Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Sparse High-Dimensional Graphical Model Learning For Dependent Time Series

Jitendra K. Tugnait

TL;DR

This work provides sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size.

Abstract

We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem based on frequency-domain sufficient statistic for the observed time series is presented. We investigate an alternating direction method of multipliers (ADMM) approach for optimization of the sparse-group lasso penalized log-likelihood. We provide sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size. This results also yields a rate of convergence. We also empirically investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.

On Sparse High-Dimensional Graphical Model Learning For Dependent Time Series

TL;DR

This work provides sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size.

Abstract

We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem based on frequency-domain sufficient statistic for the observed time series is presented. We investigate an alternating direction method of multipliers (ADMM) approach for optimization of the sparse-group lasso penalized log-likelihood. We provide sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size. This results also yields a rate of convergence. We also empirically investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.
Paper Structure (28 sections, 122 equations, 5 figures)

This paper contains 28 sections, 122 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contributions
  4. Relationship to Prior Conference Publications
  5. Outline and Notation
  6. Background and Problem Formulation
  7. Sufficient Statistic and Log-Likelihood
  8. Complex Gaussian Vectors
  9. Wirtinger Calculus
  10. Whittle Likelihood
  11. Problem Formulation
  12. Penalized Log-Likelihood
  13. Optimization via ADMM
  14. ADMM Algorithm
  15. Update (a)
  16. ...and 13 more sections

Figures (5)

  • Figure 1: $F_1$-score for synthetic data example. The label "GMS" refers to the approach of Jung2015a.
  • Figure 2: Average timing per run for synthetic data example. The label "GMS" refers to the approach of Jung2015a.
  • Figure 3: IID modeling-based weighted adjacency matrices. The red squares (in dotted lines) show the communities -- they are not part of the adjacency matrices.
  • Figure 4: Weighted adjacency matrices for dependent time series modeling: $M=4$. The red squares (in dotted lines) show the communities -- they are not part of the adjacency matrices.
  • Figure 5: Weighted adjacency matrices for financial time series, $p=97$, $n=1258$. (a) IID model approach, (b) Proposed approach, $M=4$. The red squares (in dashed lines) show the 11 GISC sectors -- they are not part of the adjacency matrices.