Professor, Department of Applied Mathematics & StatisticsUniversity of California - Santa Cruz, CA
Latent Structure in Non-Stationary Time Series
The class of time-varying autoregressions (TVAR) constitutes a suitable class of models to describe long series that exhibit non-stationarities. Signals experiencing changes in frequency content over time are often appropriately modeled via relatively long AR processes with time-varying coefficients. Using a particular DLM (dynamic linear model) representation of TVAR models, time-domain decompositions of the series into latent components with dynamic, correlated structures are obtained. Methodological aspects of such decompositions and interpretability of the underlying processes are discussed in the study of EEG (electroencephalogram) traces. Multiple EEG signals recorded under different ECT (electroconvulsive therapy) conditions are analyzed using TVAR models. Decompositions of these series and summaries of the evolution of functions of the TVAR parameters over time, such as characteristic frequency, amplitude and modulus trajectories of the latent, often quasi-periodic processes, are helpful in obtaining insights into the common structure driving the multiple series. From the scientific viewpoint, characterizing the system structure underlying the EEG signals is a key factor in assessing the efficacy of ECT treatments. Factor models that assume a time-varying AR structure on the factors and dynamic regression models that account for time-varying instantaneous lead/lag and amplitude structures across the multiple series are also explored. Issues of posterior inference and implementation of these models using Markov chain Monte Carlo (MCMC) methods are discussed. Decompositions of the scalar components of multivariate time series are presented. Similar to the univariate case, the state-space representation of a VAR$(p)$ model implies that each univariate element of a vector process can be decomposed into a sum of latent processes where every characteristic modulus and frequency component appears in the decomposition of each univariate series, while the phase and amplitude of each latent component vary in magnitude across the univariate elements. Simulated data sets and portions of a multi-channel EEG data set are analyzed here in order to illustrate the multivariate decomposition techniques.