Signal Selection for Estimation and Identification in Networks of Dynamic Systems: A Graphical Model Approach

Abstract

Network systems have become a ubiquitous modeling tool in many areas of science where nodes in a graph represent distributed processes and edges between nodes represent a form of dynamic coupling. When a network topology is already known (or partially known), two associated goals are (i) to derive estimators for nodes of the network which cannot be directly observed or are impractical to measure; and (ii) to quantitatively identify the dynamic relations between nodes. In this article we address both problems in the challenging scenario where only some outputs of the network are being measured and the inputs are not accessible. The approach makes use of the notion of d-separation for the graph associated with the network. In the considered class of networks, it is shown that the proposed technique can determine or guide the choice of optimal sparse estimators. The article also derives identification methods that are applicable to cases where loops are present providing a different perspective on the problem of closed-loop identification. The notion of d-separation is a central concept in the area of probabilistic graphical models, thus an additional contribution is to create connections between control theory and machine learning techniques.

A brief explanation of the issue of faithfulness and link orientation in network reconstruction

“…In this work we develop our results for a class of dynamic stochastic networks called Linear Dynamic Influence Models (LDIMs), which have been extensively studied [17], [19], [20]. These networks can be considered as a special case of Dynamic Structure Function [9] with unknown (not measured) forcing signals that are modeled as mutually independent stochastic processes.…”

“…For formal statements and detailed derivations of the non-causal/causal Wiener filter, refer to [8]. These concepts lead to the notions of orthogonality and "Wiener separation" [20].…”

“…Definition 9 (Wiener separation [20]): Let (H(z), e) be a LDIM with output processes (y 1 , ., y n ). We say that the process y j is Wiener separated from y i given a set of processes S ⊆ y if the non-causal (causal) Wiener filter estimating y j from y i ∪ S has a zero entry associated with y i , i.e.…”

Local Module Identification in Dynamic Networks Using Regularized Kernel-Based Methods

“…We note that from (26) that we get closed-form solutions for all λ k , k P tN j Yjuztiu, while the β k , k P tN j Yjuztiu, can be updated by solving scalar optimization problems in the domain r0, 1q, as detailed in (25). Therefore, the hyperparameters update turns out to be a computationally fast operation.…”

“…Since we are not paramterizing any modules, we do not have θ in the parameter vector η. The solutions for the β's and λ's at each iteration are given by (25) and (26) respectively. The solution toσ 2 j at each iteration is given by,…”