Recent advances in information technology have made high-dimensional non-stationary signals increasingly common in many areas. We develop a suite of models and computationally fast methods for analysis and forecasting of multiple and multivariate non-stationary time series. These approaches are based on dynamic model representations in the partial autocorrelation domain.
Chapter 1 introduces some background and discusses the limitations of current models and methods for analyzing high-dimensional non-stationary time series. In order to obtain fast and accurate modeling and inference such high-dimensional dynamic settings, a system of Bayesian lattice filtering and smoothing approaches in the PARCOR domain are proposed in this thesis. This PARCOR framework leads to lower dimensional representations, and consequently computationally faster inference, than those required by models in the time and/or frequency domains, such as state-space representations of time-varying autoregressive and vector autoregressive models, which are commonly used in practice.
Chapter 2 proposes an efficient hierarchical dynamic PARCOR model to describe the time-varying behavior of multiple time series, and develops procedures to infer the latent structure underlying multiple non-stationary time series. The performance of the proposed models and methods is tested in the context of analyzing multiple brain signals recorded simultaneously during specific experimental settings or clinical studies. The proposed approach improves the efficiency in obtaining posterior summaries of the time-frequency characteristics of the multiple time series, as well as those summarizing their common underlying structure.
Chapter 3 proposes a set of multivariate dynamic linear models (MDLMs) on the forward and backward predictions errors in the PARCOR domain along with computationally efficient methods for filtering and smoothing methods within this modeling class. The proposed framework allows us to obtain posterior estimates of the time-varying spectral densities of individual time series components, as well as posterior measurements of the time-frequency relationships across multiple components such as time-varying coherence and partial coherence. Computationally expensive schemes for posterior inference on the multivariate dynamic PARCOR model are avoided using approximate inference. The performance of the TV-VPARCOR methods is illustrated in simulation studies and in the analysis of multivariate non-stationary temporal data arising in neuroscience and in environmental applications. Model performance is evaluated using goodness-of-fit measurements in the time-frequency domain and also by assessing the quality of short-term forecasting.
Chapter 4 extends the models and methods of Chapter 3 by considering shrinkage priors on the TV-VPARCOR parameters in order to reduce overfitting. Such priors allow shrinkage of time-varying parameters to static ones, as well as shrinkage to zero for those parameters that are not statistically significant. A Markov chain Monte Carlo (MCMC) algorithm for full posterior inference is proposed. In addition, an importance sampling variational Bayes (ISVB) approach is also developed and implemented for fast and reliable approximate inference, making the dynamic TV-VPARCOR modeling and inference framework feasible for analysis of large-dimensional time series. The performance of the proposed models and methods is examined in extensive simulation studies and also in a case study involving the analysis of wind component data from several locations in Northern California.
Chapter 5 summarizes this thesis and provides some ideas for possible future work.
