Mattia Zorzi

A Maximum Entropy Enhancement for a Family of High-Resolution Spectral Estimators

Structured covariances occurring in spectral analysis, filtering and identification need to be estimated from a finite observation record. The corresponding sample covariance usually fails to possess the required structure. This is the case, for instance, in the Byrnes-Georgiou-Lindquist THREE-like tunable, high-resolution spectral estimators. There, the output covariance Σ of a linear filter is needed to initialize the spectral estimation technique. The sample covariance estimate Σ, however, is usually not compatible with the filter. In this paper, we present a new, systematic way to overcome this difficulty. The new estimate Σο is obtained by solving an ancillary problem with an entropic-type criterion. Extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances.

A New Family of High-Resolution Multivariate Spectral Estimators

In this paper, we extend the Beta divergence family to multivariate power spectral densities. Similarly to the scalar case, we show that it smoothly connects the multivariate Kullback-Leibler divergence with the multivariate Itakura-Saito distance. We successively study a spectrum approximation problem, based on the Beta divergence family, which is related to a multivariate extension of the THREE spectral estimation technique. It is then possible to characterize a family of solutions to the problem. An upper bound on the complexity of these solutions will also be provided. Finally, we will show that the most suitable solution of this family depends on the specific features required from the estimation problem.

Rational approximations of spectral densities based on the Alpha divergence

We approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by selecting the spectral density having minimum “distance” from under the constraint corresponding to imposing the given second-order statistics. We analyze the structure of the optimal solutions as the minimized “distance” varies in the Alpha divergence family. We show that the corresponding approximation problem leads to a family of rational solutions. Secondly, such a family contains the solution which generalizes the Kullback–Leibler solution proposed by Georgiou and Lindquist in 2003. Finally, numerical simulations suggest that this family contains solutions close to the non-rational solution given by the principle of minimum discrimination information.

Multivariate Spectral Estimation Based on the Concept of Optimal Prediction

In this technical note, we deal with a spectrum approximation problem arising in THREE-like multivariate spectral estimation approaches. The solution to the problem minimizes a suitable divergence index with respect to an a priori spectral density. We derive a new divergence family between multivariate spectral densities which takes root in the prediction theory. Under mild assumptions on the a priori spectral density, the approximation problem, based on this new divergence family, admits a family of solutions. Moreover, an upper bound on the complexity degree of these solutions is provided.

On the estimation of structured covariance matrices

This paper discusses a method for estimating the covariance matrix of a multivariate stationary process w generated as the output of a given linear filter fed by a stationary process y. The estimated covariance matrix must satisfy two constraints: it must be positive semi-definite and it must be consistent with the fact that w is the output of the given linear filter. It turns out that these constraints force the estimated covariance to lie in the intersection of a cone with a linear space. While imposing only the first of the two constraints is rather straightforward, guaranteeing that both are satisfied is a non-trivial issue to which quite a bit of attention has already been devoted in the literature. Our approach extends the method for estimating the Toeplitz covariance matrix of order M of a process y based on the biased spectral estimator (Stoica & Moses, 1997). This extension is based on the characterization of the output covariance matrix in terms of the filter parameters and the sequence of covariance lags of the input process. After introducing our estimation method, we propose a comparison performance between this one and other methods proposed in the literature. Simulation results show that our approach constitutes a valid estimation procedure.

AR Identification of Latent-Variable Graphical Models

The paper proposes an identification procedure for autoregressive Gaussian stationary stochastic processes under the assumption that the manifest (or observed) variables are nearly independent when conditioned on a limited number of latent (or hidden) variables. The method exploits the sparse plus low-rank decomposition of the inverse of the manifest spectral density and the efficient convex relaxations recently proposed for such decompositions.

Robust Kalman Filtering Under Model Perturbations

We consider a family of divergence-based minimax approaches to perform robust filtering. The mismodeling budget, or tolerance, is specified at each time increment of the model. More precisely, all possible model increments belong to a ball which is formed by placing a bound on the Tau-divergence family between the actual and the nominal model increment. Then, the robust filter is obtained by minimizing the mean square error according to the least favorable model in that ball. It turns out that the solution is a family of Kalman like filters. Their gain matrix is updated according to a risk sensitive like iteration where the risk sensitivity parameter is now time varying. As a consequence, we also extend the risk sensitive filter to a family of risk sensitive like filters according to the Tau-divergence family.

An interpretation of the dual problem of the THREE-like approaches

Spectral estimation can be performed using the so called THREE-like approach. Such method leads to a convex optimization problem whose solution is characterized through its dual problem. In this paper, we show that the dual problem can be seen as a new parametric spectral estimation problem. This interpretation implies that the THREE-like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities describing ARMA models, enriching in this way its meaningfulness.

A contraction analysis of the convergence of risk-sensitive filters

A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the N-fold composition of the Riccati map is strictly contractive with respect to the Riemannian metric of positive definite matrices, when N is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.

A Bayesian approach to sparse plus low rank network identification

We consider the problem of modeling multivariate stochastic processes with parsimonious dynamical models which can be represented with a sparse dynamic network with few latent nodes. This structure translates into a sparse plus low rank model. In this paper, we propose a Bayesian approach to identify such models.