[Paper Review] Bayesian time-aligned factor analysis of paired multivariate time series

Arkaprava Roy, Jana Schaich Borg+1 more|PubMed|Apr 27, 2019

Sensory Analysis and Statistical Methods33 references4 citations

TL;DR

This paper proposes TACIFA, a Bayesian dynamic factor model that jointly estimates shared and individual components in paired multivariate time series while accounting for unknown time warping via a flexible warping function. The method enables uncertainty quantification in time alignment and demonstrates strong performance in simulations and a social synchrony experiment, correctly identifying that one individual followed the other in time with high accuracy.

ABSTRACT

Many modern data sets require inference methods that can estimate the shared and individual-specific components of variability in collections of matrices that change over time. Promising methods have been developed to analyze these types of data in static cases, but only a few approaches are available for dynamic settings. To address this gap, we consider novel models and inference methods for pairs of matrices in which the columns correspond to multivariate observations at different time points. In order to characterize common and individual features, we propose a Bayesian dynamic factor modeling framework called Time Aligned Common and Individual Factor Analysis (TACIFA) that includes uncertainty in time alignment through an unknown warping function. We provide theoretical support for the proposed model, showing identifiability and posterior concentration. The structure enables efficient computation through a Hamiltonian Monte Carlo (HMC) algorithm. We show excellent performance in simulations, and illustrate the method through application to a social mimicry experiment.

Motivation & Objective

• To address the lack of methods that jointly model shared and individual variability in paired multivariate time series with time-varying lags.
• To develop a fully Bayesian framework that incorporates uncertainty in time alignment through a nonparametric warping function.
• To enable efficient posterior computation via Hamiltonian Monte Carlo (HMC) while ensuring model identifiability and posterior concentration.
• To apply the method to real-world data, particularly in social synchrony studies, where temporal misalignment between interacting individuals is common.

Proposed method

• TACIFA models paired matrices as a combination of shared factors, individual-specific factors, and error, with the shared components linked via a warping function to align time points.
• The warping function is modeled nonparametrically using B-splines with a prior that ensures monotonicity and smoothness, enabling flexible time alignment.
• A hierarchical Bayesian model is specified with conjugate priors on factor loadings and a non-centered parameterization to improve MCMC mixing.
• Posterior inference is performed using Hamiltonian Monte Carlo (HMC), which enables efficient exploration of the joint posterior distribution.
• The model enforces orthogonality between shared and individual factor loadings to ensure identifiability of components.
• The method supports uncertainty quantification through posterior predictive checks and credible bands on the warping function.

Experimental results

Research questions

• RQ1 Can a Bayesian dynamic factor model effectively separate shared and individual components in paired multivariate time series with unknown time alignment?
• RQ2 How can uncertainty in time warping be properly quantified and integrated into a factor analysis framework?
• RQ3 Does the proposed model outperform two-stage alignment and inference approaches in terms of prediction accuracy and coverage?
• RQ4 To what extent does the model recover the true temporal lag between interacting individuals in a social synchrony experiment?
• RQ5 How do the number of relevant features affect the estimated similarity between paired time series?

Key findings

• The estimated warping function in the human mimicry dataset consistently fell below the identity line, indicating that one individual followed the other throughout the experiment, correctly capturing the temporal lag.
• TACIFA achieved lower out-of-sample prediction mean squared error (MSE) of 4.25 and 2.21 for the two individuals, compared to around 9 for two-stage approaches.
• The method achieved 95% and 98% frequentist coverage within 95% posterior predictive credible bands, indicating well-calibrated uncertainty quantification.
• The shared factor space contained 13 important components, consistent with expectations from facial symmetry and coordinated head movements.
• Similarity measures (Syn) increased from 0.80 to 0.85 as more features related to head movement were included, confirming that relevant features drive similarity.
• The model successfully identified minimal individual-specific variation, as expected in a coordinated mimicry task, with low importance assigned to individual factor loadings.