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Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

Oleksandr Masyutka, Mikhail Moklyachuk, Maria Sidei

TL;DR

This work tackles mean-square optimal extrapolation of linear functionals of a multidimensional stationary sequence when observations are available with additive noise and observations may be missing. It first develops a Hilbert space projection framework to obtain the exact optimal linear estimator under spectral certainty, including the spectral characteristic and mean-square error. It then introduces a minimax-robust approach to handle spectral uncertainty, identifying least-favorable spectral densities and minimax spectral characteristics via convex optimization and subdifferential methods across several admissible density classes. The results yield computable formulas for robust estimation and clarify how uncertainty in spectral densities affects estimation accuracy in high-dimensional settings.

Abstract

This paper focuses on the problem of the mean square optimal estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of the sequence 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.

Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

TL;DR

This work tackles mean-square optimal extrapolation of linear functionals of a multidimensional stationary sequence when observations are available with additive noise and observations may be missing. It first develops a Hilbert space projection framework to obtain the exact optimal linear estimator under spectral certainty, including the spectral characteristic and mean-square error. It then introduces a minimax-robust approach to handle spectral uncertainty, identifying least-favorable spectral densities and minimax spectral characteristics via convex optimization and subdifferential methods across several admissible density classes. The results yield computable formulas for robust estimation and clarify how uncertainty in spectral densities affects estimation accuracy in high-dimensional settings.

Abstract

This paper focuses on the problem of the mean square optimal estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of the sequence 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.

Paper Structure

This paper contains 6 sections, 14 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Hilbert space projection method of extrapolation of stationary sequences with missing observations
  3. Minimax approach to extrapolation problem for stationary sequences with missing observations
  4. Least favorable spectral densities in the class $D=D_0 \times D_V^U$
  5. Least favorable spectral densities in the class $D=D_{\varepsilon}\times D_{1\delta}$
  6. Conclusions

Key Result

Theorem 2.1

Let $\{\vec{\xi}(j), j\in \mathbb{Z}\}$ and $\{\vec{\eta}(j), j\in \mathbb{Z}\}$ be multidimensional stationary stochastic sequences with the spectral density matrices $F(\lambda), F_{\xi\eta}(\lambda), F_{\eta\xi}(\lambda), G(\lambda)$ and let the minimality condition (minimal) be satisfied. The s

Theorems & Definitions (19)

  • Theorem 2.1
  • Corollary 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.1
  • Definition 3.3
  • ...and 9 more