UC Santa Barbara

Functional data analysis (FDA) is the statistical methodology that analyzes datasets whose data points are functions measured over some domain, and is specially useful to model random processes over a continuum. This thesis develops a novel methodology to address the general problem of covariance modeling for multivariate functional data, and functional Gaussian graphical models in particular. The resulting methodology is applied to neuroimaging data from the Human Connectome Project (HCP).

First of all, a novel structural assumption for the covariance operator of multivariate functional data is introduced. The assumption, termed partial separability, leads to a novel Karhunen-Lo`eve-type expansion for such data and is motivated by empirical results from the HCP data. The optimality and uniqueness of partial separability are discussed as well as an extension to multiclass datasets. The out-of-sample predictive performance of partial separability is assessed through the analysis of functional brain connectivity during a motor task.

Next, the partial separability structure is shown to be particularly useful to pro- vide well-defined functional Gaussian graphical models. The first one is concerned with estimating conditional dependencies, while the second one estimates the difference be- tween two functional graphical models. In each case, the models can be identified with a sequence of finite-dimensional graphical models, each of identical fixed dimension. Em- pirical performance of the methods for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during motor tasks.