Comparing different interpolation methods Kuhnt on two-dimensional - PowerPoint PPT Presentation

Metamodells for CE Mühlenstädt, Comparing different interpolation methods Kuhnt on two-dimensional test functions Introduction Interpolation Kriging TPS Thomas Mühlenstädt, Sonja Kuhnt NNI KI Comparison TU Dortmund Summary References ENBIS EMSE, St. Etienne 01.07.2009

Metamodells for CE Mühlenstädt, Introduction 1 Kuhnt Introduction Interpolation methods 2 Interpolation Kriging Kriging TPS NNI Thin plate spline KI Natural neighbor interpolation Comparison Kernel interpolation Summary References Comparison 3 Summary 4

Computer experiments Metamodells for CE Mühlenstädt, Kuhnt Simulation model for real world phenomena Fang et al. (2006) Introduction Deterministic Interpolation Computational expensive Kriging TPS NNI Needed: Easy to calculate surrogate KI Comparison Setting: Summary References Design D = { � x 1 , . . . ,� x n } ,� x i = ( x i , 1 , x i , 2 ) . One-dimensional output y 1 , . . . , y n y i = f ( � x i ) , f unknown

Multivariate interpolation Metamodells for CE Mühlenstädt, Kuhnt Introduction Treated approaches: Interpolation Kriging Kriging (Gaussian Random Fields) TPS NNI KI Thin plate spline ( TPS ) Comparison Natural neighbor interpolation ( NNI ) Summary References Kernel interpolation ( KI )

Kriging Metamodells for CE Mühlenstädt, Kuhnt Santner et al. (2003) Introduction Y ( � x i ) = g β ( � x i ) + Z ( � x i ) , 1 ≤ i ≤ n , Interpolation Kriging TPS NNI KI g β ( � x i ) regression part (here: g β = β ∈ R ) Comparison x ) ∼ ( 0 , σ 2 ) normally distributed Summary Z ( � References Z ( � x 1 ) and Z ( � x 2 ) ,� x 1 � = � x 2 explicitly dependent: cor θ ( Z ( � x 1 ) , Z ( � x 2 )) → 1 for � x 1 → � x 2

Kriging Metamodells for CE Mühlenstädt, Kuhnt Introduction � 2 � Interpolation � 2 � Kriging � � cor ( Z ( � x i ) , Z ( � x i ′ )) = exp − � x i , d − x i ′ , d θ i TPS NNI d = 1 KI Comparison Estimation of parameters β, θ, σ 2 : REML Summary References Optimization of Log-likelihood: by R command optim for 50 different initial values

Thin plate spline ( TPS ) Metamodells for CE Mühlenstädt, Kuhnt Generalization of cubic splines Micula (2002) Introduction Solves the following optimization problem: Interpolation Kriging TPS Search f ∗ , such that I ( f ) is minimized NNI KI in a suitable functional space Comparison under the constraint of interpolation Summary References � ∂ 2 f ( � � 2 � 2 � 2 � � ∂ 2 f ( � ∂ 2 f ( � � x ) x ) x ) I ( f ) = + 2 + d � x ∂ x 2 ∂ x 2 ∂ x 1 ∂ x 2 R 2 1 2

Thin plate spline ( TPS ) Metamodells for CE Mühlenstädt, Kuhnt Introduction Can be solved explicitly: Interpolation Kriging n TPS � NNI f ∗ ( � x ) = λ i φ ( � � x − � x i � 2 ) + λ n + 1 + λ n + 2 x 1 + λ n + 3 x 2 KI i = 1 Comparison Summary φ ( r ) = r 2 log ( r ) References Computation λ 1 , . . . , λ n + 3 : system of linear equations

Natural neighbor interpolation ( NNI ) Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation Kriging Weighted mean of the y -values TPS Sibson (1980) NNI KI Strictly local method Comparison Uses the Voronoi diagramm for weighting Summary References

Natural neighbor interpolation ( NNI ) Metamodells for CE 1 Mühlenstädt, Kuhnt Introduction Interpolation Kriging TPS NNI KI 0.5 Comparison Summary References 0 0.5 1

Natural neighbor interpolation ( NNI ) Metamodells for CE 1 Mühlenstädt, Kuhnt Introduction Interpolation Kriging TPS NNI KI 0.5 ● Comparison Summary References 0 0.5 1

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt Weight locally fitted linear functions under the Introduction constraint of interpolation Mühlenstädt and Kuhnt (2009) Interpolation Split up the convex hull of the design into simplices S j Kriging TPS (Delaunay triangulation) NNI KI Fit a linear function ˆ y j ( x ) to each simplex S j Comparison Summary Weight the linear functions: References � N j = 1 g j ( � x )ˆ y j ( � x ) 1 , g j ( � x ) = � N � 2 x j j = 1 g j ( � i = 0 � � i − � x � 2 x ) 2

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1 Introduction Interpolation ● ● Kriging 0.5 TPS NNI KI f(x) Comparison 0 Summary ● ● References −0.5 ● −1 −1 −0.5 0 0.5 1 x

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1 Introduction Interpolation ● ● ● ● Kriging 0.5 TPS NNI KI f(x) Comparison 0 Summary ● ● ● ● References −0.5 ● ● −1 −1 −0.5 0 0.5 1 x

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function

Kernel interpolation ( KI ) Metamodells for CE Mühlenstädt, Kuhnt 1.0 1 Introduction Interpolation 0.8 ● ● ● ● ● ● Kriging 0.5 TPS NNI 0.6 KI weight f(x) Comparison 0 0.4 Summary ● ● ● ● ● ● References −0.5 ● ● ● 0.2 0.0 −1 −1 −0.5 0 0.5 1 x linear function