Nonlinear identifiability of directed acyclic graphs with partial excitation and measurement

TL;DR

This paper investigates the conditions under which directed acyclic graphs with nonlinear edge functions can be uniquely identified from partial excitation and measurement data, providing theoretical criteria for identifiability.

Contribution

It establishes necessary and sufficient conditions for the identifiability of nonlinear DAGs with partial data, including special results for tree structures and generic nonlinear networks.

Findings

Identification requires excitation of sources and measurement of sinks.

A DAG is identifiable with a given pattern if and only if it is identifiable with full measurement.

For trees, any identification pattern guarantees identifiability.

Abstract

We analyze the identifiability of directed acyclic graphs in the case of partial excitation and measurement. We consider an additive model where the nonlinear functions located in the edges depend only on a past input, and we analyze the identifiability problem in the class of pure nonlinear functions satisfying $f(0)=0$. We show that any identification pattern (set of measured nodes and set of excited nodes) requires the excitation of sources, measurement of sinks and the excitation or measurement of the other nodes. Then, we show that a directed acyclic graph (DAG) is identifiable with a given identification pattern if and only if it is identifiable with the measurement of all the nodes. Next, we analyze the case of trees where we prove that any identification pattern guarantees the identifiability of the network. Finally, by introducing the notion of a generic nonlinear network matrix, we provide sufficient conditions for the identifiability of DAGs based on the notion of vertex-disjoint paths.