# Adaptive Bayesian Power Spectrum Analysis of Multivariate Nonstationary Time Series

## TL;DR

This paper presents a Bayesian nonparametric method for analyzing multivariate nonstationary time series by adaptively partitioning data into segments with potentially different spectral properties, capturing both abrupt and gradual changes.

## Contribution

It introduces a novel Bayesian framework using reversible jump MCMC and Hamiltonian Monte Carlo for flexible, nonparametric spectral analysis with adaptive segmentation.

## Key findings

- Effective in simulation studies
- Successfully applied to EEG sleep data
- Captured both abrupt and slow spectral changes

## 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 slow-varying changes in spectral matrices. Empirical performance is evaluated in simulation studies and illustrated through analyses of electroencephalography during sleep and of the El Ni\~no-Southern Oscillation.