Efficient Bayesian PARCOR approaches for dynamic modeling of multivariate time series

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.

Bayesian Circular Lattice Filters for Computationally Efficient Estimation of Multivariate Time-Varying Autoregressive Models

“…Thus, the usage is limited to those time series with a small number of series and TV-VAR models of low orders. Zhao and Prado (2020) proposed a multivariate BLF in which the computational cost increases linearly with the model order, but the computational cost of their approach still increases exponentially with the number of series. In contrast, we propose methods that avoid cumbersome matrix calculations in the BLFs.…”

“…We consider 500 bivariate time series of length T = 1034 simulated from a TV-VAR model (Zhao and Prado, 2020) as follows:…”

“…In addition, these models are also well-suited for measuring the effect of policy changes on other factors of the economy (Huerta and Lopes, 2000;Primiceri, 2005;Nakajima et al, 2011;Hunter et al, 2017). Another example in which these models are useful is multi-channel electroencephalography (EEG) data, where the data are analyzed through their time-frequency representation to reveal how the neuronal activity in one area of the human brain may influence another (Ombao et al, 2005;Zhao and Prado, 2020;Zhao, 2022). In the era of big data, the number of series and series length within a multivariate time series have been increasing due to technological advances.…”

“…In contrast, in the frequency domain, the spectral density and the coherence can often reveal features of each time series that are not readily apparent in the time domain. For example, frequency domain approaches to multi-channel EEG data can aid in the understanding of connective activities within the brain by estimating their spectrum and coherence (Ombao et al, 2001(Ombao et al, , 2005Zhao and Prado, 2020;Zhao, 2022). Some of the current methods for TV-VAR models assume that the standard deviations of the innovations evolve as geometric random walks and, therefore, belong to the class of stochastic volatility models (Shephard, 2005;Primiceri, 2005;Del Negro and Primiceri, 2015;Nakajima et al, 2011).…”

“…However, this method is computationally expensive due to sequential computation of matrix inversions. Zhao and Prado (2020) extended the Bayesian lattice filter (BLF) of Yang et al (2016) to TV-VAR models with a constant innovation covariance by implementing the multivariate Durbin-Levison algorithm (Brockwell and Davis, 2009) and using MDLMs on the forward and backward time-varying partial autocorrelation coefficient matrices (TV-VPARCOR).…”