Adaptive Bayesian Spectral Analysis of High-Dimensional Nonstationary Time Series

Abstract

This article introduces a nonparametric approach to spectral analysis of a high-dimensional multivariate nonstationary time series. The procedure is based on a novel frequency-domain factor model that provides a flexible yet parsimonious representation of spectral matrices from a large number of simultaneously observed time series. Real and imaginary parts of the factor loading matrices are modeled independently using a prior that is formulated from the tensor product of penalized splines and multiplicative gamma process shrinkage priors, allowing for infinitely many factors with loadings increasingly shrunk toward zero as the column index increases. Formulated in a fully Bayesian framework, the time series is adaptively partitioned into approximately stationary segments, where both the number and locations of partition points are assumed unknown. Stochastic approximation Monte Carlo techniques are used to accommodate the unknown number of segments, and a conditional Whittle likelihood-based Gibbs sampler is developed for efficient sampling within segments. By averaging over the distribution of partitions, the proposed method can approximate both abrupt and slowly varying changes in spectral matrices. Performance of the proposed model is evaluated by extensive simulations and demonstrated through the analysis of high-density electroencephalography. Supplementary materials for this article are available online.

Keywords:

• Factor analysis
• High-dimensional time series
• Locally stationary process
• Multiplicative gamma process
• Spectral analysis
• Stochastic approximation Monte Carlo

Supplementary Materials

Supplementary materials are available online, including a pdf file that includes details of the sampling scheme, a convergence assessment tool for SAMC and its application to the TMS-evoked EEG data analysis, and additional simulation results. Matlab code for implementing the proposed method is provided.

Additional information

Funding

This work was funded in part by PSC-CUNY Research Award 63069-0051, a Eugene M. Lang Junior Faculty Research Fellowship, and National Institutes of Health grants R01GM113243, R01MH113827, and R21MH119543.