  To capture the slowly time-varying spectral content of real-world
time-series, a common paradigm is to partition the data into approximately
stationary intervals and perform inference in the time-frequency domain.
However, this approach lacks a corresponding nonstationary time-domain
generative model for the entire data and thus, time-domain inference occurs in
each interval separately. This results in distortion/discontinuity around
interval boundaries and can consequently lead to erroneous inferences based on
any quantities derived from the posterior, such as the phase. To address these
shortcomings, we propose the Piecewise Locally Stationary Oscillation (PLSO)
model for decomposing time-series data with slowly time-varying spectra into
several oscillatory, piecewise-stationary processes. PLSO, as a nonstationary
time-domain generative model, enables inference on the entire time-series
without boundary effects and simultaneously provides a characterization of its
time-varying spectral properties. We also propose a novel two-stage inference
algorithm that combines Kalman theory and an accelerated proximal gradient
algorithm. We demonstrate these points through experiments on simulated data
and real neural data from the rat and the human brain.
