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Design and Analysis of Complex Computer Models

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Advances in Computational Modeling and Simulation

Part of the book series: Lecture Notes in Mechanical Engineering ((LNME))

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Abstract

This chapter presents a review of some state-of-the-art statistical techniques for analyzing real computer experiments which play a significant role in various scientific research and industrial applications. In computer experiments, emulators (i.e. surrogate models) are often used to rapidly approximate the outcomes and reduce the computational expense. Gaussian process (GP) models, also known as Kriging, are a common choice of emulators, and optimal experimental designs should be used to improve their accuracy. Specifically, space-filling designs are widely used in the literature, which proved to be efficient under GP models. In this chapter, we review different types of GP models as well as various kinds of space-filling designs. We further provide a practical tutorial on how to construct space-filling designs and fit GP emulators to analyze real computer experiments.

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References

  1. Bhattacharjee NV, Ranjan P, Mandal A, Tollner E (2019) A history matching approach for calibrating hydrological models. Environ Ecol Stat 26:87–105

    Article  MathSciNet  Google Scholar 

  2. Butler NA (2001) Optimal and orthogonal Latin hypercube designs for computer experiments. Biometrika 88(3):847–857

    Google Scholar 

  3. Chen R-B, Hsieh D-N, Hung Y, Wang W (2013) Optimizing Latin hypercube designs by particle swarm. Stat Comput 23(5):663–676

    Article  MathSciNet  Google Scholar 

  4. Chen Y, Welling M, Smola A (2010) Super-samples from kernel herding. In: UAI

    Google Scholar 

  5. Cioppa TM, Lucas TW (2007) Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics 49(1):45–55

    Google Scholar 

  6. Cui J, Krems RV (2015) Gaussian process model for collision dynamics of complex molecules. Phys Rev Lett 115(7)

    Google Scholar 

  7. Deisenroth M, Fox D, Rasmussen C (2015) Gaussian processes for data-efficient learning in robotics and control. IEEE Trans Pattern Anal Mach Intell 37:408–423

    Article  Google Scholar 

  8. Deng F, Deng Q, Shen S (2018) A three-dimensional mixed finite element for flexoelectricity. J Appl Mech 85(3)

    Google Scholar 

  9. Doob JL, Doob JL (1953) Stochastic Processes. Wiley, Probability and Statistics Series

    Google Scholar 

  10. Ek CH, Torr PH, Lawrence ND (2008) Gaussian process latent variable models for human pose estimation. In: Popescu-Belis A, Renals S, Bourlard H (eds) Machine learning for multimodal interaction. Berlin, Heidelberg, Springer Berlin Heidelberg, pp 132–143

    Google Scholar 

  11. Fang K-T (1980) The uniform design: application of number-theoretic methods in experimental design. Acta Math Appl Sin 3(4):9

    Google Scholar 

  12. Fang KT, Li R, Sudjianto A (2005) Design and modeling for computer experiments. Chapman and Hall/CRC

    Google Scholar 

  13. Fang KT, Li R, Sudjianto A (2005) Design and modeling for computer experiments. CRC Press

    Google Scholar 

  14. Fang KT, Lin DK, Winker P, Zhang Y (2000) Uniform design: theory and application. Technometrics 42(3):237–248

    Google Scholar 

  15. Fang K-T, Ma C-X, Winker P (2002) Centered \({L}_{2}\)-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs. Math Comput 71(237):275–296

    Article  MathSciNet  Google Scholar 

  16. Georgiou SD (2009) Orthogonal Latin hypercube designs from generalized orthogonal designs. J Stat Plan Inference 139(4):1530–1540

    Google Scholar 

  17. Georgiou SD, Efthimiou I (2014) Some classes of orthogonal Latin hypercube designs. Stat Sin 24(1):101–120

    Google Scholar 

  18. Gramacy RB, Lee HK (2007) Bayesian treed gaussian process models with an application to computer modeling. J Am Stat Assoc 103:1119–1130

    Google Scholar 

  19. Gramacy RB (2019) laGP: large-scale spatial modeling via local approximate gaussian processes in R. J Stat Softw 72(1):1–46

    MathSciNet  Google Scholar 

  20. Gramacy RB (2020) Surrogates: gaussian process modeling, design and optimization for the applied sciences. Chapman Hall/CRC, Boca Raton, Florida. http://bobby.gramacy.com/surrogates/

  21. Gramacy RB, Apley DW (2015) Local gaussian process approximation for large computer experiments. J Comput Graph Stat 24(2):561–578

    Article  MathSciNet  Google Scholar 

  22. Han G, Santner TJ, Notz WI, Bartel DL (2009) Prediction for computer experiments having quantitative and qualitative input variables. Technometrics 51(3):278–288

