An Empirical Study of the Prediction Performance of Space-filling Designs

Rachel T. Johnson

Douglas C. Montgomery

Bradley Jones

Computer Models

• Widely used in engineering design

• Use continues to grow

• May have lots of variables, many responses

• Can have long run times

• Complex output results

• Need efficient methods for designing the experiment and analyzing the results

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Computer Simulation Types

Discrete Event

Simulation(DES)

Computational Fluid

Dynamics(CFD)

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Computer Simulation Models

Stochastic Simulation Models

Deterministic Simulation Models

Comparing Designs

• Space-filling designs compared– Sphere packing (SP)– Latin Hypercube (LH)– Uniform (U) – Gaussian Process Integrated Mean Square Error

(GP IMSE)

• Assumed surrogate model– Gaussian Process model

• Comparisons based on prediction

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Designs

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-1

-0.5

0

0.5

1

X2

-1 -0.5 0 0.5 1

X1

Sphere Packing Sphere Packing:• Johnson et al. (1990)• Maximizes the minimum distance between pairs of design points

-1

-0.5

0

0.5

1

X2

-1 -0.5 0 0.5 1

X1

Maximum Entropy

-1

-0.5

0

0.5

1

X2

-1 -0.5 0 0.5 1

X1

Latin HypercubeLatin Hypercube:• McKay et al. (1979)• A random n x s matrix, in which columns are a random permutation of {1, . . ., n}

-1

-0.5

0

0.5

1

X2

-1 -0.5 0 0.5 1

X1

Uniform

Uniform:• Feng (1980)• A set of n points uniformly scattered within the design space

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

X2

-0.75 -0.5 -0.25 0 0.25 0.5

X1

GASP IMSE

GP IMSE:• Sacks et al. (1989) • Minimizes the integrated mean square error of the Gaussian Process model

-1

-0.5

0

0.5

1

X2

-1 -0.5 0 0.5 1

X1

I - OptimalI - Optimal:• Box and Draper (1963)• Minimizes the average prediction variance (of a linear regression model) over a design region

Maximum Entropy:• Shewry and Wynn (1987) • Maximizes the information contained in the distribution of a data set

Model Fitting Techniques

• Gaussian Process Model

– Assumes a normal distribution

– Interpolator based on correlation between points – Prediction variance calculated as

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Evaluation of Designs for the GP Model

• Recall the prediction variance:

• Prediction variance dependent on: – Design – Value of unknown θ– Sample size– Dimension of x

• Design of Experiments can be used to evaluate the designs

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Example FDS Plots

Sphere Packing

Uniform

GP IMSE

Latin Hypercube

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Maximum Entropy

Research Questions

• Does it matter in practice what design you choose? – Is there a dominating experimental design that performs better in terms

of model fitting and prediction?

• What is the role of sample size in experimental designs used to fit the GASP model? – At what point does prediction error variance and other measures of

prediction performance become reasonably small with respect to N, the sample size chosen?

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Empirical Comparison

• Used test functions to act as surrogates to simulation code

• Evaluated designs based on RMSE and AAPE

• Interested in effect of: – Design type – Sample size – Dimension of design – *Gaussian Correlation Function

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Comparison Procedure

• Step 1: Choose a test function

• Step 2: Choose a sample size and space-filling design

• Step 3: Create the design with the number of factors equal to the number in the test function chosen in step 1) and the specifications set in step 2)

• Step 4: Using the test function in 1) find the values that correspond to each row in the design

• Step 5: Fit the GASP model

• Step 6: Generate a set of 40,000 uniformly random selected points in the design space and compare the predicted value (generated by the fitted GASP model) to the actual value (generated by the test function) at these points. Compute the Root mean square error (RMSE).

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Test Function #4

Test Function #3

Test Function #2

Test Function #1

12

Test Functions

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Test Function #1 Results

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RM

SE

-1

0

1

2

3

4

5

6

7

8

9

10

11

GP

IMS

E

LHD

SP U

GP

IMS

E

LHD

SP U

GP

IMS

E

LHD

SP U

SS_10 SS_20 SS_40

Design within Sample Size

Test Function #2 Results

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RM

SE

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

GP

IM

SE

LH

D

SP U

GP

IM

SE

LH

D

SP U

GP

IM

SE

LH

D

SP U

SS_10 SS_20 SS_40

Design within Sample Size

Test Function #3 Results

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RM

SE

0

5

10

15

20

GP

IMS

E

LHD

SP U

GP

IMS

E

LHD

SP U

GP

IMS

E

LHD

SP U

SS_10 SS_20 SS_40

Design within Sample Size

Test Function #2 Results

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RM

SE

0

10

20

30

40

50

GP

IMS

E

LHD

SP U

GP

IMS

E

LHD

SP U

GP

IMS

E

LHD

SP U

SS_10 SS_20 SS_40

Design within Sample Size

ANOVA Analysis

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-4

-2

0

2

4

Log(

RM

SE

)

-0.3

1293

±0.1

7581

5

TF 1

TF 2

TF 3

TF 4

TF 1

TF

GP

IMS

E

LHD S

P U

GP IMSE

Design

10 15 20 25 30 35 40

23.333

Sample Size

Jackknife Plots

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-5

0

5

10

Tes

t F

un

ctio

n 2

-5 0 5 10

Test Function 2

Jackknife Predicted

-10

-5

0

5

10

Test

Fun

ctio

n 2

-10 -5 0 5 10

Test Function 2

Jackknife Predicted

Test Function #2 – LHD with a sample size of 20

Test Function #2 – LHD with a sample size of 50

Conclusions

• No one design type is better than the others• Increasing sample size decreases RMSE• There is a strong interaction between sample size and

test function type – the more complex the function the more runs required

• Jackknife plot is an excellent indicator of a “good fit”

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References

• Fang, K.T. (1980). “The Uniform Design: Application of Number-Theoretic Methods in Experimental Design,” Acta Math. Appl. Sinica. 3, pp. 363 – 372.

• Johnson, M.E., Moore, L.M. and Ylvisaker, D. (1990). “Minimax and maxmin distance design,” Journal for Statistical Planning and Inference 26, pp. 131 – 148.

• McKay, N. D., Conover, W. J., Beckman, R. J. (1979). “A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code,” Technometrics 21, pp. 239 – 245.

• Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989). “Design and Analysis of Computer Experiments,” Statistical Science 4(4), pp. 409 – 423.

• Shewry, M.C. and Wynn, H.P. (1987). “Maximum entropy sampling,” Journal of Applied Statistics 14, pp. 898 – 914.

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Correlation Function

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Gaussian Correlation Function Cubic Correlation Function

Sample Size

Design

SS 50

SS 40

SS 20

SS 10

RMSE

GCF

RMSE

CCF

RMSE

GCF

RMSE

CCF

RMSE

GCF

RMSE

CCF

RMSE

GCF

RMSE

CCF

4

3

2

1

0

RM

SE

Boxplot of RMSE