The Kullback-Leibler pseudo-distance, or divergence, can be used as a criterion for spectral approximation. Unfortunately this criterion is not convex over the most general classes of rational spectra. In this work it will be shown that divergence minimization is equivalent to a costrained entropy minimization problem, whose concave structure can be exploited in order to guarantee global convergence in the most general case.

In this paper, we consider the matricial version of generalized moment problem with degree constraint. Specifically we focus on computing the solution that minimize the Kullback-Leibler criterion. Several strategies to find such optimum via descent methods are considered and their convergence studied. In particular a parameterization with better numerical properties is derived from a spectral factorization problem. Such parameterization, in addition to guaranteeing descent methods to be globally convergent, it appears to be very reliable in practice.

Spectral moment interpolation find application in a wide array of use cases: robust control, system identification, model reduction to name the most notable ones. This thesis aims to expand the theory of such methods in three different directions. The first main contribution concerns the practical applicability. From this point of view various solving algorithm and their properties are considered. This study lead to identify a globally convergent method with excellent numerical properties. The second main contribution is the introduction of an extended interpolation problem that allows to model ARMA spectra without any explicit information of zero’s positions. To this end it was necessary for practical reasons to consider an approximated interpolation insted. Finally, the third main contribution is the application to some problems such as graphical model identification and ARMA spectral approximation.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.

Lindquist, Anders

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.

Wahlberg, Bo

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.

ARMA Identification of Graphical Models2013In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 58, no 5, p. 1167-1178Article in journal (Refereed)

Abstract [en]

Consider a Gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components. Such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence. This leads to a sparsity pattern in the inverse of the matrix-valued spectral density. Such graphical models find applications in speech, bioinformatics, image processing, econometrics and many other fields, where the problem to fit an autoregressive (AR) model to such a process has been considered. In this paper we take this problem one step further, namely to fit an autoregressive moving-average (ARMA) model to the same data. We develop a theoretical framework and an optimization procedure which also spreads further light on previous approaches and results. This procedure is then applied to the identification problem of estimating the ARMA parameters as well as the topology of the graph from statistical data.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.

Lindquist, Anders

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.

Wahlberg, Bo

KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.

Consider a Gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components. Such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence. This leads to a sparsity pattern in the inverse of the matrix-valued spectral density. Such graphical models find applications in speech, bioinformatics, image processing, econometrics and many other fields, where the problem to fit an autoregressive (AR) model to such a process has been considered. In this paper we take this problem one step further, namely to fit an autoregressive moving-average (ARMA) model to the same data. We develop a theoretical framework which also spreads further light on previous approaches and results.

Given output data of a stationary stochastic process estimates of the covariances parameters can be obtained. These estimates can be used to determine ARMA models to approximatly fit the data by matching the covariances exactly. However, the estimates of the covariances may contain large errors, especially if they are determined from short data sequences, and thus it makes sense to match the covariances only in an approximative way. Here we consider a convex method for solving an approximative covariance interpolation problem while maximizing the entropy and penalize the quadratic deviation from the nominal covariances.

7.

Enqvist, Per

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.

Avventi, Enrico

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.

Given output data of a stationary stochastic process estimates of covariance and cepstrum parameters can be obtained. These estimates can be used to determine ARMA models to approximately fit the data by matching the parameters exactly. However, the estimates of the parameters may contain large errors, especially if they are determined from short data sequences, and thus it makes sense to match the parameters in an approximate way. Here we consider a convex method for solving an approximate linear and logarithmic spectrum interpolation problem while maximizing the entropy and penalize the quadratic deviation from the nominal parameters.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.

Avventi, Enrico

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.

Wahlberg, Bo

KTH, School of Electrical Engineering (EES), Automatic Control.