# Learning linear modules in a dynamic network with missing node observations

arXiv: https://arxiv.org/abs/2208.10995 · HTML: https://mlsanity.com/paper/2208.10995/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 23 Aug 2022.

## Builds on (most cited first, 10 of 10)

- [Sequential Monte Carlo Methods for System Identification](https://mlsanity.com/paper/1503.06058/) (2015)
- [Prediction error identification of linear dynamic networks with rank-reduced noise](https://mlsanity.com/paper/1711.06369/) (2017)
- [Learning linear modules in a dynamic network using regularized kernel-based methods](https://mlsanity.com/paper/2005.06266/) (2020)
- [Sparse plus Low rank Network Identification: A Nonparametric Approach](https://mlsanity.com/paper/1510.02961/) (2015)
- [A local direct method for module identification in dynamic networks with correlated noise](https://mlsanity.com/paper/1908.00976/) (2019)
- [Robust EM kernel-based methods for linear system identification](https://mlsanity.com/paper/1411.5915/) (2014)
- [A practical method for the consistent identification of a module in a dynamical network](https://mlsanity.com/paper/1811.01943/) (2018)
- [An empirical Bayes approach to identification of modules in dynamic networks](https://mlsanity.com/paper/1705.01337/) (2017)
- [A frequency domain approach for local module identification in dynamic networks](https://mlsanity.com/paper/2105.10901/) (2021)
- [A scalable multi-step least squares method for network identification with unknown disturbance topology](https://mlsanity.com/paper/2106.07548/) (2021)

## Abstract (arXiv)

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.