Filtrated Common Functional Principal Components for Multivariate Functional data

Local field potentials (LFPs) are signals that measure electrical activity in localized cortical regions from multiple implanted tetrodes in the human or animal brain. They can be treated as multivariate functional data (i.e., curves observed at many tetrodes spread across a patch on the surface of the cortex). Most multivariate functional data contain both global features (which are shared in common to all curves) as well isolated features (common only to a small subset of curves). The goal is this paper is to develop a procedure for capturing this common features. We propose a novel tree-structured functional principal component (filt-fPC) model through low-dimensional functional representation, specifically via filtration. A popular approach to dimension reduction of functional data is functional principal components analysis (fPCA). Ordinary fPCA can only capture the major information of one population, but fail to reveal the similarity of variation pattern of different groups, which is potentially related to functional connectivity of brain. One major advantage of the proposed filt-fPC method is the ability to extracting components that are common to multiple groups, and meanwhile preserves the idiosyncratic individual features of different groups, leading to a parsimonious and interpretable low dimensional representation of multivariate functional data. Another advantage is that the extracted functional principal components satisfy the orthonormal property for each set, making filt-fPC scores easy to be obtained. The proposed filt-fPC method was employed to study the impact of a shock (induced stroke) on the functional organization structure of the rat brain. Finally we point to further directions as this filtration idea can also be generalized to other functional statistical models, such as functional regression, classification and functional times series models.