    Article  MathSciNet  Google Scholar 

  23. Hensman J, Fusi N, Lawrence ND (2013) Gaussian processes for big data. In: Technical report UAI-P-2013-PG-282-290, twenty-ninth conference on uncertainty in artificial intelligence (UAI2013)

    Google Scholar 

  24. Jin R, Chen W, Sudjianto A (2005) An efficient algorithm for constructing optimal design of computer experiments. J Stat Plan Inference 134(1):268–287

    Article  MathSciNet  Google Scholar 

  25. Johnson ME, Moore LM, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Inference 26(2):131–148

    Google Scholar 

  26. Joseph VR, Gul E, Ba S (2015) Maximum projection designs for computer experiments. Biometrika 102(2):371–380

    Google Scholar 

  27. Joseph VR, Hung Y (2008) Orthogonal-maximin Latin hypercube designs. Stat Sin 171–186

    Google Scholar 

  28. Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc: Series B (Statistical Methodology) 63(3):425–464

    Article  MathSciNet  Google Scholar 

  29. Kidder LE, Scheel MA, Teukolsky SA, Carlson ED, Cook GB (2000) Black hole evolution by spectral methods. Phys Rev D 62:084032

    Google Scholar 

  30. Kruckow M, Tauris T, Langer N, Kramer M, Izzard R (2018) Progenitors of gravitational wave mergers: binary evolution with the stellar grid-based code combine. Mon Not R Astron Soc 481:1908–1949

    Article  Google Scholar 

  31. Leary S, Bhaskar A, Keane A (2003) Optimal orthogonal-array-based Latin hypercubes. J Appl Stat 30(5):585–598

    Article  MathSciNet  Google Scholar 

  32. Liefvendahl M, Stocki R (2006) A study on algorithms for optimization of Latin hypercubes. J Stat Plan Inference 136(9):3231–3247

    Article  MathSciNet  Google Scholar 

  33. Lin CD, Mukerjee R, Tang B (2009) Construction of orthogonal and nearly orthogonal Latin hypercubes. Biometrika 96(1):243–247

    Google Scholar 

  34. Lin L, Bingham D, Broekgaarden F, Mandel I (2021) Uncertainty quantification of a computer model for binary black hole formation

    Google Scholar 

  35. Liu H, Ong YS, Shen X, Cai J (2020) When gaussian process meets big data: a review of scalable GPS. IEEE Trans Neural Netw Learn Syst 31:4405–4423

    Google Scholar 

  36. MacDonald B, Ranjan P, Chipman H (2015) GPfit: an R package for fitting a gaussian process model to deterministic simulator outputs. J Stat Softw 64(12):1–23

    Article  Google Scholar 

  37. Mak S, Joseph VR (2018) Support points. Ann Stat

    Google Scholar 

  38. McKay MD, Beckman RJ, Conover WJ (1979) Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245

    Google Scholar 

  39. Morris MD, Mitchell TJ (1995) Exploratory designs for computational experiments. J Stat Plan Inference 43(3):381–402

    Google Scholar 

  40. Nocedal J, Wright S (2006) Numerical optimization. Springer series in operations research and financial engineering. Springer, New York

    Google Scholar 

  41. Qian PZ, Wu H, Wu CJ (2008) Gaussian process models for computer experiments with qualitative and quantitative factors. Technometrics 50(3):383–396

    Google Scholar 

  42. Roustant O, Ginsbourger D, Deville Y (2012) DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by kriging-based metamodeling and optimization. J Stat Softw 51(1):1–55

    Article  Google Scholar 

  43. Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 409–423

    Google Scholar 

  44. Brenneman WA, Ba S, Myers WR (2015) Slhd: maximin-distance (sliced) Latin hypercube designs. Technometrics 57(4):479–487

    Article  MathSciNet  Google Scholar 

  45. Steinberg DM, Lin DK (2006) A construction method for orthogonal Latin hypercube designs. Biometrika 93(2):279–288

    Google Scholar 

  46. Sun F, Liu MQ, Lin DK (2009) Construction of orthogonal Latin hypercube designs. Biometrika 96(4):971–974

    Google Scholar 

  47. Sun F, Liu MQ, Lin DK (2010) Construction of orthogonal Latin hypercube designs with flexible run sizes. J Stat Plan Inference 140(11):3236–3242

    Google Scholar 

  48. Sun F, Tang B (2017) A general rotation method for orthogonal Latin hypercubes. Biometrika 104(2):465–472

    MathSciNet  MATH  Google Scholar 

  49. Sung CL, Hung Y, Rittase W, Zhu C, Jeff Wu CF (2017) A generalized gaussian process model for computer experiments with binary time series. J Am Stat Assoc 115:945–956

    Google Scholar 

  50. Swiler LP, Hough PD, Qian P, Xu X, Storlie C, Lee H (2014) Surrogate models for mixed discrete-continuous variables. In: Constraint programming and decision making, Studies in computational intelligence, pp 181–202. Springer

