Latent position network models with applications in time series analysis

In this dissertation, we develop a novel framework for modeling multivariate count time series with a network structure, focusing on the use of latent position models (LPM) to uncover hidden relationships among time series. Traditional approaches to time series analysis within network frameworks have primarily concentrated on continuous data, leaving a gap in the literature for count data, particularly when the network structure evolves over time. This work addresses this gap by proposing statistical models that reveal completely unknown latent network structures underlying count time series data, thus enabling a deeper understanding of the complex connection patterns between time series. The proposed models provide a comprehensive framework that captures both static and dynamic interdependencies, offering theoretical advancements as well as practical tools for visualizing and interpreting complex relationships in time series data. The core objective of this research is to develop methodologies for estimating embedded network structures within time series data informed by latent factors. To illustrate the effectiveness of these approaches, we apply them to two datasets: burglary counts in Chicago and mumps cases in England, contributing to fields such as epidemiology and criminology. The findings demonstrate the models’ ability to reveal relationships between burglary occurrences across regions in Chicago and the fixed or temporal patterns in the spread of mumps infection across regions in England, uncovering connections that are not readily observable using conventional time series analysis techniques. We demonstrate network inference from the proposed modeling frameworks using optimization methods, such as the L-BFGS algorithm, and sampling methods such as the Hamiltonian Monte Carlo method. These approaches efficiently estimate model parameters, such as the latent positions of nodes within the network. A collection of simulation studies are presented to evaluate the performance of the proposed models across various settings. These studies demonstrate the models’ capacity to produce consistent parameter estimates and to uncover underlying network structures in both static and dynamic contexts. The simulation results further support the model’s applicability to real-world data, illustrating how they can uncover hidden interdependencies in network-embedded count time series data.