Analysis of High-Dimensional Time Series with Applications on Brain Signals

Neuronal populations behave in a coordinated manner both during resting-state and while executing tasks such as learning and memory retention. One of the major challenges to analyzing brain signals such as electroencephalograms (EEGs) and functional magnetic resonance imaging (fMRI)is high dimensionality. There can be hundreds of channels in a typical EEG recording, and the number of voxels in a fMRI recording can be hundreds of thousands. We developed computationally efficient and theoretically justified tools for analyzing high dimensional brain signals. Our approach is to extract the optimal lower dimensional representations for each brain region and then characterize and estimate connectivity between regions through these factors. This approach is motivated by the fact that electroencephalograms (EEGs) from many channels within each region exhibit a high degree of multicollinearity and synchrony thereby suggesting that it would be sensible to extract summary factors for each region. We focus on two types of linear filters. These methods were compared through simulations under different conditions and the results provide insights on advantages and limitations of each. We also performed exploratory analysis of resting state EEG data and fMRI data. The spectral properties of the factors were estimated and connectivity between regions via the factors using coherence measures were computed. We implemented these methods in a Matlab toolbox XHiDiTS. The toolbox was utilized to investigate consistency of these factors across all epochs during the entire resting-state period. In order to quantify the effective connectivity among sources in a densely voxelated (high-dimensional) cortical surface, we developed the source-space factor vector autoregressive (VAR) model. The first step in our procedure is to estimate cortical activity from multichannel electroencephalograms (EEG) using anatomically constrained brain imaging methods. Following parcellation of the cortical surface into disjoint regions of interest (ROIs), latent factors within each ROI are computed using principal component analysis. These factors are ROI specific low-rank approximations (or representations) which allow for efficient estimation of connectivity in the high-dimensional cortical source space. The second step is to model effective connectivity between ROIs by fitting a VAR model jointly on all the latent processes. The different cortical sources within a ROI may share common factors as each source is a mixture of these VAR factors. From this commonality we derive the connectivity between the sources. Measures of cortical connectivity, in particular partial directed coherence (PDC), are formulated using the VAR parameters. We illustrate the proposed model to investigate connectivity and interactions between cortical ROIs during resting state