Wolfgang Scherrer

Consistency and relative efficiency of subspace methods

We give a consistency proof for two subspace methods. We then show the asymptotic equivalence of a special subspace method and the initial estimate proposed by Hannan and Rissanen. Finally, a simulation study comparing two subspace methods and the maximum-likelihood method is performed.

Statistical analysis of novel subspace identification methods

Abstract In this paper four subspace algorithms which are based on an initial estimate of the state are considered. Three novel algorithms are introduced and compared with an algorithm which is essentially equal to the N4SID algorithm by Van Overschee and De Moor. For the algorithms considered a consistency result is proved. In a simulation study the relative (statistical) efficiency of these algorithms in relation to the maximum likelihood algorithm is investigated.

Consistency and asymptotic normality of some subspace algorithms for systems without observed inputs

Asymptotic normality for a class of subspace algorithms, which estimate the state in a first step, is derived. Expressions for the asymptotic variance are given. Linear systems with unobserved white noise inputs are considered. A class of subspace estimates for the system matrices obtained by estimating the state in the first step is analyzed. The main result presented here states asymptotic normality of subspace estimates. In addition, a consistency result for the system matrix estimates is given. An algorithm to compute the asymptotic variances of the estimates is presented. In a final section the implications of the result are discussed.

A Structure Theory for Linear Dynamic Errors-in-Variables Models

We deal with problems connected with the identification of linear dynamic systems in situations when inputs and outputs may be contaminated by noise. The case of uncorrelated noise components and the bounded noise case is considered. If also the inputs may be contaminated by noise, a number of additional complications in identification arise, in particular the underlying system is not uniquely determined from the population second moments of the observations. A description of classes of observationally equivalent systems is given, continuity properties of mappings relating classes of observationally equivalent systems to the spectral densities of the observations are derived and the classes of spectral densities corresponding to a given maximum number of outputs are studied.

Consistency of system identification by global total least squares

Global total least squares (GTLS) is a method for the identification of linear systems where no distinction between input and output variables is required. This method has been developed within the deterministic behavioural approach to systems. In this paper we analyse statistical properties of this method when the observations are generated by a multivariable stationary stochastic process. In particular, sufficient conditions for the consistency of GTLS are derived. This means that, when the number of observations tends to infinity, the identified deterministic system converges to the system that provides an optimal appoximation of the data generating process. The two main results are the following. GTLS is consistent if a guaranteed stability bound can be given a priori. If this information is not available, then consistency is obtained if GTLS is applied to the observed data extended with zero values in past and future.

On the Impact of Weighting Matrices in Subspace Algorithms

Abstract In this paper some results concerning the asymptotic distribution of the estimates of the transfer function obtained by using subspace methods are given. The discussion centers on the estimation of the transfer function rather than on the estimation of the system matrices, since the transfer function is a system invariant. Therefore the accuracy of the transfer function estimate can be directly used to evaluate the relative efficiency of different methods. The present paper uses the techniques and the asymptotic expressions derived in (Bauer et al., 1999). In particular the effects of certain weighting matrices used in the algorithms are investigated.

The Analysis of the Asymptotic Variance of Subspace Algorithms

Abstract This paper provides a central limit theorem for certain subspace estimators of the parameters of linear dynamic systems. The asymptotic variance of the estimators is calculated in two examples in order to assess their relative efficiency in comparison with maximum likelihood estimates. In addition in one example a simulation study investigating the bias due to truncation is presented.

Identification of linear systems from noisy data

Linear dynamic errors-in-variables models with mutually uncorrelated noise components are considered. A main complication in identification is that the systems are not uniquely determined from the (ensemble) second moments of the observations. The authors analyze certain properties of the set of all observationally equivalent systems. In addition, they describe the sets of spectral densities corresponding to a given Frisch corank. The results obtained are of importance for developing and analyzing identification algorithms.<<ETX>>

On The Parametrization Of Multivariate Garch Models

This paper deals with issues of structure and parametrization of VECH models proposed in Bollerslev, Engle, and Wooldridge (1988) and Baba, Engle, Kraft, and Kroner (BEKK) models. Both general models and also restricted versions such as the widely used diagonal VECH (DVECH) and factor generalized autoregressive conditional heteroskedastic (F-GARCH) models are discussed. A simple algorithm is presented that checks whether a given VECH model may be cast as a BEKK model. It is shown that in the bivariate case BEKK models are as general as VECH models. In higher dimensional cases however, VECH models allow for more flexibility. In addition, a parametrization for a generic, i.e., open and dense, class of BEKK models is given, and the frequently cited parametrization by Engle and Kroner (1995, Econometric Theory 11, 122–150) is analyzed. Two shortcomings of the latter are pointed out. Finally, parametrizations for BEKK(p,q,K) models with K ≤ n, including DVECH, F-GARCH, and a generalization of the latter, are discussed.This research was supported by the project P17065 “Identification of multivariate dynamic systems with a focus on dimension reduction†of the Austrian Science Foundation (FWF).

Time series models for short term forecasting of ozone in the eastern part of Austria

Ozone and several meteorological variables are monitored routinely at several monitoring sites in Austria. Enhanced ozone formation is mainly observed in the eastern part of the country, where peaks occur in the lee of the conurbation area of Vienna. The time series structure of ozone data within this area is analyzed by using autoregressive type models with an exogenous part (ARX) . In particular, separate ARX models for each time of the day, threshold ARX models for defined ozone regimes, as well as periodic ARX models are applied. The analysis of the estimated coefficients gives some insight into the dynamics of ozone and its explanation by meteorological variables. Copyright