Abstract
The problem of optimal linear estimation of functionals depending on the unknown values of a random fieldζ(t,x), which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the unit sphere Sn with respect to spatial argumentxєSn. Estimates are based on observations of the fieldζ(t,x) +Θ(t,x) at points (t,x) :t< 0;xєSn, whereΘ(t,x) is an uncorrelated withζ(t,x) random field, which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the sphereSnwith respect to spatial argumentxєSn. 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.
Highlights
Cosmological Principle: the Universe is, in the large, homogeneous and isotropic
Following the method we find the optimal linear estimate Aζ as projection of Aζ on the closed linear subspace H−(ζ + θ) generated by values of the field ζ(t, x) + θ(t, x) at points (t, x), t < 0, x ∈ Sn, in the space H = L2(Ω, F, P)
The spectral characteristic h(F ) and the mean square error ∆(F ) of the optimal linear estimate of the functional Aζ from observations of the field ζ(t, x) at points (t, x), t < 0, x ∈ Sn are determined by formulas (hlm(F ))⊤ = (Alm(λ))⊤ − (Cml (λ))⊤(Fm(λ))−1, (14)
Summary
Cosmological Principle (first coined by Einstein): the Universe is, in the large, homogeneous and isotropic
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