Statistics and Computing

, Volume 27, Issue 4, pp 1083–1097 | Cite as

Computer experiments with functional inputs and scalar outputs by a norm-based approach

  • Thomas MuehlenstaedtEmail author
  • Jana Fruth
  • Olivier Roustant


A framework for designing and analyzing computer experiments is presented, which is constructed for dealing with functional and scalar inputs and scalar outputs. For designing experiments with both functional and scalar inputs, a two-stage approach is suggested. The first stage consists of constructing a candidate set for each functional input. During the second stage, an optimal combination of the found candidate sets and a Latin hypercube for the scalar inputs is sought. The resulting designs can be considered to be generalizations of Latin hypercubes. Gaussian process models are explored as metamodel. The functional inputs are incorporated into the Kriging model by applying norms in order to define distances between two functional inputs. We propose the use of B-splines to make the calculation of these norms computationally feasible.


Space-filling design Gaussian process Maximin design 



The authors thank Andon Iyassu and Leo Geppert for their help on editing. This paper is based on investigations of the collaborative research centre SFB 708, project C3. The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for funding.

Supplementary material

11222_2016_9672_MOESM1_ESM.r (59 kb)
Supplementary material 1 (r 59 KB)
11222_2016_9672_MOESM2_ESM.r (14 kb)
Supplementary material 2 (r 14 KB)


  1. Bastos, L.S., O’Hagan, A.: Diagnostics for gaussian process emulators. Technometrics 51(4), 425–438 (2009)MathSciNetCrossRefGoogle Scholar
  2. Bayarri, M.J., Berger, J.O., Cafeo, J., Garcia-Donato, G., Liu, F., Palomo, J., Parthasarathy, R.J., Paulo, R., Sacks, J., Walsh, D.: Computer model validation with functional output. Ann. Stat. 35, 1874–1906 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. Currin, C., Mitchell, T., Morris, M., Ylvisaker, D.: Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments. J. Am. Stat. Assoc. 86(416), 953–963 (1991)MathSciNetCrossRefGoogle Scholar
  4. de Boor, C.: A Practical Guide to Splines. Springer, New York (2001)zbMATHGoogle Scholar
  5. den Hertog, D., Kleijnen, J.P.C., Siem, A.Y.D.: The correct kriging variance estimated by bootstrapping. J. Oper. Res. Soc. 57(4), 400–409 (2005)zbMATHCrossRefGoogle Scholar
  6. Dixon, L.C.W., Szego, G.P.: The global optimization problem: an introduction. Towards Glob. Optim. 2, 1–15 (1978)Google Scholar
  7. Fang, K.-T., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments, Computer Science and Data Analysis Series. Chapman & Hall/CRC, New York (2006)zbMATHGoogle Scholar
  8. Johnson, M., Moore, L., Ylvisaker, D.: Minimax and maximin distance designs. J. Stat. Plan. Inference 26, 131–148 (1990)Google Scholar
  9. Jones, D., Schonlau, M., Welch, W.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13, 455–492 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. Leitenstorfer, F., Tutz, G.: Generalized monotonic regression based on b-splines with an application to air pollution data. Biostatistics 8, 654–673 (2007)zbMATHCrossRefGoogle Scholar
  11. Loeppky, J.L., Sacks, J., Welch, W.J.: Choosing the sample size of a computer experiment: A practical guide. Technometrics 51(4), 366–376 (2009)MathSciNetCrossRefGoogle Scholar
  12. McKay, M., Beckman, R., Conover, W.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)MathSciNetzbMATHGoogle Scholar
  13. Morris, M.: Gaussian surrogates for computer models with time-varying inputs and ouptputs. Technometrics 54, 42–50 (2012)MathSciNetCrossRefGoogle Scholar
  14. Morris, M.: Maximin distance optimal designs for computer experiments with time-varying inputs and outputs. J. Stat. Plan. Inference 144, 63–68 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Morris, M., Mitchell, T.: Exploratory designs for computational experiments. J. Stat. Plan. Inference 43, 381–402 (1995)zbMATHCrossRefGoogle Scholar
  16. R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2013). ISBN 3-900051-07-0Google Scholar
  17. Ramsay, J., Silverman, B.: Functional Data Analysis. Springer, New York (1997)zbMATHCrossRefGoogle Scholar
  18. Rasmussen, C., Williams, C.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  19. Roustant, O., Ginsbourger, D., Deville, Y.: DiceKriging, DiceOptim: Two R packages for the analysis of computer experiments by kriging-based metamodeling and optimization. J. Stat. Softw. 51(1), 1–55 (2012)CrossRefGoogle Scholar
  20. Sacks, J., Schiller, S., Welch, W.: Design for computer experiments. Technometrics 31, 41–47 (1989a)MathSciNetCrossRefGoogle Scholar
  21. Sacks, J., Welch, W., Mitchell, T., Wynn, H.: Design and analysis of computer experiments. Stat. Sci. 4, 409–435 (1989b)MathSciNetzbMATHCrossRefGoogle Scholar
  22. Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer Series in Statistics. Springer, New York (2003)zbMATHCrossRefGoogle Scholar
  23. Shi, J.Q., Shoi, T.: Gaussian Process Regression Analysis for Functional Data. Chapman & Hall, London (2011)Google Scholar
  24. Shi, J.Q., Wang, B., Murray-Smith, R., Titterington, D.M.: Gaussian process functional regression modeling for batch data. Biometrics 63(3), 714–723 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. Tan, M.H.Y.: Minimax designs for finite design regions. Technometrics 55(3), 346–358 (2013)MathSciNetCrossRefGoogle Scholar
  26. ul Hassan, H., Fruth, J., Güner, A., Tekkaya, A.E.: Finite element simulations for sheet metal forming process with functional input for the minimization of springback. In: IDDRG conference 2013, pp. 393–398. (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Thomas Muehlenstaedt
    • 1
    Email author
  • Jana Fruth
    • 2
  • Olivier Roustant
    • 3
  1. 1.W. L. Gore & AssociatesNewarkUSA
  2. 2.Faculty of StatisticsTU DortmundDortmundGermany
  3. 3.Ecole Nationale Superieure des Mines de Saint-Etienne, FAYOL-EMSE, LSTISaint-EtienneFrance

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