Filtering Problem for Stationary Sequences with Missing Observations

Mikhail Moklyachuk,Maria Sidei

doi:10.19139/soic.v4i4.241

Abstract

We deal with the problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a stationary stochastic sequence from observations of the sequence with a stationary noise sequence. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived 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 of spectral uncertainty, where spectral densities are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some special sets of admissible densities.

Highlights

Estimation of unknown values of stochastic sequences is of a great interest both in the theory of random processes and applications of this theory to the data analysis
Moklyachuk [19] – [22] problems of linear optimal estimation of the functionals which depend on the unknown values of stationary sequences and processes were investigated
∑In this paper we investigate the problem of the mean-square optimal estimation of the functional Aξ = a(j)ξ(−j) which depends on the unknown values of a stationary sequence {ξ(j), j ∈ Z} from observations j ∈Z S

Summary

Introduction

Estimation of unknown values of stochastic sequences is of a great interest both in the theory of random processes and applications of this theory to the data analysis. V. Poor [6] the minimax extrapolation and filtering problems for stationary sequences were investigated with the help of convex optimization methods. Poor [6] the minimax extrapolation and filtering problems for stationary sequences were investigated with the help of convex optimization methods This approach makes it possible to find equations that determine the least favorable spectral densities for different classes of densities. Moklyachuk [19] – [22] problems of linear optimal estimation of the functionals which depend on the unknown values of stationary sequences and processes were investigated. The problem is investigated in the case of spectral certainty, where both spectral densities of the sequences ξ(j) and η(j) are known In this case we derive formulas for calculating the spectral characteristic and the meansquare error of the optimal linear estimates using the method of projection in the Hilbert space of random variables with finite second moments proposed by Kolmogorov [15]. Formulas for determination the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the functional are proposed for some specific classes of admissible spectral densities

Hilbert space projection method of filtering

Minimax-robust method of filtration

Conclusions

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