Identifiability in Continuous Lyapunov Models

TL;DR

This paper investigates when the parameters of continuous Lyapunov models can be uniquely identified from data, establishing a graph-theoretic criterion based on the absence of directed two-cycles.

Contribution

It provides a necessary and sufficient condition for global identifiability of the drift matrix in terms of graph simplicity, and classifies small graph cases.

Findings

Global identifiability holds if and only if the graph has no directed two-cycles.

A necessary condition for generic identifiability is established.

Small graph classifications up to 5 nodes are provided.

Abstract

The recently introduced graphical continuous Lyapunov models provide a new approach to statistical modeling of correlated multivariate data. The models view each observation as a one-time cross-sectional snapshot of a multivariate dynamic process in equilibrium. The covariance matrix for the data is obtained by solving a continuous Lyapunov equation that is parametrized by the drift matrix of the dynamic process. In this context, different statistical models postulate different sparsity patterns in the drift matrix, and it becomes a crucial problem to clarify whether a given sparsity assumption allows one to uniquely recover the drift matrix parameters from the covariance matrix of the data. We study this identifiability problem by representing sparsity patterns by directed graphs. Our main result proves that the drift matrix is globally identifiable if and only if the graph for the sparsity pattern is simple (i.e., does not contain directed two-cycles). Moreover, we present a necessary condition for generic identifiability and provide a computational classification of small graphs with up to 5 nodes.