Analyzing spectral properties of multivariate time series is important in many fields, from sleep medicine to climatology. Existing methods either assume smooth spectral changes over time or require common break points for all components, which limits accuracy. This work fills the gap by allowing different components of the spectral matrix to evolve independently, capturing both abrupt and gradual changes.
The time series is approximated by a piecewise stationary process with a random number of segments and break points. For each segment, the local spectrum is modeled via a modified Cholesky decomposition, where components are expanded in periodic even and odd spline bases. The Bayesian scheme uses reversible-jump MCMC to change the number of segments and Hamiltonian MCMC to update parameters within the model. Averaging over the posterior distribution of partitions allows approximating both abrupt and slow changes.
In simulations with a piecewise stationary three-dimensional process, the proposed method showed lower mean squared error compared to piecewise vector autoregression and ANOVA smoothing splines. For a slowly varying two-dimensional process, the method outperformed piecewise vector autoregression and was comparable to ANOVA splines. When estimating the number of segments in a complex case, the method more often selected the correct number (posterior probability 0.794) and was computationally faster than SSAMC. On real EEG data, the method detected a sharp transition between sleep stages with probability 0.94 near the visually determined boundary. For El Niño data, the method revealed a possible spectral change in the 1980s with posterior probability 0.652 for two segments.
This work expands the possibilities of analyzing multivariate nonstationary signals by allowing spectral components to change independently and capturing both abrupt and gradual changes. This distinguishes the method from neighboring approaches that either fix common break points or require smoothness. Practical significance is confirmed on medical and climatic data, paving the way for more accurate diagnostics and monitoring.
The method can be extended to high-dimensional multivariate series through the inclusion of frequency-domain principal components, as well as to replicated time series for analyzing associations with clinical outcomes. Adaptation to rapidly but smoothly varying spectra is possible, and other bases can be used for specific tasks.
The method will influence EEG analysis in sleep medicine, climatology (El Niño), neuroscience, and other fields where the dynamics of spectral relationships are important.
Development of a version for high-dimensional and replicated data, as well as improved handling of rapid smooth changes.
In statistics, the problem of efficiently analyzing multivariate nonstationary time series with heterogeneous component dynamics remains open. This method contributes to a solution, but questions of scalability and interpretation remain.
In EEG analysis, the method automatically identified the transition from slow-wave to REM sleep within a quarter of a second, comparable to the performance of an experienced clinician.
The idea of adaptively partitioning a signal into segments with different dynamics echoes the analysis of brain waves in the movie Inception, where characters track changes in states of consciousness through activity patterns.