Kriging and Radial Basis Functions

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SIAM OP05 Kriging and RBFs 3InterpolationΦα + Fβ = y , Φ ij = φ j (θ,x i ), F i,: = f(x i ) Tm equations with m+q unknowns.Both in Kriging and general RBF approach: supply with F T α =0( )( ) ( )Φ F α y=(∗)F ⊤ 0 β 0Φ∈R m×mis symmetric. Full rank if the x i are distinct.F ∈ R m×q ,q
SIAM OP05 Kriging and RBFs 5• General RBFsΦ is not necessarily definite, but Powell 1 shows thata number of popular RBFs satisfysign ( v T Φ v ) = (−1) µfor all v ∈ V µ ,v≠0V µ = {v∈R m ∑: mi=1 v ip(x i )=0for any p ∈ Π µ−1 }We assume that {f j } comprises a basis of Π µ−1 .Then V µ ⊆N(F T ), the nullspace of F T . An orthonormalbasis of N can be found as N in the completeQR factorization of F ,( ) ( )RF = Q N = QR0Name φ(r), r ≥ 0 µGaussian e −r2 0inverse multiquadric (r 2 +1) −1/2 0linear r 1multiquadric (r 2 +1) 1/2 1thin plate spline r 2 log r 2cubic r 3 21M.J.D. Powell: 5 lectures on radial basis functions. Report IMM-REP-2005-03, IMM, DTU, 2005.SIAM OP05 Kriging and RBFs 6s(x) = α T φ(θ,x)+β T f(x).Φα+Fβ = ySeek α ∈ V µ ⊆N(F T ):α=Nα NN T ΦNα N +N T Fβ = N T y, (−1) µ ˜Φ αN = y NThe matrix ˜Φ =(−1) µ N T Φ N is symmetric and positive definite.The regression coefficients are found fromRβ = Q ( )T y−ΦNα N
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