Connectivity Structure of Systems
This work provides a theoretical limitation for causal structure learning in linear systems, which is relevant to researchers in system identification and causal inference.
The paper investigates whether the structure of a linear system (its causal graph) can be identified from input/output data. It proves that without strong additional assumptions, the structure is not determined by input/output behavior, and even assumptions of non-influence are insufficient for parameter identification.
In this paper, we consider to what degree the structure of a linear system is determined by the system's input/output behavior. The structure of a linear system is a directed graph where the vertices represent the variables in the system and an edge (x,y) exists if x directly influences y. In a number of studies, researchers have attempted to identify such structures using input/output data. Thus, our main aim is to consider to what degree the results of such studies are valid. We begin by showing that in many cases, applying a linear transformation to a system will change the system's graph. Furthermore, we show that even the graph's components and their interactions are not determined by input/output behavior. From these results, we conclude that without further assumptions, very few aspects, if any, of a system's structure are determined by its input/output relation. We consider a number of such assumptions. First, we show that for a number of parameterizations, we can characterize when two systems have the same structure. Second, in many applications, we can use domain knowledge to exclude certain interactions. In these cases, we can assume that a certain variable x does not influence another variable y. We show that these assumptions cannot be sufficient to identify a system's parameters using input/output data. We conclude that identifying a system's structure from input/output data may not be possible given only assumptions of the form x does not influence y.