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

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Metamodells for CE Mhlenstdt, Comparing different interpolation methods Kuhnt on two-dimensional test functions Introduction Interpolation Kriging TPS Thomas Mhlenstdt, Sonja Kuhnt NNI KI Comparison TU Dortmund Summary

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Comparing different interpolation methods

  • n two-dimensional test functions

Thomas Mühlenstädt, Sonja Kuhnt

TU Dortmund

ENBIS EMSE, St. Etienne 01.07.2009

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

1

Introduction

2

Interpolation methods Kriging Thin plate spline Natural neighbor interpolation Kernel interpolation

3

Comparison

4

Summary

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Computer experiments

Simulation model for real world phenomena

Fang et al. (2006)

Deterministic Computational expensive Needed: Easy to calculate surrogate Setting: Design D = { x1, . . . , xn}, xi = (xi,1, xi,2). One-dimensional output y1, . . . , yn yi = f( xi), f unknown

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Multivariate interpolation

Treated approaches: Kriging (Gaussian Random Fields) Thin plate spline (TPS) Natural neighbor interpolation (NNI) Kernel interpolation (KI)

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kriging

Santner et al. (2003)

Y( xi) = gβ( xi) + Z( xi), 1 ≤ i ≤ n, gβ( xi) regression part (here: gβ = β ∈ R) Z( x) ∼ (0, σ2) normally distributed Z( x1) and Z( x2), x1 = x2 explicitly dependent: corθ(Z( x1), Z( x2)) → 1 for x1 → x2

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kriging

cor(Z( xi), Z( xi′)) = exp

  • −

2

  • d=1

θi

  • xi,d − xi′,d
  • 2
  • Estimation of parameters β, θ, σ2: REML

Optimization of Log-likelihood: by R command optim for 50 different initial values

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Thin plate spline (TPS)

Generalization of cubic splines

Micula (2002)

Solves the following optimization problem: Search f ∗, such that I(f) is minimized in a suitable functional space under the constraint of interpolation I(f) =

  • R2
  • ∂2f(

x) ∂x2

1

2 + 2 ∂2f( x) ∂x1∂x2 2 +

  • ∂2f(

x) ∂x2

2

2 d x

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Thin plate spline (TPS)

Can be solved explicitly: f ∗( x) =

n

  • i=1

λiφ( x − xi2) + λn+1 + λn+2x1 + λn+3x2 φ(r) = r 2log(r) Computation λ1, . . . , λn+3: system of linear equations

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Natural neighbor interpolation (NNI)

Weighted mean of the y-values

Sibson (1980)

Strictly local method Uses the Voronoi diagramm for weighting

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Natural neighbor interpolation (NNI)

0.5 1 0.5 1

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Natural neighbor interpolation (NNI)

  • 0.5

1 0.5 1

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

Weight locally fitted linear functions under the constraint of interpolation

Mühlenstädt and Kuhnt (2009)

Split up the convex hull of the design into simplices Sj (Delaunay triangulation) Fit a linear function ˆ yj(x) to each simplex Sj Weight the linear functions: N

j=1 gj(

x)ˆ yj( x) N

j=1 gj(

x) , gj( x) = 1 2

i=0

xj

i −

x2

2

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

x f(x)

  • −1

−0.5 0.5 1 −1 −0.5 0.5 1

  • linear function

weight 0.0 0.2 0.4 0.6 0.8 1.0

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Kernel interpolation (KI)

Properties of the Kernel interpolation: Continuous Differentiable (also at observation points) Able to predict outside of observation range Exactly reproduces linear functions

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Comparison of interpolation methods

Which interpolation method should be used? 5 analytical 2-dim. examples Aim: high prediction power Prediction power: root mean square error (RMSE) Different experimental designs

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Designs for computer experiments

Difference to ’standard’ DoE: no random error Example for a ’good’ space filling designs: maximin latin hypercube design

Fang et al. (2006)

Also often encountered: factorial designs For sequential procedures: designs with clusters

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Used Designs

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Root Mean square error(RMSE)

Criterium for comparing different predictions: RMSE(y, ˆ y, r1, . . . , rm) :=

  • 1

m

m

  • i=1

(y( ri) − ˆ y( ri))2

  • r1, . . . ,

rm points in the design space

  • r1, . . . ,

rm 10000 points of a Sobol’ Uniform Sequence

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Implementation aspects

Easy: Thin plate spline Acceptable:

Kriging Kernel interpolation (if Delaunay triangulation is available)

Difficult: Natural neighbor interpolation (Calculation of the Voronoi diagram constrained to design space)

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Computation times

Design TPS KI Kriging NNI DMm

10

10 15 13 - 22 839 DMm

20

13 27 69 - 162 1536 DMm

30

15 39 178 - 448 2239 Computation times in seconds For 10000 prediction points Based on the maximin latin hypercube designs

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Die force

Model for the effective die force in sheet metal forming processes depending on friction and blankholder force Nearly linear in one variable f( x) = 0.9996(1090.91 + 4x1x2) exp

  • x1

π 2

  • ,

D = [0.05, 0.2] × [5, 30]

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Die force

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Hump function

Standard example from

  • ptimization literature

Extreme values on boundary f(x, y) = 1.0316 + 4x2 − 2.1x4 + 1 3x6 + xy − 4y2 + 4y4, D = [−5, 5]2

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Hump function

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Matlabs peaks function

Good example for a hilly contour f( x) =3(1 − x1)2 exp

  • −x2

1 − (x2 + 1)2

− 10 x1 5 − x3

1 − x5 2

  • exp
  • −x2

1 − x2 2

  • −

1 3 exp

  • −(x1 + 1)2 − x2

2

  • , D = [−2, 2]2
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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Matlabs peaks function

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Not smooth

Assumption of smoothness

  • ften unrealistic

Continuous but not differentiable f(x, y) = |x2 + sin (0.5πy) − y|, D = [0, 1]2.

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Not smooth

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Sibson’s function

Proposed by Sibson (1980) for illustrating NNI High complexity for size of sample f( x) = cos

  • 4π
  • (x1 − 0.25)2 + (x2 − 0.25)2
  • ,

D = [0, 1]2

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Sibson’s function

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Summary

No overall winner Decision depends on design Kriging often very efficient, especially for higher sample sizes, designs with clusters KI and TPS good for small sample sizes NNI not recommendable

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Metamodells for CE Mühlenstädt, Kuhnt Introduction Interpolation

Kriging TPS NNI KI

Comparison Summary References

Literatur

Fang, K.-T., Li, R., and Sudjianto, A. (2006). Design and Modeling for Computer

  • Experiments. Computer Science and Data Analysis Series. Chapman &

Hall/CRC, New York. Mühlenstädt, T. and Kuhnt, S. (2009). Kernel interpolation. Technical report, Faculty of Statistics, Technische Universität Dortmund, Dortmund, Germany. Micula, G. (2002). A variational approach to spline functions theory. General Mathematics, 10(1):21–50. Santner, T., Williams, B., and Notz, W. (2003). The Design and Analysis of Computer Experiments. Springer Series in Statistics. Springer Verlag, New York. Sibson, R. (1980). A brief description of natural neighbor interpolation. In Barnett, V., editor, Interpreting multivariate data, pages 21–36. Wiley.