  A Bayesian lattice filtering and smoothing approach is proposed for fast and
accurate modeling and inference in multivariate non-stationary time series.
This approach offers computational feasibility and interpretable time-frequency
analysis in the multivariate context. The proposed framework allows us to
obtain posterior estimates of the time-varying spectral densities of individual
time series components, as well as posterior measurements of the time-frequency
relationships across multiple components, such as time-varying coherence and
partial coherence.
  The proposed formulation considers multivariate dynamic linear models (MDLMs)
on the forward and backward time-varying partial autocorrelation coefficients
(TV-VPARCOR). Computationally expensive schemes for posterior inference on the
multivariate dynamic PARCOR model are avoided using approximations in the MDLM
context. Approximate inference on the corresponding time-varying vector
autoregressive (TV-VAR) coefficients is obtained via Whittle's algorithm. A key
aspect of the proposed TV-VPARCOR representations is that they are of lower
dimension, and therefore more efficient, than TV-VAR representations. The
performance of the TV-VPARCOR models is illustrated in simulation studies and
in the analysis of multivariate non-stationary temporal data arising in
neuroscience and environmental applications. Model performance is evaluated
using goodness-of-fit measurements in the time-frequency domain and also by
assessing the quality of short-term forecasting.
