Filtering of Multidimensional Stationary Sequences with Missing Observations

TL;DR

This paper develops methods for optimal linear estimation of functionals of multidimensional stationary sequences with missing data and noise, providing formulas for errors and spectral characteristics under spectral certainty and uncertainty.

Contribution

It introduces formulas for mean-square errors and spectral characteristics of optimal estimates, including minimax robust methods for uncertain spectral densities.

Findings

Formulas for calculating mean-square errors and spectral characteristics under spectral certainty.

Minimax estimation formulas for uncertain spectral densities.

Identification of least favorable spectral densities for robust estimation.

Abstract

The problem of mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence from observations of the sequence with a noise and missing observations is considered. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of the functionals are proposed under the condition of spectral certainty, where spectral densities of the sequences are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.