Fredrik Tjärnström

Using the bootstrap to estimate the variance in the case of undermodeling

This paper deals with the problem of estimating the variance of an undermodeled model. Undermodeling means that the model class used is not flexible enough to describe the underlying system. The proposed solution to the problem is an algorithm that is based on the bootstrap. A simulation example shows that the variance estimates based on the proposed algorithm are in very good agreement with Monte Carlo simulations.

Variance Properties of a Two-step ARX Estimation Procedure

In this contribution we discuss some variance properties of a two-step ARX estimation scheme. An expression for the co-variance of the final low order model is calculated and it is discussed how one should minimize this covariance. The implication of the results is that identification of the dynamics of a system could very easily be performed with standard linear least squares (two times), even if the measurement noise is heavily colored. We also show a numerical example, where this two-step estimation scheme gives a variance which is close (but not equal) to the the Cramèr-Rao lower bound. Moreover, we show that the point estimate of the covariance is close to the one obtained through Monte Carlo simulations.

Estimating the variance in case of undermodeling using bootstrap

Simulation based methods have gained interest in the signal processing community. In this article we propose an algorithm to estimate the probability density function of some statistic associated with an identified model in the case of undermodeling. With this algorithm, we are thus able to estimate the variance error of any statistic associated with the model. We also give a simulation example, which shows that the estimates are in very good agreement with Monte Carlo simulations.

A mixed probabilistic/bounded-error approach to parameter estimation in the presence of amplitude bounded white noise

In this paper, we propose a mixed approach to identification of linear dynamic systems corrupted by additive, amplitude bounded white noise. To come up with an estimate of the system, we mix the information from a prediction error estimate and a set membership estimate. This approach robustifies the parameter estimates, so that they benefit from both the prediction error and the set membership features. The proposed estimation technique shows good performance on a number of simulation examples, which are performed using a wide range of noise sources.

Computing Uncertainty Regions with Simultaneous Confidence Degree using Bootstrap

We discuss the importance of constructing confidence regions of simultaneous confidence degree for certain statistics, e.g., the frequency function. In this contribution we show how bootstrap can be used to obtain reliable confidence regions of simultaneous confidence degree, independently of how many confidence regions we calculate. The procedure is illustrated by comparison with classical methods and Monte Carlo simulations. We will also provide an evaluation of the quality of the obtained confidence regions.

Variance Aspects of L2 Model Reduction when Undermodeling - the Output Error Case

In this contribution, variance properties of L2 model reduction are studied. That is, given an estimated model of high order we study the resulting variance of an L2 reduced approximation. The main result of the paper is showing that estimating a low order output error (OE) model via L2 model reduction of a high order model gives a smaller variance compared to estimating a low order model directly from data in the case of undermodeling. This has previously been shown to hold for FIR (Finite Impulse Response) models, but is in this paper extended to general linear OE models.

Comparison of methods for probabilistic uncertainty bounding

The problem of computing probabilistic uncertainty regions for the frequency responses of identified models is studied. A novel method for uncertainty bounding that uses bootstrap is presented and compared to a classical method using estimated covariance information. It is shown that, with bootstrap, it is possible to compute realistic uncertainty regions that closely resemble those obtainable through Monte Carlo simulations.

A NONPARAMETRIC APPROACH TO MODEL ERROR MODELING

To validate an estimated model and evaluate its reliability is an important part of the system identification process. Recent work on model validation has shown that the use of explicit model error models provide a better way of visualizing the possible deficiencies of the nominal model. Previous contributions have mainly focused on parametric black-box models for estimating the error model. However, this requires that a correct model order for the error model has to be selected. Here we suggest an adaptive and nonparametric frequency-domain method that estimates the frequency response of the model error by an automatic procedure. A benefit with this approach is that the tuning can be done locally, i.e., that different resolutions can be used in different frequency bands. The ideas are based on local polynomial regression and utilize a statistical criterion for selecting the optimal resolution.

L2 Model Reduction and Variance Reduction

Abstract In this contribution we demonstrate that estimating a low order model (leaving some dynamics unmodeled) by L2 model reduction of a higher order estimated model may give smaller variance and mean square error than directly estimating it from the same data that produced the high order model. It will also be shown in a quite general case that the reduced model will reach the Cramer-Rao bound if no undermodeling is present. From the derivations of this result it follows that L2 model reduction is optimal, meaning that the reduced model possesses the lowest possible variance.

Modeling and Identification of a Mechanical Industrial Manipulator

This paper covers modeling and identification of one joint of an industrial robot manipulator including flexibilities. It is shown how models can be built in the Modelica graphical environment and how these models can be transformed into a mathematical state space description which directly can be used for identification. A motivation to use linear models for modeling of the robot arm is given. This includes an analysis of the nonlinearities in the input/output data from the actual robot using a special kind of input signal. Identification and validation of the physically parameterized models are also covered.