Local module identification in dynamic networks with correlated noise: the full input case

The identification of local modules in dynamic networks with known topology has recently been addressed by formulating conditions for arriving at consistent estimates of the module dynamics, typically under the assumption of having disturbances that are uncorrelated over the different nodes. The conditions typically reflect the selection of a set of node signals that are taken as predictor inputs in a MISO identification setup. In this paper an extension is made to arrive at an identification setup for the situation that process noises on the different node signals can be correlated with each other. In this situation the local module may need to be embedded in a MIMO identification setup for arriving at a consistent estimate with maximum likelihood properties. This requires the proper treatment of confounding variables. The result is an algorithm that, based on the given network topology and disturbance correlation structure, selects an appropriate set of node signals as predictor inputs and outputs in a MISO or MIMO identification setup. As a first step in the analysis, we restrict attention to the (slightly conservative) situation where the selected output node signals are predicted based on all of their in-neighbor node signals in the network.Comment: Extended version of paper submitted to the 58th IEEE Conf. Decision and Control, Nice, 201

Learning linear modules in a dynamic network using regularized kernel-based methods

In order to identify one system (module) in an interconnected dynamic network, one typically has to solve a Multi-Input-Single-Output (MISO) identification problem that requires identification of all modules in the MISO setup. For application of a parametric identification method this would require estimating a large number of parameters, as well as an appropriate model order selection step for a possibly large scale MISO problem, thereby increasing the computational complexity of the identification algorithm to levels that are beyond feasibility. An alternative identification approach is presented employing regularized kernel-based methods. Keeping a parametric model for the module of interest, we model the impulse response of the remaining modules in the MISO structure as zero mean Gaussian processes (GP) with a covariance matrix (kernel) given by the first-order stable spline kernel, accounting for the noise model affecting the output of the target module and also for possible instability of systems in the MISO setup. Using an Empirical Bayes (EB) approach the target module parameters are estimated through an Expectation-Maximization (EM) algorithm with a substantially reduced computational complexity, while avoiding extensive model structure selection. Numerical simulations illustrate the potentials of the introduced method in comparison with the state-of-the-art techniques for local module identification.Comment: 15 pages, 7 figures, Submitted for publication in Automatica, 12 May 2020. Final version of paper submitted on 06 January 2021 (To appear in Automatica

A scalable multi-step least squares method for network identification with unknown disturbance topology

Identification methods for dynamic networks typically require prior knowledge of the network and disturbance topology, and often rely on solving poorly scalable non-convex optimization problems. While methods for estimating network topology are available in the literature, less attention has been paid to estimating the disturbance topology, i.e., the (spatial) noise correlation structure and the noise rank in a filtered white noise representation of the disturbance signal. In this work we present an identification method for dynamic networks, in which an estimation of the disturbance topology precedes the identification of the full dynamic network with known network topology. To this end we extend the multi-step Sequential Linear Regression and Weighted Null Space Fitting methods to deal with reduced rank noise, and use these methods to estimate the disturbance topology and the network dynamics in the full measurement situation. As a result, we provide a multi-step least squares algorithm with parallel computation capabilities and that rely only on explicit analytical solutions, thereby avoiding the usual non-convex optimizations involved. Consequently we consistently estimate dynamic networks of Box Jenkins model structure, while keeping the computational burden low. We provide a consistency proof that includes path-based data informativity conditions for allocation of excitation signals in the experimental design. Numerical simulations performed on a dynamic network with reduced rank noise clearly illustrate the potential of this method.Comment: 17 pages, 4 figures, resubmitted to Automatica on 23 November 2021, provisionally accepte

Path-based data-informativity conditions for single module identification in dynamic networks

For consistent or minimum variance estimation of a single module in a dynamic network, a predictor model has to be chosen with selected inputs and outputs, composed of a selection of measured node signals and possibly external excitation signals. The predictor model has to be chosen in such a way that consistent estimation of the target module is possible, under the condition that we have data-informativity for the considered predictor model set. Consistent and minimum variance estimation of target modules is typically obtained if we follow a direct method of identification and predictor model selection, characterized by the property that measured node signals are the prime predictor input signals. In this paper the concept of data-informativity for network models will be formalized, and for the direct method the required data-informativity conditions will be specified in terms of path-based conditions on the graph of the network model, guaranteeing data-informativity in a generic sense, i.e. independent on numerical values of the network transfer functions concerned

