Bayesian analysis of nonstationary periodic time series

Identifying the periodicities present in a cyclical process allows us to gain knowledge about the sources of variability that drive that phenomenon. For instance, respiratory traces obtained from a plethysmograph used on rodents in experimental sleep apnea research reveal many sudden changes in their periodic features as the rat spontaneously changes its breathing pattern during its sleep-wake activities. Similarly, human temperature, as measured by a wearable sensing device over several days at relatively high temporal resolution (e.g. 5 minutes), may be subject to a different periodic behaviour during the night when the individual transitions between different stages of sleep. While the theory and methods for analyzing the periodicities of time series data are relatively well-developed for the case of stationary time series, the task of modelling time series that undergo regime shifts in periodicity, amplitude and phase remains challenging because the timing of the changes and the relevant periodicities are usually unknown (both in value and number). This thesis introduces new methodologies for the automated analysis of non-stationary periodic time series. In the first part of this research, we present a novel Bayesian approach for analyzing time series data that exhibit regime shifts in periodicity, amplitude and phase, where we approximate the time series using a piece-wise oscillatory model with unknown periodicities, and our goal is to estimate the change-points while simultaneously identifying the changing periodicities in the data. Our proposed methodology is based on a trans-dimensional Markov chain Monte Carlo (MCMC) algorithm that simultaneously updates the change-points and the periodicities relevant to any segment between them. We show that the proposed methodology successfully identifies time changing oscillatory behaviour in two applications which are relevant to e-Health and sleep research, namely the occurrence of ultradian oscillations in human skin temperature during the time of night rest, and the characterization of instances of sleep apnea in plethysmographic respiratory traces. In addition to detecting temporal changes, it may also be of interest to recognize the recurrence of a relevant periodic pattern. In the second half of this thesis, we consider periodic phenomena, whose behaviour switches over time, as realizations of a hidden Markov model where the number of states is unknown along with the relevant periodicities, the role of which varies over the different states. Flexibility on the number of states is achieved by using Bayesian nonparametric techniques that address the stochastic switching dynamics of the time series via a hierarchical Dirichlet process that captures the temporal mode persistence of the hidden states. The variable dimensionality regarding the number of periodicities that characterizes the different regimes is addressed by developing an appropriate trans-dimensional MCMC sampler. We illustrate the use of our proposed approach in a case study relevant to respiratory research, namely the detection of recurring instances of sleep apnea in human respiratory traces.