Kriging and Radial Basis Functions

SIAM OP05 Kriging and RBFs 7• Unified approacha ∈ R m . a = Qa R + Na N , a R = Q T a, a N = N T aAssume that y = Fβ +z, z = Nz NE [ z z T] = σ 2 NAN T , A = D˜ΦD, D = diag(˜Φ −1/2ii )A is symmetric, positive definite, and A ii =1. Valid correlation matrix.A = DC T CDBLUE:min ‖v‖ 2 2 s.t. Fβ +NDC T v = y Solution: C T v =(−1) µ D −1 N T yv,gbv ∈ R m−q has variance σ 2 I. Estimate: σ 2 = ‖v‖ 2 2/(m − q)}Assume Gaussian process: θ ∗ = argmin θ{Γ(θ) ≡ det A(θ) · σ 2(m−q) (θ)SIAM OP05 Kriging and RBFs 8Theorem: For φ(r) ∈{r, r 3 ,r 2 log r}, ω∈R + :s(ωθ,x) = s(θ,x),Γ(ωθ) = Γ(θ)Remarks:“Scalar” θ has no effect.“Full” θ : Finding θ ∗ reduces to a problem in R d−1+ instead of R d +