Optimal Average Case Estimation in Hilbert Norms |
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Authors: | Bolesław Kacewicz |
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Affiliation: | (1) Department of Applied Mathematics, University of Mining and Metallurgy, Al. Mickiewicza 30, paw. A3/A4, III p., pok. 301, 30-059 Cracow, Poland. kacewicz@uci.agh.edu.pl., PL |
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Abstract: | In contrast to the worst case approach, the average case setting provides less conservative insight into the quality of estimation
algorithms. In this paper we consider two local average case error measures of algorithms based on noisy information, in Hilbert
norms in the problem element and information spaces. We define the optimal algorithm and provide formulas for its two local
errors, which explicitly exhibit the influence of factors such as information, information (measurement) errors, norms in
the considered spaces, a subset where approximations are allowed, and “unmodeled dynamics.” Based on the error expression,
we formulate in algebraic language the problem of selecting the optimal approximating subspace. The solution is given along
with the specific formula for the error, which depends on the eigenvalues of a certain matrix defined by information and norms
under consideration.
Date received: November 25, 1999. Date revised: May 30, 2000. |
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Keywords: | , Identification algorithms, Average case setting, Local errors, Optimal algorithm, Optimal subspace, |
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