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.
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References
Polyak B.T., Nazin S.A., Durieub C., Walterc E.: Ellipsoidal parameter or state estimation under model uncertainty. Automatica 40, 1171–1179 (2004)
Kiselev, O.N., Polyak, B.T.: Ellipsoidal estimation based on a generalized criterion. Remote Control 52, 1281–1292 (1991)
Barnett, V.: The ordering of multivariate data. J. R. Stat. Soc. Ser. A (General) 139, 318–355 (1976)
Tukey, J.W.: Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, pp. 523–531. Montreal, Canada (1975)
Titterington, D.M.: Estimation of correlation coefficients by ellipsoidal trimming. Appl. Stat. 27, 227–234 (1978)
Oja, H.: Descriptive statistics for multivariate distributions. Stat. Probab. Lett. 1, 327–332 (1983)
Liu, R.J.: On a notion of data depth based on random simplices. Ann. Stat. 18, 405–414 (1990)
Zuo, Y., Serfling, R.: General notions of statistical depth function. Ann. Stat. 28, 461–482 (2000)
Petunin, Yu.I., Rublev, B.V.: Pattern recognition with the help quadratic discriminant function. J. Math. Sci. 97, 3959–3967 (1999)
Lyashko, S.I., Klyushin, D.A., Alexeenko, V.V.: Multivariate ranking using elliptical peeling. Cybern. Syst. Anal. 49, 511–516 (2013)
Hill, B.: Posteriori distribution of percentiles: Bayes’ theorem for sampling from a population. J. Am. Stat. Assoc. 63:677–691 (1968).
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Lyashko, S.I., Klyushin, D.A., Semenov, V.V., Prysiazhna, M.V., Shlykov, M.P. (2016). Nonparametric Ellipsoidal Approximation of Compact Sets of Random Points. In: Goldengorin, B. (eds) Optimization and Its Applications in Control and Data Sciences. Springer Optimization and Its Applications, vol 115. Springer, Cham. https://doi.org/10.1007/978-3-319-42056-1_11
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DOI: https://doi.org/10.1007/978-3-319-42056-1_11
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