar but some of the predictors are functional. The general goal is to gain understanding on the inference of the models based on partially observed and error-contaminated functional data. Distinctions will be made between dense functional data, usually obtained from images, and sparse functional data, usually obtained from longitudinal studies. The specific topics include the consideration of (i) a functional generalized linear model for dense functional data using a penalized likelihood approach, (ii) dimension reduction methodologies based on sliced inverse regression and sliced average variance estimation, and (iii) a functional generalized linear model for sparse functional data using an approximated quasi-likelihood approach. New approaches will be proposed in the consideration of these problems, and asymptotic theories will be proved to validate the approaches. The sparse functional generalized linear model will be considered in a framework of joint modeling between a longitudinal life style profile and an endpoint health outcome. This involves the study of a new type of error-in-variable problem, which is expected to extend the horizon of longitudinal-data modeling.
An important current focal point of statistical research is the so-called high-dimensional data analysis. Indeed, high-dimensional data are a fact of life. This is evidenced by our increasing need for larger storage devices on our computers. Roughly speaking, functional data are high-dimensional data which can be approximated by smooth curves or functions. Such data are abundant in scientific investigations, and it is of crucial importance to be able to effectively analyze such data. The PI will investigate approaches that will fundamentally contribute to the practice of functional data analysis. Direct applications of the research can be found in areas including image analysis, bioinformatics, and medicine. Research-level classes on functional data analysis based on this research will be offered at both University of Georgia and University of Michigan.