    Google Scholar 

  51. Tang B (1993) Orthogonal array-based Latin hypercubes. J Am Stat Assoc 88(424):1392–1397

    Article  MathSciNet  Google Scholar 

  52. Velásquez RM, Lara JV (2020) Forecast and evaluation of covid-19 spreading in USA with reduced-space gaussian process regression. Chaos Solitons Fractals 136:109924

    Google Scholar 

  53. Vernon I, Goldstein M, Bower R (2009) Galaxy formation: a bayesian uncertainty analysis. Bayesian Anal 5:619–669

    MathSciNet  MATH  Google Scholar 

  54. Wang H, Xiao Q, Mandal A (2020) LHD: Latin Hypercube Designs (LHDs) algorithms. R package version 1.2.0

    Google Scholar 

  55. Wang H, Xiao Q, Mandal A (2021) Musings about constructions of efficient Latin hypercube designs with flexible run-sizes

    Google Scholar 

  56. Wang L, Xiao Q, Hongquan X (2018) Optimal maximin \({L}_{1}\)-distance Latin hypercube designs based on good lattice point designs. Ann Stat 46(6B):3741–3766

    Article  MathSciNet  Google Scholar 

  57. Warnock TT (1972) Computational investigations of low-discrepancy point sets. In: Applications of number theory to numerical analysis, pp 319–343. Elsevier

    Google Scholar 

  58. Williams D, Heng IS, Gair J, Clark JA, Khamesra B (2020) Precessing numerical relativity waveform surrogate model for binary black holes: a gaussian process regression approach. Phys Rev D 101:063011. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.063011

  59. Williams EJ (1949) Experimental designs balanced for the estimation of residual effects of treatments. Aust J Chem 2(2):149–168

    Article  MathSciNet  Google Scholar 

  60. Xiao Q, Mandal A, Lin CD, Deng X (2021) Ezgp: easy-to-interpret gaussian process models for computer experiments with both quantitative and qualitative factors. SIAM/ASA J Uncertain Quantif 9(2):333–353

    Google Scholar 

  61. Yang J, Liu MQ (2012) Construction of orthogonal and nearly orthogonal Latin hypercube designs from orthogonal designs. Stat Sin 433–442

    Google Scholar 

  62. Ye KQ (1998) Orthogonal column Latin hypercubes and their application in computer experiments. J Am Stat Assoc 93(444):1430–1439

    Google Scholar 

  63. Zhang MM, Dumitrascu B, Williamson SA, Engelhardt BE (2019) Sequential gaussian processes for online learning of nonstationary functions. arXiv:abs/1905.10003

  64. Zhang Q, Chien P, Liu Q, Xu L, Hong Y (2020) Mixed-input gaussian process emulators for computer experiments with a large number of categorical levels. J Qual Technol 1–11

    Google Scholar 

  65. Zhang Y, Tao S, Chen W, Apley DW (2020) A latent variable approach to gaussian process modeling with qualitative and quantitative factors. Technometrics 62(3):291–302

    Article  MathSciNet  Google Scholar 

  66. Zhou Q, Qian PZ, Zhou S (2011) A simple approach to emulation for computer models with qualitative and quantitative factors. Technometrics 53(3):266–273

    Google Scholar 

  67. Zhou Y, Hongquan X (2015) Space-filling properties of good lattice point sets. Biometrika 102(4):959–966

    Article  MathSciNet  Google Scholar 

  68. Zhu Y, Fujimura K (2010) A bayesian framework for human body pose tracking from depth image sequences. Sensors (Basel, Switzerland) 10:5280–5293

    Google Scholar 

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Authors and Affiliations

  1. University of Georgia, Athens, GA, 30602, Georgia

    Jeevan Jankar, Hongzhi Wang, Lauren Rose Wilkes, Qian Xiao & Abhyuday Mandal

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  1. Jeevan Jankar
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  2. Hongzhi Wang
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  4. Qian Xiao
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  5. Abhyuday Mandal
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Correspondence to Abhyuday Mandal .

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Editors and Affiliations

  1. Department of Civil Engineering, Birla Institute of Technology and Scienc, Pilani, Rajasthan, India

    Rallapalli Srinivas

  2. Department of Mathematics, Birla Institute of Technology and Scienc, Pilani, Rajasthan, India

    Rajesh Kumar

  3. Department of Biotechnology, Birla Institute of Technology and Scienc, Dubai, United Arab Emirates

    Mainak Dutta

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Jankar, J., Wang, H., Wilkes, L.R., Xiao, Q., Mandal, A. (2022). Design and Analysis of Complex Computer Models. In: Srinivas, R., Kumar, R., Dutta, M. (eds) Advances in Computational Modeling and Simulation. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-16-7857-8_2

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  • DOI: https://doi.org/10.1007/978-981-16-7857-8_2

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