(proba-stat) Identifiability of VAR(1) model in a stationary setting

A central topic in ecological research is the reconstruction of dynamical interaction networks among species in an ecosystem with abundance data. Yet a common challenge is the absence of time-series observations, restricting the data to independent samples from a wide range of sampling sites, which should be interpreted as independent and identically distributed observations from the equilibrium of the system. Aiming to incorporate data limitations, this work studies the identifiability of First-order Vector AutoRegressive (VAR(1)) models in a stationary setting where time-series observations are unavailable. Recovering the dynamic interaction graph from static data is non-trivial because the autoregressive interaction graph does not coincide with the graphical model of the steady-state distribution, rendering standard Gaussian Graphical Model tools inapplicable. To resolve this, we adopt an approach from algebraic statistics, using Jacobian matroids of the model’s parametrization map to derive sufficient graphical conditions under which distinct dynamic networks yield distinguishable stationary distributions. This offers a framework for reconstructing dynamical interactions solely from equilibrium data.