We detail and illustrate time series analysis and spectral inference in autoregressive models with a focus on the underlying latent structure and time series decompositions. A novel class of priors on parameters of latent components leads to a new class of smoothness priors on autoregressive coefficients, provides for formal inference on model order, including very high order models, and leads to 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. Applications to analysis of the frequency composition of time series, in both time and spectral domains, is illustrated in a study of a time series from astronomy. This analyses demonstrates the impact and utility of the new class of priors in addressing model order uncertainty and in allowing for unit root structure. Time domain decomposition of a time series into estimated latent components provides an important alternative view of the component spectral characteristics of a series. In addition, our data analysis illustrates the utility of the smoothness prior and allowance for unit root structure in inference about spectral densities. In particular, the framework overcomes supposed problems in spectral estimation with autoregressive models using more traditional model fitting methods.

This research was partially supported by the National Science Foundation uder grants DMS-9704432 and DMS-9707914.

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