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Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

Iryna Golichenko, Oleksandr Masyutka, Mykhailo Moklyachuk

TL;DR

The paper addresses mean-square linear extrapolation of a functional $A\zeta$ of a continuous-time, cyclostationary (periodically correlated) isotropic random field on the sphere $S_n$, using observations of $\zeta+\theta$ with $\theta$ uncorrelated noise. It develops a Hilbert-space projection framework to derive the optimal spectral characteristic and mean-square error under spectral certainty, and extends this to minimax robust extrapolation when spectral densities are uncertain, providing explicit equations for least-favorable densities across several admissible density classes. The work yields a comprehensive set of formulas for both the certainty and robustness cases, including canonical-factorization results and saddle-point characterizations, enabling practical computation of robust estimators. By representing the MSE as a linear functional in $L_1\times L_1$, the authors enable constrained optimization over spectral densities, thus offering a principled approach to robust estimation for cyclostationary isotropic fields with spherical symmetry. The results generalize classical stationary-field extrapolation to cyclostationary, sphere-based settings and have potential applications in cosmology, geophysics, and global environmental data analysis.

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 $(t,x):t<0,x\in S_{n}$, 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.

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

TL;DR

The paper addresses mean-square linear extrapolation of a functional of a continuous-time, cyclostationary (periodically correlated) isotropic random field on the sphere , using observations of with uncorrelated noise. It develops a Hilbert-space projection framework to derive the optimal spectral characteristic and mean-square error under spectral certainty, and extends this to minimax robust extrapolation when spectral densities are uncertain, providing explicit equations for least-favorable densities across several admissible density classes. The work yields a comprehensive set of formulas for both the certainty and robustness cases, including canonical-factorization results and saddle-point characterizations, enabling practical computation of robust estimators. By representing the MSE as a linear functional in , the authors enable constrained optimization over spectral densities, thus offering a principled approach to robust estimation for cyclostationary isotropic fields with spherical symmetry. The results generalize classical stationary-field extrapolation to cyclostationary, sphere-based settings and have potential applications in cosmology, geophysics, and global environmental data analysis.

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 7 sections, 14 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Spectral properties of continuous time periodically correlated isotropic on a sphere random fields
  3. Hilbert space projection method of extrapolation of continuous time periodically correlated isotropic random fields
  4. Minimax-robust method of extrapolation
  5. Least favorable spectral densities in the class $D_\varepsilon\times D_{0}$
  6. Least favorable spectral densities in the class $D_V^U\times D_{1\varepsilon}$
  7. 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 isotropic on the unit sphere ${S_n}$. Let the stationary sequences $\{\vec{\zeta}_m^l(j),j\in\mathbb Z\}$ and $\{\vec{\theta}_m^l

Theorems & Definitions (16)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 3
  • Corollary 2
  • ...and 6 more