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The problem of obtaining dynamical models directly from an observed time-series occurs in many fields of application. There are a number of possible approaches to this problem. In this volume a number of such points of view are exposed: the statistical time series approach, a theory of guaranted performance, and finally a deterministic approximation approach. This volume is an out-growth of a number of get-togethers sponsered by the Systems and Decision Sciences group of the International Institute of Applied Systems Analysis (IIASA) in Laxenburg, Austria. The hospitality and support of this organization is gratefully acknowledged. Jan Willems Groningen, the Netherlands May 1989 TABLE OF CONTENTS Linear System Identification- A Survey page 1 M. Deistler A Tutorial on Hankel-Norm Approximation 26 K. Glover A Deterministic Approach to Approximate Modelling 49 C. Heij and J. C. Willems Identification - a Theory of Guaranteed Estimates 135 A. B. Kurzhanski Statistical Aspects of Model Selection 215 R. Shibata Index 241 Addresses of Authors 246 LINEAR SYSTEM IDENTIFICATION· A SURVEY M. DEISTLER Abstract In this paper we give an introductory survey on the theory of identification of (in general MIMO) linear systems from (discrete) time series data. The main parts are: Structure theory for linear systems, asymptotic properties of maximum likelihood type estimators, estimation of the dynamic specification by methods based on information criteria and finally, extensions and alternative approaches such as identification of unstable systems and errors-in-variables. Keywords Linear systems, parametrization, maximum likelihood estimation, information criteria, errors-in-variables.

Inhaltsverzeichnis

Frontmatter

Linear System Identification — A Survey

Abstract
In this paper we give an introductory survey on the theory of identification of (in general MIMO) linear systems from (discrete) time series data. The main parts are: Structure theory for linear systems, asymptotic properties of maximum likelihood type estimators, estimation of the dynamic specification by methods based on information criteria and finally, extensions and alternative approaches such as identification of unstable systems and errors-in-variables.
M. Deistler

A Tutorial on Hankel-Norm Approximation

Abstract
A self-contained derivation is presented of the characterization of all optimal Hankel-norm approximations to a given matrix-valued transfer function. The approach involves a state-space characterization of all-pass systems as in the author’s previous work, but has been greatly simplified. A section of preliminary results is included giving general results on linear fractional transformations, Hankel operators and all-pass systems. These results then can be applied to give the characterization of all optimal Hankel-norm approximations of a given stable transfer function. Frequency response bounds for these approximations are then derived from finite rank perturbation results.
Keith Glover

A Deterministic Approach to Approximate Modelling

Abstract
In this paper we will describe a deterministic approach to time series analysis. The central problem consists of approximate modelling of an observed time series by means of a deterministic dynamical system. The quality of a model with respect to data will depend on the purpose of modelling. We will consider the purpose of description and that of prediction. We define the quality by means of complexity and misfit measures, expressed in terms of canonical parametrizations of dynamical systems. We give algorithms to determine optimal models for a given time series and investigate some consistency properties. Finally we present some simulations of these modelling procedures.
C. Heij, J. C. Willems

Identification — A Theory of Guaranteed Estimates

Abstract
This paper gives an introduction to the theory of parameter identification and state estimation for systems subjevcted to uncertainties with set-membership bounds on the unknowns.
The situation under discussion may often turn to be more a propos since here the system and the environment are modelled as truly uncertain rather than noisy. The described approach is purely deterministic.
The techniques given here are mainly aimed at problems with nonquadratic constraints with the quadratic case acting as a necessary complimentary tool. Some substantial properties of nonlinear estimation schemes are also indicated.
On the other hand the techniques involved her for the treatment of systems with nonquadratic constraints on the unknowns are proved to have some nontrivial interretalions with those developed in stochastic estimation theory. This may lead to some further estimation schemes that would combine the deterministic and the stochastic models of uncertainty.
The recurrence procedures of this paper are devised into relations that would allow numerical simulations.
A. B. Kurzhanski

Statistical Aspects of Model Selection

Abstract
Various aspects of statistical model selection are discussed from the view point of a statistician. Our concern here is about selection procedures based on the Kullback Leibler information number. Derivation of AIC (Akaike’s Information Criterion) is given. As a result a natural extension of AIC, called TIC (Takeuchi’s Information Criterion) follows. It is shown that the TIC is asymptotically equivalent to Cross Validation in a general context, although AIC is asymptotically equivalent only for the case of independent identically distributed observations. Next, the maximum penalized likelihood estimate is considered in place of the maximum likelihood estimate as an estimation of parameters after a model is selected. Then the weight of penalty is also the one to be selected. We will show that, starting from the same Kullback-Leibler information number, a useful criterion RIC (Regularization Information Criterion) is derived to select both the model and the weight of penalty. This criterion is in fact an extension of TIC as well as of AIC. Comparison of various criteria, including consistency and efficiency is summarized in Section 5. Applications of such criteria to time series models are given in the last section.
Ritei Shibata

Backmatter

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