[Paper Review] Fitting Graphical Interaction Models to Multivariate Time Series

Michael Eichler|arXiv (Cornell University)|Jun 27, 2012

Statistical Methods and Inference23 references13 citations

TL;DR

This paper proposes a parametric approach to fitting graphical interaction models to multivariate stationary time series by generalizing covariance selection models to the time series context through inverse covariance structures. It formulates the models using vector autoregressive representations under conditional independence constraints encoded by undirected graphs, enabling maximum likelihood estimation via Whittle's approximation and an iterative solution method, with illustration on a real data example.

ABSTRACT

Graphical interaction models have become an important tool for analysing multivariate time series. In these models, the interrelationships among the components of a time series are described by undirected graphs in which the vertices depict the components while the edges indictate possible dependencies between the components. Current methods for the identification of the graphical structure are based on nonparametric spectral stimation, which prevents application of common model selection strategies. In this paper, we present a parametric approach for graphical interaction modelling of multivariate stationary time series. The proposed models generalize covariance selection models to the time series setting and are formulated in terms of inverse covariances. We show that these models correspond to vector autoregressive models under conditional independence constraints encoded by undirected graphs. Furthermore, we discuss maximum likelihood estimation based on Whittle's approximation to the log-likelihood function and propose an iterative method for solving the resulting likelihood equations. The concepts are illustrated by an example.

Motivation & Objective

• To develop a parametric framework for graphical interaction models in multivariate time series, overcoming limitations of nonparametric spectral methods.
• To generalize covariance selection models to the time series domain by embedding conditional independence constraints in undirected graphs.
• To enable standard model selection strategies by formulating the model in terms of inverse covariances and vector autoregressive representations.
• To provide a computationally feasible estimation procedure using Whittle's approximation to the log-likelihood function.
• To demonstrate the method’s applicability through a real data example illustrating graphical structure identification.

Proposed method

• The model is formulated using inverse covariance matrices to encode conditional independence relationships among time series components.
• Conditional independence constraints are represented by an undirected graph, where edges indicate non-zero partial correlations.
• The joint distribution is modeled as a multivariate normal process with a precision matrix structured according to the graphical model.
• The likelihood is approximated using Whittle's method, which simplifies the computation of the log-likelihood for stationary time series.
• An iterative algorithm is proposed to solve the likelihood equations, enabling maximum likelihood estimation of the model parameters.
• The model is shown to be equivalent to a vector autoregressive process under the specified conditional independence constraints.

Experimental results

Research questions

• RQ1 How can graphical interaction models be extended to multivariate time series in a parametric framework that supports model selection?
• RQ2 What is the relationship between conditional independence structures in time series and vector autoregressive representations?
• RQ3 Can Whittle's approximation be effectively used to estimate graphical models in the time series setting?
• RQ4 How can the precision matrix be structured to reflect graphical dependencies in multivariate time series?
• RQ5 What computational method enables efficient estimation of the model parameters under the proposed parametric formulation?

Key findings

• The proposed parametric model generalizes covariance selection to time series by embedding conditional independence constraints in an undirected graph via the inverse covariance matrix.
• The model is mathematically equivalent to a vector autoregressive process under the specified conditional independence constraints.
• Maximum likelihood estimation is feasible using Whittle's approximation, which enables efficient computation of the log-likelihood.
• An iterative algorithm is developed to solve the likelihood equations, ensuring convergence to the maximum likelihood estimates.
• The method supports standard model selection strategies, which were previously infeasible with nonparametric spectral estimation.
• A real data example demonstrates the method’s ability to recover meaningful graphical structures from multivariate time series.