Bayesian Spectral Modeling for Multiple Time Series

We develop a novel Bayesian modeling approach to spectral density estimation for multiple time series. The log-periodogram distribution for each series is modeled as a mixture of Gaussian distributions with frequency-dependent weights and mean functions. The implied model for the log-spectral density is a mixture of linear mean functions with frequency-dependent weights. The mixture weights are built through successive differences of a logit-normal distribution function with frequency-dependent parameters. Building from the construction for a single spectral density, we develop a hierarchical extension for multiple time series. Specifically, we set the mean functions to be common to all spectral densities and make the weights specific to the time series through the parameters of the logit-normal distribution. In addition to accommodating flexible spectral density shapes, a practically important feature of the proposed formulation is that it allows for ready posterior simulation through a Gibbs sampler with closed form full conditional distributions for all model parameters. The modeling approach is illustrated with simulated datasets and used for spectral analysis of multichannel electroencephalographic recordings, which provides a key motivating application for the proposed methodology.