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Spectral Moment Problems: Generalizations, Implementation and Tuning
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology , 2011. , xii, 10 p.
Series
Trita-MAT. OS, ISSN 1401-2294 ; 11:06
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-39026ISBN: 978-91-7501-087-8 (print)OAI: oai:DiVA.org:kth-39026DiVA: diva2:438977
Public defence
2011-09-16, Sal F2, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20110906Available from: 2011-09-06 Created: 2011-09-06 Last updated: 2011-09-08Bibliographically approved
List of papers
1. Fast, Globally Converging Algorithms for Spectral Moments Problems
Open this publication in new window or tab >>Fast, Globally Converging Algorithms for Spectral Moments Problems
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-39040 (URN)
Note
QC 20110907Available from: 2011-09-08 Created: 2011-09-07 Last updated: 2011-09-08Bibliographically approved
2. Approximative Linear and Logarithmic Interpolation of Spectra
Open this publication in new window or tab >>Approximative Linear and Logarithmic Interpolation of Spectra
2009 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2009. 25 p.
Series
TRITA-MAT. OS, ISSN 1401-2294 ; 09:02
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-39044 (URN)
Note

QC 20110907

Available from: 2011-09-08 Created: 2011-09-07 Last updated: 2014-09-24Bibliographically approved
3. ARMA Identification of Graphical Models
Open this publication in new window or tab >>ARMA Identification of Graphical Models
2013 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 58, no 5, 1167-1178 p.Article in journal (Refereed) Published
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.

Keyword
Autoregressive moving-average (ARMA) modeling, conditional independence, graphical models, system identification
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-39065 (URN)10.1109/TAC.2012.2231551 (DOI)000318542200006 ()2-s2.0-84886418860 (Scopus ID)
Funder
Swedish Research Council
Note

Updated from "Preprint" to "Article" QC 20130627

Available from: 2011-09-08 Created: 2011-09-07 Last updated: 2017-12-08Bibliographically approved
4. Divergence-based spectral approximation with degree constraint as a concave optimization problem
Open this publication in new window or tab >>Divergence-based spectral approximation with degree constraint as a concave optimization problem
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-39066 (URN)
Note
QC 20110907Available from: 2011-09-08 Created: 2011-09-07 Last updated: 2011-09-08Bibliographically approved

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