  In order to identify a system (module) embedded in a dynamic network, one has
to formulate a multiple-input estimation problem that necessitates certain
nodes to be measured and included as predictor inputs. However, some of these
nodes may not be measurable in many practical cases due to sensor selection and
placement issues. This may result in biased estimates of the target module.
Furthermore, the identification problem associated with the multiple-input
structure may require determining a large number of parameters that are not of
particular interest to the experimenter, with increased computational
complexity in large-sized networks. In this paper, we tackle these problems by
using a data augmentation strategy that allows us to reconstruct the missing
node measurements and increase the accuracy of the estimated target module. To
this end, we develop a system identification method using regularized
kernel-based methods coupled with approximate inference methods. Keeping a
parametric model for the module of interest, we model the other modules as
Gaussian Processes (GP) with a kernel given by the so-called stable spline
kernel. An Empirical Bayes (EB) approach is used to estimate the parameters of
the target module. The related optimization problem is solved using an
Expectation-Maximization (EM) method, where we employ a Markov-chain Monte
Carlo (MCMC) technique to reconstruct the unknown missing node information and
the network dynamics. Numerical simulations on dynamic network examples
illustrate the potentials of the developed method.
