Adaptive Bayesian Time–Frequency Analysis of Multivariate Time Series

ABSTRACT

This article introduces a nonparametric approach to multivariate time-varying power spectrum analysis. The procedure adaptively partitions a time series into an unknown number of approximately stationary segments, where some spectral components may remain unchanged across segments, allowing components to evolve differently over time. Local spectra within segments are fit through Whittle likelihood-based penalized spline models of modified Cholesky components, which provide flexible nonparametric estimates that preserve positive definite structures of spectral matrices. The approach is formulated in a Bayesian framework, in which the number and location of partitions are random, and relies on reversible jump Markov chain and Hamiltonian Monte Carlo methods that can adapt to the unknown number of segments and parameters. By averaging over the distribution of partitions, the approach can approximate both abrupt and slowly varying changes in spectral matrices. Empirical performance is evaluated in simulation studies and illustrated through analyses of electroencephalography during sleep and of the El Niño-Southern Oscillation. Supplementary materials for this article are available online.

Keywords:

• Locally stationary process
• Modified Cholesky decomposition
• Nonstationary multivariate time series
• Penalized splines
• Reversible jump Markov chain Monte Carlo
• Spectral analysis

Acknowledgment

The authors thank the referees, associate editor, and editor for providing insightful comments that greatly improved the article. The work presented in this article was developed as part of the first author's PhD dissertation at Temple University.

Additional information

Funding

This work was supported by NIH grant R01GM113243. The EEG data considered in the article are available through PhysioNet, which was supported by NIH grant R01GM104987.