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We propose a Bayesian hidden Markov model for analyzing time series and sequential data where a special structure of the transition probability matrix is embedded to model explicit-duration semi-Markovian dynamics. Our formulation allows for the development of highly flexible and interpretable models that can integrate available prior information on state durations while keeping a moderate computational cost to perform efficient posterior inference. We show the benefits of choosing a Bayesian approach for HSMM estimation over its frequentist counterpart, in terms of model selection and out-of-sample forecasting, also highlighting the computational feasibility of our inference procedure whilst incurring negligible statistical error. The use of our methodology is illustrated in an application relevant to e-Health, where we investigate rest-activity rhythms using telemetric activity data collected via a wearable sensing device. This analysis considers for the first time Bayesian model selection for the form of the explicit state dwell distribution. We further investigate the inclusion of a circadian covariate into the emission density and estimate this in a data-driven manner., Comment: Paper to appear in Bayesian Analysis

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). \ud \ud 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. \ud \ud 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.

The prefrontal cortex (PFC) is a region of the brain that in humans is involved in the production of higher-order functions such as cognition, emotion, perception, and behavior. Neurotransmission in the PFC produces higher-order functions by integrating information from other areas of the brain. At the foundation of neurotransmission, and by extension at the foundation of higher-order brain functions, are an untold number of coordinated molecular processes involving the DNA sequence variants in the genome, RNA transcripts in the transcriptome, and proteins in the proteome. These "multiomic" foundations are poorly understood in humans, perhaps in part because most modern studies that characterize the molecular state of the human PFC use tissue obtained when neurotransmission and higher-order brain functions have ceased (i.e., the postmortem state). Here, analyses are presented on data generated for the Living Brain Project (LBP) to investigate whether PFC tissue from individuals with intact higher-order brain function has characteristic multiomic foundations. Two complementary strategies were employed towards this end. The first strategy was to identify in PFC samples obtained from living study participants a signature of RNA transcript expression associated with neurotransmission measured intracranially at the time of PFC sampling, in some cases while participants performed a task engaging higher-order brain functions. The second strategy was to perform multiomic comparisons between PFC samples obtained from individuals with intact higher-order brain function at the time of sampling (i.e., living study participants) and PFC samples obtained in the postmortem state. RNA transcript expression within multiple PFC cell types was associated with fluctuations of dopaminergic, serotonergic, and/or noradrenergic neurotransmission in the substantia nigra measured while participants played a computer game that engaged higher-order brain functions. A subset of these associations - termed the "transcriptional program associated with neurotransmission" (TPAWN) - were reproduced in analyses of brain RNA transcript expression and intracranial neurotransmission data obtained from a second LBP cohort and from a cohort in an independent study. RNA transcripts involved in TPAWN were found to be (1) enriched for RNA transcripts associated with measures of neurotransmission in rodent and cell models, (2) enriched for RNA transcripts encoded by evolutionarily constrained genes, (3) depleted of RNA transcripts regulated by common DNA sequence variants, and (4) enriched for RNA transcripts implicated in higher-order brain functions by human population genetic studies. In PFC excitatory neurons of living study participants, higher expression of the genes in TPAWN tracked with higher expression of RNA transcripts that in rodent PFC samples are markers of a class of excitatory neurons that connect the PFC to deep brain structures. TPAWN was further reproduced by RNA transcript expression patterns differentiating living PFC samples from postmortem PFC samples, and significant differences between living and postmortem PFC samples were additionally observed with respect to (1) the expression of most primary RNA transcripts, mature RNA transcripts, and proteins, (2) the splicing of most primary RNA transcripts into mature RNA transcripts, (3) the patterns of co-expression between RNA transcripts and proteins, and (4) the effects of some DNA sequence variants on RNA transcript and protein expression. Taken together, this report highlights that studies of brain tissue obtained in a safe and ethical manner from large cohorts of living individuals can help advance understanding of the multiomic foundations of brain function.