ARMAX identification of low rank graphical models

TL;DR

This paper develops a maximum likelihood-based method for identifying low rank graphical models from noisy data, addressing the challenge of measurement noise obscuring the low rank structure in large-scale systems.

Contribution

It introduces a novel approach combining maximum entropy covariance extension and ARMAX modeling to accurately identify low rank processes under measurement noise.

Findings

Method reliably estimates parameters in noisy conditions

Algorithm effectively filters noise and recovers low rank structure

Proven identifiability and consistency of the approach

Abstract

In large-scale systems, complex internal relationships are often present. Such interconnected systems can be effectively described by low rank stochastic processes. When identifying a predictive model of low rank processes from sampling data, the rank-deficient property of spectral densities is often obscured by the inevitable measurement noise in practice. However, existing low rank identification approaches often did not take noise into explicit consideration, leading to non-negligible inaccuracies even under weak noise. In this paper, we address the identification issue of low rank processes under measurement noise. We find that the noisy measurement model admits a sparse plus low rank structure in latent-variable graphical models. Specifically, we first decompose the problem into a maximum entropy covariance extension problem, and a low rank graphical estimation problem based on an autoregressive moving-average with exogenous input (ARMAX) model. To identify the ARMAX low rank graphical models, we propose an estimation approach based on maximum likelihood. The identifiability and consistency of this approach are proven under certain conditions. Simulation results confirm the reliable performance of the entire algorithm in both the parameter estimation and noisy data filtering.