Graphoid-based methodologies in modeling, analysis, identification and control of networks of dynamic systems

Abstract

There is extensive literature on the description of joint probability distributions via graphs, where each node represents a random variable and the edges describe a form of coupling among the variables. The connections in these graphical models do not necessarily represent forms of input/output relations among the variables involved. Instead, they typically represent a convenient factorization of their joint probability distribution. For these factorizations to be mathematically meaningful, the underlying graph structure is required to have no directed cycles. Thus, graphical models of random variables typically deal with directed acyclic graphs (DAGs). Conversely, in the area of dynamic systems, and especially control theory, it is common to find network models involving stochastic processes that influence each other according to a directed graph. In this case the graph connections do indeed represent a input/output relations and feedback loops may be present, as well. As a consequence, the network structures underlying graphical models of random variables and networks of dynamic systems differ fundamentally. Furthermore, it is not a straightforward task to unify or reconcile these two semantically different graph descriptions. Indeed, introducing a notion of factorization in networks of dynamic systems would present technical difficulties due not only to the potential presence of loops, but also to the fact that stochastic processes involve an infinite number of random variables. Despite these differences, it has become evident during recent years that methodologies can be borrowed from probabilistic graphical models and used for the analysis, identification and control design in the domain of networks of dynamic systems and viceversa. Some of these methodologies can be imported with no significant modifications, while others need to be substantially revisited. This article is an attempt to bridge the conceptual and methodological gap between graphical models of random variables and networks of dynamic systems, creating a single unified language and theoretical framework for these two different classes. This is achieved by drawing parallels between similar approaches and highlighting their main differences.