Functional Data Analysis: From Univariate to High-Dimensional Functional Data

This project deals with data that are in the form of functions, images, and shapes. A common feature of such data is that they are infinite dimensional, which distinguishes them from traditional data. Methodology and theory for these data is termed "functional data analysis," an area that is rapidly evolving. The infinite dimensional nature of functional data calls for sampling and dimension reduction approaches, which are challenging due to the large sizes of the data sets. This research tackles these challenges by developing novel dimension reduction methodology to model univariate functional data (Projects 1 and 2), as well as next generation functional data (Projects 3 and 4). The guiding principle is to accomplish dimension reduction and flexible modeling simultaneously. The proposed research, motivated by potential applications to research on aging, biomedical research, and neuroimaging, includes theory, methodology, data analysis, and applications. The project is also driven by the pressing need to involve more statisticians in brain research and to train the next generation of statisticians to tackle the challenges of big data. Software development is a key component of this project, and the accompanying computer code will be integrated into an existing open-source package, PACE, available for public access. Functional data analysis is a fascinating area in statistics that deals with a sample of random functions. However, measurements of these random functions, realistically, can only be taken at discrete time points or grids, and the measurements often are contaminated by noise. Current approaches in functional data analysis are typically tailored toward specific sampling plans for the measurements. In Project 1, a unifying approach, both in theory and in implementation, for functional regression is proposed. The focus is on dimension reduction models for functional responses and functional/vector covariates, where spline basis functions will be used to estimate the unknown nonparametric components functions. Project 2 deals with a new class of functional survival models that accommodate censored univariate response data with functional/vector covariates. These models differ fundamentally from existing survival models, and can handle censored response data in contrast to current functional (generalized) linear models. Projects 3 and 4 deal with multivariate functional data, with Project 3 focusing on measures of disparity or synchronization for pairs of functional data, and Project 4 focusing on reconfiguration of high-dimensional multivariate functional data. New functional measures of distance or synchronization are proposed in Project 3 using a novel concept that aims at concordance of the derivatives of functional data. These new measures facilitate the reconfiguration of high-dimensional functional data in Project 4, so that functions in nearby regions of the reconfigured space are smoothly connected, thereby overcoming the curse of high dimensionality. The proposed reconfiguration methods not only extend the multidimensional scaling method from multivariate to functional data, but also turn the curse of high dimensionality into a blessing under mild assumptions.

This project deals with data that are in the form of functions, images, and shapes. A common feature of such data is that they are infinite dimensional, which distinguishes them from traditional data. Methodology and theory for these data is termed "functional data analysis," an area that is rapidly evolving. The infinite dimensional nature of functional data calls for sampling and dimension reduction approaches, which are challenging due to the large sizes of the data sets. This research tackles these challenges by developing novel dimension reduction methodology to model univariate functional data (Projects 1 and 2), as well as next generation functional data (Projects 3 and 4). The guiding principle is to accomplish dimension reduction and flexible modeling simultaneously. The proposed research, motivated by potential applications to research on aging, biomedical research, and neuroimaging, includes theory, methodology, data analysis, and applications. The project is also driven by the pressing need to involve more statisticians in brain research and to train the next generation of statisticians to tackle the challenges of big data. Software development is a key component of this project, and the accompanying computer code will be integrated into an existing open-source package, PACE, available for public access. Functional data analysis is a fascinating area in statistics that deals with a sample of random functions. However, measurements of these random functions, realistically, can only be taken at discrete time points or grids, and the measurements often are contaminated by noise. Current approaches in functional data analysis are typically tailored toward specific sampling plans for the measurements. In Project 1, a unifying approach, both in theory and in implementation, for functional regression is proposed. The focus is on dimension reduction models for functional responses and functional/vector covariates, where spline basis functions will be used to estimate the unknown nonparametric components functions. Project 2 deals with a new class of functional survival models that accommodate censored univariate response data with functional/vector covariates. These models differ fundamentally from existing survival models, and can handle censored response data in contrast to current functional (generalized) linear models. Projects 3 and 4 deal with multivariate functional data, with Project 3 focusing on measures of disparity or synchronization for pairs of functional data, and Project 4 focusing on reconfiguration of high-dimensional multivariate functional data. New functional measures of distance or synchronization are proposed in Project 3 using a novel concept that aims at concordance of the derivatives of functional data. These new measures facilitate the reconfiguration of high-dimensional functional data in Project 4, so that functions in nearby regions of the reconfigured space are smoothly connected, thereby overcoming the curse of high dimensionality. The proposed reconfiguration methods not only extend the multidimensional scaling method from multivariate to functional data, but also turn the curse of high dimensionality into a blessing under mild assumptions.