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Filtering of Continuous Time Periodically Correlated Isotropic Random Fields

Iryna Golichenko, Oleksandr Masyutka, Mikhail Moklyachuk

Abstract

The problem of optimal linear estimation of functionals depending on the unknown values of a random field $ζ(t,x)$, which is mean-square continuous periodically correlated with respect to time argument $t\in\mathbb R$ and isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $ζ(t,x)+θ(t,x)$ at points $(t,x):t\leq 0,x\in S_{n}$, where $θ(t,x)$ is an uncorrelated with $ζ(t,x)$ random field, which is mean-square continuous periodically correlated with respect to time argument $t\in\mathbb R$ and isotropic on the sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

Filtering of Continuous Time Periodically Correlated Isotropic Random Fields

Abstract

The problem of optimal linear estimation of functionals depending on the unknown values of a random field , which is mean-square continuous periodically correlated with respect to time argument and isotropic on the unit sphere with respect to spatial argument . Estimates are based on observations of the field at points , where is an uncorrelated with random field, which is mean-square continuous periodically correlated with respect to time argument and isotropic on the sphere with respect to spatial argument . Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

Paper Structure

This paper contains 6 sections, 6 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Spectral properties of periodically correlated isotropic on a sphere random fields
  3. Hilbert space projection method of filtering
  4. Minimax-robust method of filtering
  5. The least favorable spectral densities in the class $D_{0}\times D_V^U$
  6. Conclusions

Key Result

Theorem 1

Let $\{\zeta(t,x), t\in \mathbb{R}, x\in S_n\}$ and $\{\theta(t,x), t\in \mathbb{R}, x\in S_n\}$ be mutually uncorrelated random fields, which are periodically correlated with respect to time argument $t\in\mathbb R$ and isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n

Theorems & Definitions (8)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Corollary 1