Nonparametric Ellipsoidal Approximation of Compact Sets of Random Points

Sergey I. Lyashko (), Dmitry A. Klyushin (), Vladimir V. Semenov (), Maryna V. Prysiazhna () and Maksym P. Shlykov ()
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Sergey I. Lyashko: Kiev National Taras Shevchenko University
Dmitry A. Klyushin: Kiev National Taras Shevchenko University
Vladimir V. Semenov: Kiev National Taras Shevchenko University
Maryna V. Prysiazhna: Kiev National Taras Shevchenko University
Maksym P. Shlykov: Kiev National Taras Shevchenko University

A chapter in Optimization and Its Applications in Control and Data Sciences, 2016, pp 327-340 from Springer

Abstract: Abstract One of the main problems of stochastic control theory is the estimation of attainability sets, or information sets. The most popular and natural approximations of such sets are ellipsoids. B.T. Polyak and his disciples use two kinds of ellipsoids covering a set of points—minimal volume ellipsoids and minimal trace ellipsoids. We propose a way to construct an ellipsoidal approximation of an attainability set using nonparametric estimations. These ellipsoids can be considered as an approximation of minimal volume ellipsoids and minimal trace ellipsoids. Their significance level depends only on the number of points and only one point from the set lays on a bound of such ellipsoid. This unique feature allows to construct a statistical depth function, rank multivariate samples and identify extreme points. Such ellipsoids in combination with traditional methods of estimation allow to increase accuracy of outer ellipsoidal approximations and estimate the probability of attaining a target set of states.

Keywords: Ellipsoidal approximation; Attainability set; Information set; Nonparametric estimation; Extreme point; Confidence ellipse (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-42056-1_11

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DOI: 10.1007/978-3-319-42056-1_11

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