AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series

Abstract

We propose a method for analyzing possibly nonstationary time series by adaptively dividing the time series into an unknown but finite number of segments and estimating the corresponding local spectra by smoothing splines. The model is formulated in a Bayesian framework, and the estimation relies on reversible jump Markov chain Monte Carlo (RJMCMC) methods. For a given segmentation of the time series, the likelihood function is approximated via a product of local Whittle likelihoods. Thus, no parametric assumption is made about the process underlying the time series. The number and lengths of the segments are assumed unknown and may change from one MCMC iteration to another. The frequentist properties of the method are investigated by simulation, and applications to electroencephalogram and the El Niño Southern Oscillation phenomenon are described in detail.

Keywords:

• Electroencephalogram
• El Niño Southern Oscillation
• Locally stationary time series
• Reversible jump Markov chain Monte Carlo
• Whittle likelihood.

Acknowledgments

We thank Dr. Li Qin for providing the IEEG data and the Associate Editor and the referees for their helpful comments. O. Rosen was supported in part by NSF grant DMS-0804140 and by the National Security Agency under grant H98230-12-1-0246. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. D. S. Stoffer was supported in part by NSF grant DMS-0805050. S. Wood was supported by the Australian Research Council and Elders Australia Limited through Linkage Project LP0989778.