Sparse plus low rank network identification: A nonparametric approach

Abstract

Modeling and identification of high-dimensional stochastic processes is ubiquitous in many fields. In particular, there is a growing interest in modeling stochastic processes with simple and interpretable structures. In many applications, such as econometrics and biomedical sciences, it seems natural to describe each component of that stochastic process in terms of few factor variables, which are not accessible for observation, and possibly of few other components of the stochastic process. These relations can be encoded in graphical way via a structured dynamic network, referred to as "sparse plus low-rank (S+L) network" hereafter. The problem of finding the S+L network as well as the dynamic model can be posed as a system identification problem. In this paper, we introduce two new nonparametric methods to identify dynamic models for stochastic processes described by a S+L network. These methods take inspiration from regularized estimators based on recently introduced kernels (e.g. "stable spline", "tuned-correlated" etc.). Numerical examples show the benefit to introduce the S+L structure in the identification procedure.

Allocation of Excitation Signals for Generic Identifiability of Dynamic Networks

“…Corollary 1. The minimal number K of external excitation signals that guarantees the generic identifiability of a directed network model set Σ is bounded as (19) where κ( G) is the minimal number of disjoint pseudotrees that cover all the parameterized edges in E p .…”

Identification of Sparse Reciprocal Graphical Models

“…It can be shown that strong duality holds between (5) and (6), so that (5) and (6) are equivalent. In what follows we assume thatΣ > 0 as it is a necessary condition for problem (5) to be feasible. In the case thatΣ is not positive definite, we can consider a positive definite banded block-circulant matrix sufficiently close toΣ which can be obtained by solving a structured covariance estimation problem, see [23], [24].…”

“…Graphical representations provide an immediate visual intuition on the data interdependence. The simplest D. Alpago, M. Zorzi and A. Ferrante are with the Department of Information Engineering, University of Padova, Padova, Italy; email: alpagodani@dei.unipd.it (D. Alpago) zorzimat@dei.unipd.it (M. Zorzi) augusto@dei.unipd.it (A. Ferrante) graphical model is an undirected graph that can be associated with a Gaussian random vector [3], [4], [5]: nodes correspond to the components of the random vector, and there is an edge between two nodes if the corresponding components are conditionally dependent given all the others. Very often data are given as time-series and can thus be modelled as stochastic processes.…”

Robust Identification of “Sparse Plus Low-rank” Graphical Models: An Optimization Approach

“…However, almost all of the interdependence between these variables is explained by the two latent variables x 7 , x 8 so that an illuminant structure emerges when we integrate these two variables with the observed ones. It is worth noting that this idea has been exploited also for Bayesian networks, [10].…”