Modeling Multiple-Subject and Discrete-Valued High-Dimensional Time Series

This thesis focuses on two separate topics in modeling of high-dimensional time series (HDTS) with several structures and their various applications. The first topic is on modeling HDTS from multiple subjects. Here, the structure of interest includes model components that are shared by all subjects and that are individual to subjects or their groups. A running theme in this modeling is the heterogeneity of subjects. Dealing with heterogeneous data has been of particular interest recently in social, health, behavioral, and other sciences. The second topic is on modeling HDTS that are discrete-valued, including binary, categorical, and non-negative count observations. Compared with continuous time series modeling where autoregressive-type models dominate, there are no generally preferred models in the discrete setting. The models considered in this thesis are based on latent Gaussian processes, which drive the dynamics of the observed discrete series. The models have the advantages of allowing negative autocorrelations, and flexible choices of marginal distributions of discrete observations.

The presentation will focus on one project in the first topic, namely, integrative dynamic factor models (DFMs) for multiple subjects in several groups. The models have components that allow one to explore the inter-differences across subjects (and groups). At the same time, the intra-differences can be investigated by reconstructing the individual temporal dynamics of different subjects. A flexible identifiability condition on the factor covariance is adopted, which expands the scope of heterogeneity and contributes to better model interpretation and forecasting results. From a methodological standpoint, a novel algorithm that combines non-iterative block segmentation, efficient rank selection, and variants of PCA for multiple subjects, is suggested. Simulations under various scenarios and analysis of resting-state functional MRI data collected from multiple subjects are conducted. If time permits, connections to the other project in the first topic, namely, vector autoregressive modeling (multi-VAR), will be also made.

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