Professor, Department of Mathematics and StatisticsUniversity of New Mexico
Bayesian Analysis of Latent Structure in Time Series Models
The analysis and decomposition of time series is considered under autoregressive models with a new class of prior distributions for parameters defining latent components. The approach induces a new class of smoothness priors on autoregressive coefficients, provides for formal inference on model order, including very high order models, and permits for the incorporation of uncertainty about model order into summary inferences. The class of prior models also allows for subsets of unit roots, and hence leads to inference on sustained though stochastically time-varying periodicities in time series and for formal treatment of initial values as latent variables. As the prior modeling induces complicated forms for prior distributions on the usual ``linear'' autoregressive parameters, exploration of posterior distributions naturally involves in iterative stochastic simulation with a Gibbs sampling format. Conditional posterior distributions are available in closed form, except for those corresponding to the parameters defining the quasi-cyclical components of the model. In this case and to assess for the induced changes of dimensionality, a reversible jump Markov chain Monte Carlo step is implemented. This methodology overcomes supposed problems in spectral estimation with autoregressive models using more traditional model fitting methods. Detailed simulation studies are presented to evaluate the efficiency of the sampler at detecting model order, wavelengths and amplitudes of cyclical components, and unitary roots or "spikes" in the spectral density at key frequencies. Analysis, decomposition and forecasting of several series is illustrated with applications to EEG studies, discovering underlying periodicities in astronomical series and climate change issues. Additionally, an extension is proposed for continuous autoregressive models that permits analysis of unequally-spaced time series. This new model, that falls within the class of dynamic linear models, overcome some of the difficulties in both embedding and fitting models defined through stochastic differential equation methods. Specifically, model structure incorporates spacings through the likelihood function and, as before, priors are specified on relevant parameters defining latent components. Simulation from the posterior distributions is implemented through component-wise random walk Metropolis steps with a reversible jump. Efficiency of the method is explored as for the standard autoregressive model with applications to irregularly sampled oxygen-isotope records.