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Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

Oleksandr Masyutka, Mikhail Moklyachuk, Maria Sidei

TL;DR

The paper addresses mean-square optimal linear interpolation of functionals that depend on unknown values of a multidimensional continuous-time stationary process, using observations with additive noise and with missing data. It develops a Hilbert-space projection method for the spectral-certainty case and derives the spectral characteristic and mean-square error of the optimal estimator. For spectral-uncertainty, it employs a minimax robust framework to identify least-favorable spectral densities and the corresponding minimax spectral characteristics, including detailed equations for several admissible-density classes. The results provide explicit procedures to compute robust estimates via a saddle-point approach and Lagrange-multipliers, enabling practical robust interpolation in presence of spectral uncertainty and missing observations.

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

Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

TL;DR

The paper addresses mean-square optimal linear interpolation of functionals that depend on unknown values of a multidimensional continuous-time stationary process, using observations with additive noise and with missing data. It develops a Hilbert-space projection method for the spectral-certainty case and derives the spectral characteristic and mean-square error of the optimal estimator. For spectral-uncertainty, it employs a minimax robust framework to identify least-favorable spectral densities and the corresponding minimax spectral characteristics, including detailed equations for several admissible-density classes. The results provide explicit procedures to compute robust estimates via a saddle-point approach and Lagrange-multipliers, enabling practical robust interpolation in presence of spectral uncertainty and missing observations.

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

Paper Structure

This paper contains 6 sections, 6 theorems, 121 equations.

Table of Contents

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

Key Result

Theorem 2.1

Let $\{\vec{\xi}(t), t\in \mathbb{R}\}$ and $\{\vec{\eta}(t), t\in \mathbb{R}\}$ be uncorrelated multidimensional stationary stochastic processes with the spectral density matrices $F(\lambda)$ and $G(\lambda)$ which satisfy the minimality condition (minimal). The spectral characteristic $h(\lambda

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Theorem 5.1