Identification of Non-causal Graphical Models

TL;DR

The problem to estimate non-causal graphical models whose edges encode smoothing relations among the variables is considered and a new covariance extension problem is proposed and the solution minimizing the transportation distance with respect to white noise process is a double-sided autoregressive non-causal graphical model.

Abstract

The paper considers the problem to estimate non-causal graphical models whose edges encode smoothing relations among the variables. We propose a new covariance extension problem and show that the solution minimizing the transportation distance with respect to white noise process is a double-sided autoregressive non-causal graphical model. Then, we generalize the paradigm to a class of graphical autoregressive moving-average models. Finally, we test the performance of the proposed method through some numerical experiments.

Figures (3)

• Figure 1: Graph corresponding to Model (\ref{['non_causal_AR']}) with $H(z)$ defined in (\ref{['defH']}). Every node represents a component of process $y$. We have: two edges between $y_1$ and $y_3$ because $[H(z)]_{13}\neq 0$; two edges between $y_2$ and $y_3$ because $[H(z)]_{23}\neq 0$.
• Figure 2: Monte Carlo experiments in the AR case using different data lengths ($N=500$, $N=1000$ and $N=2000$).
• Figure 3: Monte Carlo experiments in the ARMA case using different data lengths ($N=500$, $N=1000$ and $N=2000$).

Theorems & Definitions (1)

• Definition 1