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Quality Estimation of Approximate Models

Author: F. Tjärnström
Published: Licentiate Thesis No. 810, Feb 2000.
Research area: Identification
Keywords: model reduction, bootstrap, model error modeling

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This thesis discusses three different topics: model error modeling, bootstrap, and model reduction. These subjects may at first sight seem to be quite far away from each other. However, there are some connections between them, the most important one being uncertainty estimation.

Model error modeling is actually a tool for model validation. The idea is to construct a model of the model errors that are present in the nominal model, and present them in an easily interpreted way. When the error models are linear, we prefer to present the result in the frequency domain. We discuss different ways of estimating such models, as well as how the "size" of such models should be presented and interpreted. Examples illustrate how some model errors could be accepted although they may be large. This is partly in contrast with traditional model validation tools, that more have the character of telling whether we have any model errors or not.

In some situations it is very difficult to calculate the uncertainties present in an estimate. One would therefore like to repeat the experiment several times to get better knowledge about it. Bootstrap mimics this, since it simulates new data from the original sample and thus makes it possible to repeat a similar experiment again. We describe how bootstrap can be used in a system identification experiment. The most interesting results are that we are able to estimate the variance error of undermodeled models and that it is possible to construct several confidence regions where we are in control of the simultaneous confidence degree (this is, regions which all cover their respective parameters with a certain confidence degree).

The last chapter is focused on quantifying the variance reduction that occurs in model reduction. We specifically look at $L_2$ model reduction and show that estimating the model in two steps, first a high order model which is then subjected to $L_2$ model reduction, in some situations give the same variance as estimating the model directly. We also show that it might even be better to estimate the model in two steps in some specific cases. From the calculations of these results it also follows that $L_2$ model reduction is optimal in reducing the variance of the estimate.


    author = "Tj{\"{a}}rnstr{\"{o}}m, Fredrik",
    title = "Quality Estimation of Approximate Models",
    school = "Department of Electrical Engineering, Link{\"{o}}ping University",
    year = "2000",
    month = Feb,
    address = "SE-581 83 Linköping, Sweden",
    type = "Licentiate Thesis No. 810",