Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

Mikhail Moklyachuk,Maria Sidei,Oleksandr Masyutka

doi:10.19139/soic.v7i1.527

Abstract

This paper focuses on the problem of the mean square optimal estimation of linear functionals which dependon the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of these quence with an additive stationary noise sequence. The aim of the paper is to develop methods of finding the optimal estimates of the functionals in the case of missing observations. The problem is investigated in the case of spectral certainty where the spectral densities of the sequences are exactly known. 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.The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences 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 of the optimal estimates of functionals are proposed for some special sets of admissible densities.

Highlights

The problem of estimation of the unknown values of stochastic processes is of constant interest in the theory and applications of stochastic processes
The spectral characteristic h(eiλ) and the mean-square error ∆(h; F, G, Fξη, Fηξ) of the optimal linear estimate of the functional Aξ⃗ which depends on the unknown values of the sequence ξ⃗(j) based on observations of the sequence ξ⃗(j) + ⃗η(j) at points j ∈ Z−\S can be calculated by formulas (10), (11)
The minimax spectral characteristic of the optimal estimate of the functional Aξ⃗ is determined by the formula (18)

Summary

Introduction

The problem of estimation of the unknown values of stochastic processes is of constant interest in the theory and applications of stochastic processes. The spectral characteristic h(eiλ) and the mean-square error ∆(h; F, G, Fξη, Fηξ) of the optimal linear estimate of the functional Aξ⃗ which depends on the unknown values of the sequence ξ⃗(j) based on observations of the sequence ξ⃗(j) + ⃗η(j) at points j ∈ Z−\S can be calculated by formulas (10), (11). Lemma 3.2 Spectral densities F 0(λ) ∈ DF , G0(λ) ∈ DG satisfying the minimality condition (12) are the least favorable in the class D = DF × DG for the optimal linear extrapolation of the functional Aξ⃗ based on observations of the uncorrelated sequences if the Fourier coefficients (15) of functions (F 0(λ) + G0(λ))−1, F 0(λ)(F 0(λ) + G0(λ))−1, F 0(λ)(F 0(λ) + G0(λ))−1G0(λ) define operators B0, R0, Q0 which determine a solution of the constrained optimization problem max (⟨R⃗a, B−1R⃗a⟩ + ⟨Q⃗a, ⃗a⟩) = ⟨R0⃗a, (B0)−1R0⃗a⟩ + ⟨Q0⃗a, ⃗a⟩. The minimax spectral characteristic of the optimal estimate of the functional Aξ⃗ is determined by the formula (18)

Conclusions

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