On Conditional Independence Graph Learning From Multi-Attribute Gaussian Dependent Time Series

TL;DR

This paper develops a unified theoretical framework for learning conditional independence graphs from multi-attribute Gaussian time series, using frequency domain penalized likelihood methods, with proven consistency and empirical validation.

Contribution

It introduces a comprehensive analysis of multi-attribute graph learning with both convex and non-convex penalties, avoiding incoherence conditions and providing consistency results.

Findings

Proven high-dimensional consistency of the proposed methods.

Effective graph recovery without incoherence assumptions.

Empirical validation on synthetic and real data.

Abstract

Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. Existing methods for graph estimation for such data are based on single-attribute models where one associates a scalar time series with each node. In multi-attribute graphical models, each node represents a random vector or vector time series. In this paper we provide a unified theoretical analysis of multi-attribute graph learning for dependent time series using a penalized log-likelihood objective function formulated in the frequency domain using the discrete Fourier transform of the time-domain data. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We do not impose any incoherence or irrepresentability condition for our convergence results. We also empirically investigate selection of the tuning parameters based on the Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.