Maximum Likelihood Estimation of Dynamic Sub-Networks with Missing Data

TL;DR

This work presents a maximum likelihood framework for identifying sub-networks within large dynamic networks by exploiting separation conditions that render the sub-network's likelihood dependent only on its own parameters. The approach uses ARMAX-based network modeling and a graph-based partition into target A, separator C, and remainder B, enabling estimation from local data and preserving privacy across organizational boundaries. Theoretical results establish conditions under which the true ML estimator is consistent for the sub-network, while a numerical example demonstrates that the method can match PEM performance with fewer observed signals and can fully identify the sub-network when full identification of the entire network is infeasible. The paper also discusses practical considerations like nonconvex optimization, initialization strategies, and future work to extend consistency to approximate ML and relate to other consistent sub-network identification methods.

Abstract

Maximum likelihood estimation is effective for identifying dynamical systems, but applying it to large networks becomes computationally prohibitive. This paper introduces a maximum likelihood estimation method that enables identification of sub-networks within complex interconnected systems without estimating the entire network. The key insight is that under specific topological conditions, a sub-network's parameters can be estimated using only local measurements: signals within the target sub-network and those in the directly connected to the so-called separator sub-network. This approach significantly reduces computational complexity while enhancing privacy by eliminating the need to share sensitive internal data across organizational boundaries. We establish theoretical conditions for network separability, derive the probability density function for the sub-network, and demonstrate the method's effectiveness through numerical examples.

Figures (1)

• Figure 1: Block diagram for the example network; We use $v^i = H^ie^i$.