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.
翻译:为了确定一个嵌入动态网络的系统(模块),人们必须制定多重投入估算问题,从而需要对某些节点进行测量并将其作为预测输入。然而,由于传感器的选择和布置问题,有些节点在许多实际情况下可能无法衡量。这可能导致对目标模块的偏差估计。此外,与多投入结构有关的识别问题可能需要确定大量对实验者来说并不特别感兴趣的参数,而大型网络的计算复杂性则增加。在本文中,我们通过使用数据增强战略来解决这些问题,从而使我们能够重建缺失的节点测量,并提高估计目标模块的准确性。为此,我们开发了一种系统识别方法,使用基于常规内核的内核法以及近似推论的方法。保持一个有关模块的分级模型,我们将其他模块建为高斯进程(GOussian Process),由所谓的稳定螺旋内核内核给出一个内核内核。一个Emprical Bayes (EBEB) 方法,用以重建缺少的节节点测量目标模块的动态网络参数。为此,我们用了一个不甚明的模型来解释模型。