Estimation Problems for Periodically Correlated Isotropic Random FieldsEstimation Problems for Random Fields

Spectral theory of isotropic random fields in Euclidean space developed by M. I. Yadrenko is exploited to find a solution to the problem of optimal linear estimation of the functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A\zeta ={\sum\limits_{t=0}^{\infty}}\,\,\,{\int_{S_n}} \,\,a(t,x)\zeta (t,x)\,m_n(dx) $$\end{document}which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere Sn in Euclidean space En random field ζ(t, x), t ∈ Z, x ∈ Sn. Estimates are based on observations of the field ζ(t, x) + θ(t, x) at points (t, x), t = − 1, − 2, ..., x ∈ Sn, where θ(t, x) is an uncorrelated with ζ(t, x) periodically correlated with respect to time isotropic on the sphere Sn random field. Formulas for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional Aζ are obtained. The least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional Aζ are determined for some special classes of spectral densities.