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An Empirical Study of the Prediction Performance of Space-filling Designs Rachel T. Johnson Douglas C. Montgomery Bradley Jones

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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 2 Johnson QPRC 2009

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Computer Simulation Types Discrete Event Simulation (DES) Computational Fluid Dynamics (CFD) 3 Johnson QPRC 2009 Computer Simulation Models Stochastic Simulation Models Deterministic Simulation Models

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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 Johnson QPRC

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Designs Johnson QPRC Sphere Packing Sphere Packing: Johnson et al. (1990) Maximizes the minimum distance between pairs of design points Maximum Entropy Latin Hypercube Latin Hypercube: McKay et al. (1979) A random n x s matrix, in which columns are a random permutation of {1,..., n} Uniform Uniform: Feng (1980) A set of n points uniformly scattered within the design space GASP IMSE GP IMSE: Sacks et al. (1989) Minimizes the integrated mean square error of the Gaussian Process model I - Optimal I - 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

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Model Fitting Techniques Gaussian Process Model –Assumes a normal distribution –Interpolator based on correlation between points –Prediction variance calculated as 6 Johnson QPRC 2009

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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 7 Johnson QPRC 2009

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Example FDS Plots Sphere Packing Uniform GP IMSE Latin Hypercube 8 Johnson QPRC 2009 Maximum Entropy

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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? Johnson QPRC

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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 Johnson QPRC

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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). Johnson QPRC

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Test Function #4 Test Function #3 Test Function #2 Test Function #1 12 Test Functions Johnson QPRC 2009

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Test Function #1 Results Johnson QPRC

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Test Function #2 Results Johnson QPRC

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Test Function #3 Results Johnson QPRC

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Test Function #2 Results Johnson QPRC

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ANOVA Analysis Johnson QPRC

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Jackknife Plots Johnson QPRC Test Function #2 – LHD with a sample size of 20 Test Function #2 – LHD with a sample size of 50

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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 Johnson QPRC

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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 – Johnson QPRC 2009

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Correlation Function Johnson QPRC Gaussian Correlation FunctionCubic Correlation Function

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