Local module identification in dynamic networks using regularized kernel-based methods

In order to identify a specific system (module) of interest embedded in a dynamic network, one typically has to formulate a multi-input single-output (MISO) identification problem which requires to identify all modules in the MISO structure, and determine their model order. While the former task poses the problem of estimating a large number of parameters that are of no interest to the experimenter, the latter task may result computationally challenging in large-size networks. To avoid these issues and increase the accuracy of the identified module of interest, we use regularized kernel-based methods. Keeping a parametric model for the module of interest, we model the impulse response of the remaining modules in the MISO structure as zero mean Gaussian vectors with covariance matrix (kernel) given by the first-order stable spline kernel, accounting also for the noise model affecting the output of the target model. Using an Empirical Bayes (EB) approach, the target-module parameters are estimated by maximizing the marginal likelihood of the module output. The related optimization problem is solved using the Expectation-Maximization (EM) algorithm. Numerical experiments illustrate the potentials of the introduced method in comparison with the state-of-the-art techniques for local identification

A regularized kernel-based method for learning a module in a dynamic network with correlated noise

In this paper, we consider the problem of identifying one system (module) embedded in a dynamic network that is disturbed by colored process noise sources, which can possibly be correlated. To achieve this using the direct method for single module identification, we need to formulate a Multi-Input-Multi-Output (MIMO) estimation problem which requires model order selection step for each module in the setup and estimation of large number of parameters. This results in a larger variance in the estimates and an increase in computation complexity. Therefore, we extend the Empirical Bayes Direct Method [1], which handles the above mentioned problems for a Multi-Input-Single-Output (MISO) setup to a MIMO setting by suitably modifying the framework. We keep a parametric model for the desired target module and model the impulse response of all the other modules as independent zero mean Gaussian process governed by a first-order stable spline kernel. The parameters of the target module are obtained by maximizing the marginal likelihood of the output using the Empirical Bayes (EB) approach. To solve this, we use the Expectation Maximization (EM) algorithm which offers computational advantages. Numerical simulation illustrate the advantages of the developed method over existing classical methods

Learning local modules in dynamic networks without prior topology information

Recently different identification methods have been developed for identifying a single module in a dynamic network. In order to select an appropriate predictor model one typically needs prior knowledge on the topology (interconnection structure) of the dynamic network, as well as on the correlation structure of the process disturbances. In this paper we present a new approach that incorporates the estimation of this prior information into the identification, leading to a fully data-driven approach for estimating the dynamics of a local module. The developed algorithm uses non-causal Wiener filters and a series of convex optimizations with parallel computation capabilities to estimate the topology, which subsequently is used to build the appropriate input/output setting for a predictor model in the local direct method under correlated process noise. A regularized kernel-based method is then employed to estimate the dynamic of the target module. This leads to an identification algorithm with attractive statistical properties that is scalable to handle larger-scale networks too. Numerical simulations illustrate the potential of the developed algorithm

Generalized sensing and actuation schemes for local module identification in dynamic networks

For the problem of identifying a target module that is embedded in a dynamic network with known interconnection structure, different sets of conditions are available for the set of node signals to be measured and the set of excitation signals to be applied at particular node locations. In previous work these conditions have typically been derived from either an indirect identification approach, considering external excitation signals as inputs, or from a direct identification approach, considering measured node signals as inputs. While both approaches lead to different sets of (sufficient) conditions, in this paper we extend the flexibility in the sufficient conditions for selection of excitation and measured node signals, by combining both direct and indirect approaches. As a result we will show the benefits of using both external excitation signals and node signals as predictor inputs. The provided conditions allow us to design sensor selection and actuation schemes with considerable freedom for consistent identification of a target module