Minimax-robust interpolation problem for periodically correlated isotropic on a sphere random field

TL;DR

This paper develops minimax-robust methods for optimal linear estimation of functionals of periodically correlated isotropic random fields on a sphere, accounting for spectral uncertainty.

Contribution

It introduces formulas for robust spectral estimation and least favourable spectral densities for spatial-temporal fields on a sphere with spectral uncertainty.

Findings

Derived formulas for mean square errors and spectral characteristics.

Proposed minimax spectral characteristics under spectral density uncertainty.

Established methods for robust estimation of spatial-temporal fields on a sphere.

Abstract

The problem of optimal linear estimation of functionals depending on the unknown values of a spatial temporal isotropic random field $\zeta(j,x)$, which is periodically correlated with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $\zeta(j,x)+\theta(j,x)$ at points $(j,x):$ $j\in Z\backslash\{0, 1, .... , N\}$, $x\in S_{n}$, where $\theta(j,x)$ is an uncorrelated with $\zeta(t,x)$ spatial temporal isotropic random field, which is periodically correlated with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous 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 the functional are derived in the case where the spectral density matrices are exactly known. Formulas that determine the least favourable spectral density matrices and the minimax (robust) spectral characteristics are proposed in the case where the spectral density matrices are not exactly known but a class of admissible spectral density matrices is given.