Varying-Coefficient Models and Functional Data Analyses for Dynamic Networks and Wearable Device Data

As more data are observed over time, investigating the variation across time has become a vital part of analyzing such data. In this dissertation, we discuss varying-coefficient models and functional data analysis methods for temporally heterogenous data. More specifically, we examine two different types of temporal heterogeneity. The first type of temporal heterogeneity stems from temporal evolution of relational pattern over time. Dynamic networks are commonly used when relational data are observed over time. Unlike static network analysis, dynamic network analysis emphasizes the importance of recognizing temporal evolution of relationship among observations. We propose and investigate a family of dynamic network models, known as varying-coefficient exponential random graph model (VCERGM), that characterize the evolution of network topology through smoothly varying parameters. The VCERGM directly provides an interpretable dynamic network model that enables the inference of temporal heterogeneity in dynamic networks. Furthermore, we introduce a method that analyzes multilevel dynamic networks. If there exist multiple relational data observed at one time point, it is reasonable to additionally consider the variability among the repeated observations at each time point. The proposed method is an extension of stochastic blockmodels with a priori block membership and temporal random effects. It incorporates a variability among multiple relational structures at one time point and provides a richer representation of dependent engagement patterns at each time point. The method is also flexible in analyzing networks with time-varying networks. Its smooth parameters can be interpreted as evolving strength of engagement within and across blocks. The second type of temporal heterogeneity is motivated by temporal shifts in continuously observed data. When multiple curves are obtained and there exists a common curvature shared by all the observed curves, understanding the common curvature may involve a preprocessing step of managing temporal shifts among curves. We explore the properties of continuous in-shoe sensor recordings to understand the source of variability in gait data. Our case study is based on measurements of three healthy subjects. The in-shoe sensor data we explore show both phase and amplitude variabilities; we separate these sources via curve registration. We examine the correlation of temporal shifts across sensors to evaluate the pattern of phase variability shared across sensors. We apply a series of functional data analysis approaches to the registered in-shoe sensor curves to examine their association with current gold-standard gait measurement, so called ground reaction force.