Notions of Separation in Graphs of Dynamical Systems

The concept of d-Separation is a key tool to analyze stochastic models defined by probability distributions of random variables that admit a factorization described by a Directed Acyclic Graph. However, in the area of dynamical systems, and especially control theory, it is common to find network models involving stochastic processes that influence each other according to a directed network where feedback loops may be present as well. These models differ from standard probabilistic models at a fundamental level. Indeed, for a network of dynamical systems it is challenging to introduce an appropriate notion of factorization not only because of the presence of loops, but also because stochastic processes involve an infinite number of random variables. In this article, we show that the concept of d-Separation can still be applied to infer properties of least square estimators defined on subsets of stochastic processes, at least if their mutual influences are described by linear operators. Similar results have been obtained by (Koster, 1999) in the domain of Structural Equation Models for random variables. However, the scenario considered in this article involves stochastic processes and deals with several technical complications, such as noise terms potentially correlated in time and the possibility of causal estimators. The article provides a general framework to overcome all these difficulties that are not present when a graphical model just represents random variables.

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