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Robust interpolation of random fields homogeneous in time and isotropic on a sphere, which are observed with noise

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Abstract

We study the problem of optimal linear estimation of the functional

$$A_N \xi = \sum\limits_{k = 0}^{\rm N} {\int\limits_{S_n } {a(k,x)\xi (k,x)m_n (dx),} }$$

, which depends on unknown values of a random field ξ(k, x),k∃Z,x∃S n homogeneous in time and isotropic on a sphereS n, by observations of the field ξ(k,x)+η(k,x) with k∃ Z{0, 1, ...,N},x∃Sn (here, η (k, x) is a random field uncorrelated with ξ(k, x), homogeneous in time, and isotropic on a sphere Sn). We obtain formulas for calculation of the mean square error and spectral characteristic of the optimal estimate of the functionalA Nξ. The least favorable spectral densities and minimax (robust) spectral characteristics are found for optimal estimates of the functionalA Nξ.

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Authors and Affiliations

  1. Kiev University, Kiev

    M. P. Moklyachuk

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  1. M. P. Moklyachuk
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 962–970, July, 1995.

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Moklyachuk, M.P. Robust interpolation of random fields homogeneous in time and isotropic on a sphere, which are observed with noise. Ukr Math J 47, 1103–1112 (1995). https://doi.org/10.1007/BF01084906

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  • DOI: https://doi.org/10.1007/BF01084906

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