Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

TL;DR

This paper addresses the problem of optimally estimating linear functionals of multidimensional stationary processes with missing data, providing formulas for both spectral certainty and uncertainty cases.

Contribution

It introduces formulas for optimal linear estimation under spectral certainty and develops minimax methods for spectral uncertainty in multidimensional processes.

Findings

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

Minimax estimation formulas for spectral uncertainty.

Identification of least favorable spectral densities for robust estimation.

Abstract

The problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional continuous time stationary stochastic process is considered. Estimates are based on observations of the process with an additive stationary stochastic noise process at points which do not belong to some finite intervals of a real line. The problem is investigated in the case of spectral certainty, where the spectral densities of the processes are exactly known. Formulas for calculating the mean-square errors and spectral characteristics of the optimal linear estimates of functionals are proposed under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case spectral uncertainty, where spectral densities of the processes are not known exactly while some sets of admissible spectral densities of the processes are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible spectral densities