Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

Abstract

Keywords: isotropic random field, periodically correlated random field, robust estimate, mean square error, least favourable spectral density, minimax spectral characteristic.Abstract. The problem of optimal linear estimation of functionals depending on the unknown values of a random field ζ(t, x), which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the unit sphere S n with respect to spatial argument x ∈ S n . Estimates are based on observations of the field ζ(t, x) + θ(t, x) at points (t, x) : t < 0, x ∈ S n , where θ(t, x) is an uncorrelated with ζ(t, x) random field, which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the sphere S n with respect to spatial argument x ∈ S n . Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

Filtering problem for periodically correlated stochastic sequences with missing observations

“…We also refer to the book by M. P. Moklyachuk, O. Yu. Masyutka and I. I. Golichenko [25] where results of the investigation of the problem of mean square optimal estimation (forecasting, interpolation, and filtering) of linear functionals constructed from unobserved values of periodically correlated isotropic random fields are described.…”

Mykhailo Moklyachuk – to the 75th anniversary of his birth

“…The monograph [8] investigates the problems of optimal estimation by the mean-squared criterion of linear functions that depend on unknown values of periodically correlated isotropic random fields. The estimates are based on observations of fields with noise.…”