functional linear regression, graphical models and dependence modelling

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My research program will focus on developing new models and novel methodologies for functional data, with emphasis on applications to brain imaging data, air pollution data and financial data. Our first research theme concerns high dimensional functional linear regression, where we are interested in selecting and estimating the effect of multiple function-valued (also called functional) covariates on a scalar response. Repeated measurements of brain signals, time-course gene expressions and high frequency observations of air pollutants and stock indices can all be treated as function-valued data. We plan to develop methodologies to fit a functional linear model with high dimensional functional covariates. This research program will not only extend the idea of variable selection to functional regression models, but also have a significant impact on various disciplines. For example, we will investigate the effect of brain signals collected from different brain regions on prediction of the attention deficit hyperactivity disorder (ADHD) index, a summary metric of the overall level of ADHD. With this model, we are able to identify specific regions that are related to ADHD. Our second research theme focuses on graphical models for functional data. Traditional graphical models are commonly used to describe the conditional dependence structure of multivariate random variables. A graphical model consists of multiple nodes and edges; each node represents one random variables and an edge connecting two nodes indicates that these two random variables are conditionally dependent. By imposing a LASSO penalty on the precision matrix, we are able to obtain a sparse graphical structure that means there are no edges between many pairs of random variables. We will propose a functional graphical model where each node represents a random function. This model can be applied to gene expressions to see how those genes are connected to each other in regulations. Additionally, people are also interested in functional connectivity between different regions of a human brain when a task is performed. The graphical structure for functional connectivity may serve as a useful biomarker in identification of disease status. The third research theme is to model the dependence structure of multivariate functional data. Typical dependent functional data include high frequency measurements of air pollutants at different sites within a specific region. We will develop a general framework to accommodate flexible dependence structures. This development enables us to acquire a comprehensive understanding of the spatial and temporal association of air pollutants. The proposed research will make significant contributions to the field of functional data analysis through developing tools to handle high dimensional functional data. More importantly, other disciplines such as environmental science, neuroscience, finance and biology will also benefit from our research.