Modeling Complex Dependence in Multivariate Time Series, with Applications to Brain Signals

Overview

The underlying dependence structure of multivariate time series observed in practice seldom conforms to restrictive assumptions such as linearity and Gaussianity. With various hidden factors that may contribute (both individually and jointly) to the behavior of the processes of interest, modeling these complex dependencies requires development of new methodologies that are tailor-fitted to the characteristics of the data at hand. In this dissertation, we develop several approaches to address issues and limitations associated with multivariate processes generated from unknown complex systems such as brain signals.

To capture possible nonlinear interactions between channels while accounting for variation across multiple subjects, we develop the mixed-effects functional-coefficient autoregressive (MXFAR) model and the functional partial directed coherence (fPDC) metric. By assuming a reference signal that drive the dependence, the MXFAR-fPDC framework provides a tool to extract easy-to-interpret nonlinear features which can be utilized to describe difference in the brain connectivity network of multiple groups of subjects (e.g., group of individuals with mental disorder and healthy controls). In addition, we formulate a novel spectral causal measure, the spectral transfer entropy (STE) metric, that quantifies the amount and direction of information transfer between two ``frequency band''-specific oscillation, which enables to derive brain connectivity networks. The main advantage of the causal inference approach based on STE is that detected significant causal impact between band-specific signals from different channels can be associated to well-explored cognitive functions of the brain.

We then generalize this framework, which we call the spectral causation entropy (SCE), to differentiate between direct and indirect information flow between two nodes of a network by accounting for the contribution of the remaining components of the network. Moreover, we develop an amortized neural estimator for SCE that produces reliable estimates in milliseconds, making a permutation-based test for significance of direct spectral information transfer feasible. Lastly, we formulate the nonlinear vector coherence (NVC) measure to quantify the dependence between two multivariate time series in the frequency domain, and thus providing a new framework for investigating functional connectivity between brain regions (i.e., groups of channels), instead of the standard channel-to-channel approach.

We further develop a rank-based inference procedure that enables fast and distribution-free estimation of the proposed measure, as well as a fully nonparametric test for spectral independence based on NVC. The main contribution of this dissertation is the development of these four methodologies translating into accessible tools for practitioners and neurologists that can yield deeper and more clinically relevant insights from neural data.