Bayesian Model Search for Nonstationary Periodic Time Series

Abstract

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.

Time-varying auto-regressive models for count time-series

“…For example, see [15] where the likelihood is formed by taking the product of individual conditional likelihoods for a non-stationary timeseries. Other approaches can be found in [27,48] where the likelihoods for the proposed non-stationary processes were computed without any local stationary approximation. Moreover, note that such approximating stationary processes can be shown to exist under the general smoothness conditions as outlined in Theorem 1 in [12] (for tvARCH case) or Proposition 2.3 in [38](for tvGARCH case).…”

Identifying the recurrence of sleep apnea using a harmonic hidden Markov model

“…Their methodology is based on the assumption that the time series are piecewise stationary, and the underlying spectral density for each partition is smooth over frequencies. In order to deal with changes in spectral densities with sharp peaks which can be observed for some physiological data sets such as respiratory data, Hadj-Amar et al (2020) proposed a change-point analysis where they introduced a Bayesian methodology for inferring changepoints along with the number and values of the periodicities affecting each segment. While these approaches allow us to analyse the spectral changing properties of a process from a retrospective and exploratory point of view, in order to develop a more comprehensive understanding of the process driving the data, further modeling assumptions are needed that quantify the probabilistic rules governing the transitions as well as recurrence of different oscillatory dynamic patterns.…”

CoRhythMo: A Computational Framework for Modeling Biobehavioral Rhythms from Mobile and Wearable Data Streams

“…CPD has been used in monitoring medical conditions. For example, applying CPD to heart rate (HR), electrocardiogram (ECG), and electroencephalogram (EEG) has helped better diagnosis of heart disease and understand brain activity [10,[21][22][23][24]. CPD has also been applied to human activity recognition using data from smart home and mobile devices.…”

“…For modeling the variations in physiological data during each cycle, we adopt the Automatic Non-stationary Oscillatory Modelling (AutoNOM) to model non-stationary time series with a known period. AutoNOM identifies change points in each cycle and achieves piecewise fitting [10] using sinusoidal regression models simultaneously. We prefer the CPD technology used in the AutoNOM because it is more sensitive to the change of frequencies of time series, and the AutoNOM can find the best sinusoidal equations to fit the data in each segment [10].…”

“…AutoNOM identifies change points in each cycle and achieves piecewise fitting [10] using sinusoidal regression models simultaneously. We prefer the CPD technology used in the AutoNOM because it is more sensitive to the change of frequencies of time series, and the AutoNOM can find the best sinusoidal equations to fit the data in each segment [10]. The following section describes these methods in detail.…”