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Filtering Problem for Functionals of Stationary Processes with Missing Observations

Mykhailo Moklyachuk, Maria Sidei

TL;DR

The paper tackles the problem of mean-square optimal linear estimation of a functional $A\xi=\int_{R^s} a(t)\xi(-t)\,dt$ of a stationary process from noisy, partially observed data with missing observations. It first develops a Hilbert-space projection framework to yield exact spectral-characteristic formulas and MSE expressions under spectral certainty, then extends to a minimax-robust setting where spectral densities are unknown but constrained to admissible sets, identifying least favorable densities and minimax spectral characteristics. The authors derive operator-based conditions (involving $\mathbf{B},\mathbf{R},\mathbf{Q}$) and extremal equations that characterize robust estimators across several classes of density-uncertainty sets, including $L_1$ and $L_2$ neighborhoods and contamination models. These results provide practical, analytically tractable robust filtering schemes for functionals of stationary processes in the presence of missing data and spectral uncertainty.

Abstract

The problem of the mean-square optimal linear estimation of the functional $Aξ=\ \int\limits_{R^s}a(t)ξ(-t)dt,$ which depends on the unknown values of stochastic stationary process $ξ(t)$ from observations of the process $ξ(t)+η(t)$ at points $t\in\mathbb{R} ^{-} \backslash S $, $S=\bigcup\limits_{l=1}^{s}[-M_{l}-N_{l}, \, \ldots, \, -M_{l} ],$ $R^s=[0,\infty) \backslash S^{+},$ $S^{+}=\bigcup\limits_{l=1}^{s}[ M_{l}, \, \ldots, \, M_{l}+N_{l}]$ is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under the condition of spectral certainty, where spectral densities of the processes $ξ(t)$ and $η(t)$ are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly, but 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 spectral densities.

Filtering Problem for Functionals of Stationary Processes with Missing Observations

TL;DR

The paper tackles the problem of mean-square optimal linear estimation of a functional of a stationary process from noisy, partially observed data with missing observations. It first develops a Hilbert-space projection framework to yield exact spectral-characteristic formulas and MSE expressions under spectral certainty, then extends to a minimax-robust setting where spectral densities are unknown but constrained to admissible sets, identifying least favorable densities and minimax spectral characteristics. The authors derive operator-based conditions (involving ) and extremal equations that characterize robust estimators across several classes of density-uncertainty sets, including and neighborhoods and contamination models. These results provide practical, analytically tractable robust filtering schemes for functionals of stationary processes in the presence of missing data and spectral uncertainty.

Abstract

The problem of the mean-square optimal linear estimation of the functional which depends on the unknown values of stochastic stationary process from observations of the process at points , is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under the condition of spectral certainty, where spectral densities of the processes and are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly, but 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 spectral densities.
Paper Structure (7 sections, 6 theorems, 64 equations)

This paper contains 7 sections, 6 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Hilbert space projection method of filtering
  3. Minimax method of filtering
  4. Least favorable spectral densities in the class $D=D_{\varepsilon_1}^1\times D_{\varepsilon_2}^2$
  5. Least favorable spectral densities in the class $D=D_{\varepsilon_1} \times D_{\varepsilon_2}^1$
  6. Least favorable spectral densities in the class $D=D_{\varepsilon_1} \times D_{\varepsilon_2}^1$
  7. Conclusions

Key Result

Theorem 2.1

Let $\xi(t)$ and $\eta(t)$ be uncorrelated stationary processes with spectral densities $f(\lambda)$ and $g(\lambda)$ which satisfy the minimality condition (minimal). The spectral characteristic $h(e^{i\lambda})$ and the mean-square error $\Delta(f,g)$ of the optimal linear estimate of the function

Theorems & Definitions (8)

  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1