Methods for time series network analysis

Abstract

Statistical networks can encode arbitrary relationships between variables in a system. Due to this flexibility, scientific hypotheses about interactions between variables can typically be formulated as a statistical network analysis. In addition to analyzing static networks, studying how statistical networks change in response to experimental or environmental conditions is often of scientific interest. A network is typically defined as a set of vertices and edges. Specifically a network or graph, G, can be written as G = (V, E) where V = {1,...,k} are the vertices or variables and E is the edge set that encodes the relationship between variables. A common example of a statistical network is the correlation matrix, where an edge represents the correlation between variables. While analysis of networks and their changes are ubiquitous across many domains, our work is motivated specifically by applications in which networks are derived from time series data. In contrast to independent data, statistical analysis of time series data is complicated by the inherent serial correlation. In practice, the degree of this correlation is unknown and network analysis methods that can flexibly handle varying degrees of dependence are needed. We approach this problem from two angles. The first angle, used in the first two portions of this thesis, focuses on developing methods with minimal assumptions on temporal dependence. In the third portion of this thesis we approach this problem from the second angle which attempts to leverage the flexibility of deep learning methods to analyze statistical networks. In the first chapter, we propose a novel order selection method in vector autoregressive (VAR) models. Order selection is an essential step in fitting VAR models and while many order selection methods exist, all come with weaknesses. Our proposed order selection method is based on the observation that the expected squared error loss is flat once the fitted order reaches or exceeds the true order. We show that under mild assumptions on the underlying process our new order selection method consistently estimates the true order. Motivated by applications in neuroscience, the second chapter of this thesis develops a novel estimation and inference procedure for a difference in the inverse spectral densities. In neuroscience, it is often of interest to study how brain networks change in response to electrical stimulation with the hopes of developing stimulation-based treatments for neurodegenerative diseases. Furthermore, it is essential to study networks in the frequency domain as higher frequencies contain key brain connectivity information. With this in mind, we develop methods to directly estimate and perform statistical inference on a difference in inverse spectral densities. Crucially, our method relies on minimal assumptions and can flexibly handle a large range of data dependence. The last chapter of this thesis proposes a new deep learning-based change-point detection framework. The core idea behind this method is a continuous approximation of the indicator function. With this approximation, change-points can be specified as parameters of a deep learning model. Thus, change-points and model parameters can be jointly learned using stochastic optimization techniques. The proposed framework is general and can be applied to both independent and dependent data, such as time series data. Furthermore, the framework is model-agnostic and thus can be used to encode networks and study their changes.