Basis functions and Kriging - IMAGe

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Basis functions and Kriging - IMAGe

Basis functions and KrigingNUS Lecture 2 D, February 2011Douglas Nychka,National Science Foundation NUS/IMS February 2011


Outline• Kriging as a ridge regression• ridge regression as KrigingD. Nychka Basis functions and Kriging 2


KrigingWe have a covariance function ρk(x, x ′ and with λ = σ 2 /ρ a basis {φ i }for the fixed partg = fixed part +h, ĥ(x 0 ) = k 0 (K + λI) −1 (y − T ˆd)Kriging → ridge regressionSTEP 1) Build basis functions from the covariance functions and observationlocations:φ 1 (x) = ρk(x, x 1 )φ 2 (x) = ρk(x, x 2 )...φ n (x) = ρk(x, x n )STEP 2) Recognize that estimate is a sum of basis functions withcoefficients found from the data!ĥ(x) = ∑ j φ j (x)c jĉ = (K + λI) −1 (y − T ˆd)D. Nychka Basis functions and Kriging 3


Ridge regressionGiven the fixed basis {φ i }, nonparametric basis {ψ j }and penalty matrix H.ridge regression →KrigingCreate the covariance function:(1/λ)k(x, x ′ ) = ∑ j,kψ j (x)H −1jk ψ k(x ′ )Conclusion:This is a spatial process estimatewith covariance (1/λ)k for h.At the observation points the covariance matrix isXH −1 X TD. Nychka Basis functions and Kriging 5

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