Weighted Average Errors in Set-Membership Estimation

Average-case analysis provides knowledge about the quality of estimation algorithms in the case when the influence of outliers (exceptionally difficult elements) is to be neglected. This is in contrast with the worst-case analysis, where exceptionally difficult elements are of particular interest. In this paper we consider the average behavior of estimation algorithms based on corrupted information, with values in a subspace of the problem element space. We study two local average errors, with respect to probability measures defined by a class of weight functions. We define the optimal algorithm and derive exact error formulas, in Euclidean norms in problem element and information spaces. The formulas explicitly show the dependence of the errors on basic components of the problem, in particular on the weights. Attention is paid to the class of isotropic weight functions, examples of which are provided by truncated Gaussian weight functions. An extension of the results to non-Euclidean norms in the information space in a special case is shown.

Authors and Affiliations

1. Department of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, paw. A3/A4, III p., pok. 301, 000, 30-059, Cracow, Poland

   Bolesław Kacewicz

Corresponding author

Correspondence to Bolesław Kacewicz.

Date received: July 19, 2002. Date revised: January 9, 2003. This research was partly supported by AGH Grant No. 10.420.03.

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