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We propose to model time-varying periodic and oscillatory processes by means of a hidden Markov model where the states are defined through the spectral properties of a periodic regime. The number of states is unknown along with the relevant periodicities, the role and number of which may vary across states. We address this inference problem by a Bayesian nonparametric hidden Markov model assuming a sticky hierarchical Dirichlet process for the switching dynamics between different states while the periodicities characterizing each state are explored by means of a trans-dimensional Markov chain Monte Carlo sampling step. We develop the full Bayesian inference algorithm and illustrate the use of our proposed methodology for different simulation studies as well as an application related to respiratory research which focuses on the detection of apnea instances in human breathing traces.

We propose a novel Bayesian methodology for analyzing nonstationary time series that exhibit oscillatory behaviour. We approximate the time series using a piecewise oscillatory model with unknown periodicities, where our goal is to estimate the change-points while simultaneously identifying the potentially 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 detection of instances of sleep apnea in plethysmographic respiratory traces., Comment: Received 23 Oct 2018, Accepted 12 May 2019

A central challenge in computational modeling of dynamic biological systems is parameter inference from experimental time course measurements. However, one would not only like to infer kinetic parameters but also study their variability from cell to cell. Here we focus on the case where single-cell fluorescent protein imaging time series data are available for a population of cells. Based on van Kampen's linear noise approximation, we derive a dynamic state space model for molecular populations which is then extended to a hierarchical model. This model has potential to address the sources of variability relevant to single-cell data, namely, intrinsic noise due to the stochastic nature of the birth and death processes involved in reactions and extrinsic noise arising from the cell-to-cell variation of kinetic parameters. In order to infer such a model from experimental data, one must also quantify the measurement process where one has to allow for nonmeasurable molecular species as well as measurement noise of unknown level and variance. The availability of multiple single-cell time series data here provides a unique testbed to fit such a model and quantify these different sources of variation from experimental data., Comment: Published in at http://dx.doi.org/10.1214/13-AOAS669 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)