Thomas Ribarits

Data driven local coordinates for multivariable linear systems and their application to system identification

In this paper we introduce a new parametrization for state-space systems: data driven local coordinates (DDLC). The parametrization is obtained by restricting the full state-space parametrization, where all matrix entries are considered to be free, to an affine plane containing a given nominal state-space realization. This affine plane is chosen to be perpendicular to the tangent space to the manifold of observationally equivalent state-space systems at the nominal realization. The application of the parametrization to prediction error identification is exemplified. Simulations indicate that the proposed parametrization has numerical advantages as compared to e.g. the more commonly used observable canonical form.

An analysis of the parametrization by data driven local coordinates for multivariable linear systems

In this paper, we study a novel parametrization for state-space systems, namely data driven local coordinates (DDLC) which have recently been introduced and applied. Even though DDLC has meanwhile become the default parametrization used in the system identification toolbox of the software package MATLAB, an analysis of properties of DDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the DDLC construction and show a number of results which are important for actual identification such as maximum likelihood-type estimation.

An analysis of separable least squares data driven local coordinates for maximum likelihood estimation of linear systems

In this paper we study a novel parametrization for state-space systems, namely separable least squares data driven local coordinates (slsDDLC). The parametrization by slsDDLC has recently been successfully applied to maximum likelihood estimation of linear dynamic systems. In a simulation study, the use of slsDDLC has led to numerical advantages in comparison to the use of more conventional parametrizations, including data driven local coordinates (DDLC). However, an analysis of properties of slsDDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the slsDDLC construction and show a number of results which are important for actual identification, in particular for maximum likelihood estimation. We also prove that the separable least squares methodology is indeed guaranteed to be applicable to maximum likelihood estimation of linear dynamic systems in typical situations.

Parametrizations of linear systems by data driven local coordinates

Some properties of data driven local coordinates (DDLC) as introduced in Wolodkin et al. (1997) and McKelvey and Helmersson (1999) are derived. First we briefly introduce the parametrization by DDLC and echelon and Ober balanced canonical forms. Then the special case s = m = n = 1 is discussed in detail: geometrical, topological and numerical properties are examined. Finally, new geometrical and topological results for DDLC in the case of general McMillan degree n and arbitrary input and output dimensions are given.

DATA DRIVEN LOCAL COORDINATES: SOME NEW TOPOLOGICAL AND GEOMETRICAL RESULTS

Abstract Certain topological and geometrical properties of data driven local coordinates (DDLC) for state-space systems as introduced in (Wolodkin et al. , 1997) and (McKelvey and Helmersson, 1999) are derived. First the special case of SISO systems with McMillan degree n = 1 is discussed in order to provide some insights into the geometry of the DDLC construction. Then for the MIMO case with general n it is shown that the set of transfer functions corresponding to the parameter space contains a nonvoid open subset of the manifold of transfer functions of order n and that the estimation problem is locally well posed. Moreover, it is stated that the parameter space always contains points corresponding to non minimal systems and a result on the number of disconnected components of the equivalence classes in the space ℝ n2 +n ( m+s ) (obtained by an embedding of the system matrices ( A,B,C )) concludes this contribution.

The State-Space Error Correction Model: Simulations and Applications

In this paper we present a simulation study as well as two applications of a newly proposed approach towards I(1) cointegration using so-called state-space error correction models (SSECMs). SSECMs have been introduced in Ribarits and Hanzon (2014) where questions of model selection, parametrization and maximum likelihood estimation are treated in detail. The present paper shows the potential of this new approach to I (1) cointegration analysis.

A Novel Approach to Parametrization and Parameter Estimation in Linear Dynamic Systems

We describe a novel approach, called data driven local coordinates (DDLC), for parametrizing linear systems in state space form, and we analyze some of its properties which are relevant for e.g. maximum likelihood estimation. In addition we describe how this idea can be used for a concentrated likelihood function, obtained by a least squares type concentration step, which gives the so called sls (separable least squares) DDLC approach. Both approaches give favourable results in numerically optimizing the likelihood function in simulation studies.

Separable Least Squares Data Driven Local Coordinates

Abstract In this paper, the parametrization of state-space systems by data driven local coordinates as introduced by (McKelvey et al., 2003) is modified. This modification leads to an alternative analogous parametrization which can be used for a suitable concentrated likelihood-type criterion function, where the concentration step can be done by a generalized least squares step. An obvious consequence is the reduced number of parameters resulting in less computational burden, but, of course, the criterion function itself is changed by the concentration step. The resulting new parametrization is called slsDDLC, and its topological and geometrical properties are investigated in detail.

A Finite-Dimensional Hjm Model: How Important Is Arbitrage-Free Evolution?

We consider a two-factor Heath–Jarrow–Morton (HJM) model under the risk neutral measure and show that it may be decoupled into a particular dynamic Nelson–Siegel (NS) model plus a somewhat counter-intuitive adjustment (lying outside the NS family) which keeps it arbitrage-free. We assess the importance of the adjustment for arbitrage-free pricing by comparing the HJM model with a novel NS model which is selected using projection techniques. We analyze forward curves and derivative prices generated by the HJM and projected NS model and consider two real-world case studies. Our analysis shows that the influence of the adjustment term on arbitrage-free evolution is small.

ON A SIMPLE OVERLAPPING STATE-SPACE PARAMETRIZATION FOR LINEAR TIME SERIES MODELS

Abstract We consider a new state-space parametrization for linear time series models: data driven coordinates (DDC), which provides an atlas for the manifold of (stable) p x m transfer functions of fixed McMillan degree n. Hence, DDC has similar desirable properties as more traditional overlapping parametrizations and better than classical canonical forms. Moreover, the choice of charts can be done in a data-driven manner in a very simple way. Althugh not yet as good numerically as the parametrization by data driven local coordinates (DDLC), this parametrization has the advantage of not being local. The application of DDC to maximum likelihood identification is exemplified.