Minimax-robust interpolation problem for periodically correlated isotropic on a sphere random field
Iryna Golichenko, Oleksandr Masyutka, Mykhailo Moklyachuk
TL;DR
The paper addresses interpolation of a spatial-temporal isotropic field on the sphere by estimating the functional $A_N\zeta=\sum_{j=0}^{N}\int_{S_n} a(j,x)\zeta(j,x)\,m_n(dx)$ from observations of $\zeta(j,x)+\theta(j,x)$, where $\zeta$ is periodically correlated and $\theta$ is uncorrelated with $\zeta$. It develops formulas for the optimal linear estimator and its mean-square error under spectral certainty and extends to minimax-robust estimation when spectral densities $(F_m, G_m)$ are not exactly known, by characterizing least favourable densities $F_m^0(\lambda)$, $G_m^0(\lambda)$ and the minimax spectral characteristic $h^0(\lambda)$ forming a saddle point. The framework relies on the Kolmogorov Hilbert-space projection with Fourier coefficients of spectral-density matrices, leveraging the cyclostationary structure and the spherical-harmonic expansion to provide tractable equations for $h(\lambda)$ and $\Delta(h;F,G)$. The results yield robust interpolation rules for periodically correlated isotropic fields on spheres with applications to cosmology and geophysics, where spectral uncertainty is common.
Abstract
The problem of optimal linear estimation of functionals depending on the unknown values of a spatial temporal isotropic random field $ζ(j,x)$, 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 $ζ(j,x)+θ(j,x)$ at points $(j,x):$ $j\in Z\backslash\{0, 1, .... , N\}$, $x\in S_{n}$, where $θ(j,x)$ is an uncorrelated with $ζ(t,x)$ 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.