Complex Mean and Variance of Linear Regression Model for High-Noised Systems by Kriging

The aim of the paper is to derive the complex-valued least-squares estimator for bias-noise mean and variance. 1. Kriging Let us consider a stationary random process ǫ = {ǫj; N1 ∋ j ⊃ i = 1, . . . , n} with zero mean E{ǫi} = E{ǫj} = E{ǫ} = 0 and the background trend ∑ k fjkβ k = fjkβ k (some known mean function m(j) with unknown regression parameters β, where k = 1, 2, . . . ) then Vj = ǫj +m(j) = ǫj + fjkβ k e.g. j = n+ 1 and Vi = ǫi +m(i) = ǫi + fikβ k i = 1, . . . , n , where fjk (f ) is a given vector and fik (F ) is a given matrix. The unbiasedness constraint on the estimation statistics V̂j = ω i jVi E{Vj} = E{ω jVi} produces the system of N(k) equations in the n unknowns ω j fjk = ω i jfik (f ′ = ω′F ) . For white noise E{[ǫ̂j − ǫj ]} = σ + σω jρiiωi j ( E{[ǫ̂− ǫ]} = σ + σωΛω ) , where ρii (Λ) is the identity auto-correlation matrix, the minimization constraint ∂E{[ǫ̂j − ǫj ]2} ∂ω j = 2σ2ρiiω i j + 2σ 2fikμ k j = 0 , where E{[ǫ̂j − ǫj]} = σ + σω jρiiωi j + 2σ (ω jfik − fjk) } {{ } 0 μkj , let us add the n equations in the N(k) unknowns μj to the system ρiiω i j = −fikμkj equivalent to ω j = −ρfikμj substituting this term into the unbiased system fkj = fkiω i j