Graphical LASSO Based Model Selection for Time Series

TL;DR

This paper introduces a nonparametric graphical model selection method for high-dimensional stationary time series, extending graphical LASSO to time series data and providing theoretical guarantees and empirical validation.

Contribution

It generalizes graphical LASSO to time series without assuming parametric models, using spectral smoothness, and offers theoretical performance bounds and practical algorithms.

Findings

Successfully recovers the conditional independence graph from limited samples.

Provides theoretical bounds on failure probability for Gaussian processes.

Demonstrates effectiveness through numerical experiments.

Abstract

We propose a novel graphical model selection (GMS) scheme for high-dimensional stationary time series or discrete time process. The method is based on a natural generalization of the graphical LASSO (gLASSO), introduced originally for GMS based on i.i.d. samples, and estimates the conditional independence graph (CIG) of a time series from a finite length observation. The gLASSO for time series is defined as the solution of an l1-regularized maximum (approximate) likelihood problem. We solve this optimization problem using the alternating direction method of multipliers (ADMM). Our approach is nonparametric as we do not assume a finite dimensional (e.g., an autoregressive) parametric model for the observed process. Instead, we require the process to be sufficiently smooth in the spectral domain. For Gaussian processes, we characterize the performance of our method theoretically by deriving an upper bound on the probability that our algorithm fails to correctly identify the CIG. Numerical experiments demonstrate the ability of our method to recover the correct CIG from a limited amount of samples.