# Efficient Bayesian PARCOR Approaches for Dynamic Modeling of Multivariate Time Series

## TL;DR

This paper introduces a Bayesian lattice filtering approach for efficient, interpretable modeling of multivariate non-stationary time series, enabling accurate inference of spectral densities and time-frequency relationships.

## Contribution

It proposes a novel, lower-dimensional Bayesian PARCOR framework that improves computational efficiency and accuracy in multivariate time series analysis compared to traditional TV-VAR models.

## Key findings

- Effective in neuroscience and environmental data analysis
- Achieves good fit in time-frequency domain
- Provides accurate short-term forecasts

## Abstract

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.