Sufficient and Necessary Graphical Conditions for MISO Identification in Networks With Observational Data

Abstract

This article addresses the problem of consistently identifying a single transfer function in a network of dynamic systems using only observational data. It is assumed that the topology is partially known, the forcing inputs are not measured, and that only a subset of the nodes outputs is accessible. The developed technique is applicable to scenarios encompassing confounding variables and feedback loops, which are complicating factors potentially introducing bias in the estimate of the transfer function. The results are based on the prediction of the output node using the input node along with a set of additional auxiliary variables which are selected only from the observed nodes. Similar prediction error methods provide only sufficient conditions for the appropriate choice of auxiliary variables and assume <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> information about the location of strictly causal operators in the network. In this article, such an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge is not required. A most remarkable feature of our approach is that the conditions for the selection of the auxiliary variables are purely graphical. Furthermore, within single-output prediction methods, such conditions are proven to be necessary and sufficient to consistently identify all networks with a given topology. A fundamental consequence of this characterization is to enable the search of a set of auxiliary variables minimizing a suitable cost function for single-output prediction error identification. In the article, we suggest possible approaches to tackle such optimal identification problems.