Statistics and Computing

, Volume 27, Issue 2, pp 319–329 | Cite as

A standardized distance-based index to assess the quality of space-filling designs

  • François WahlEmail author
  • Cécile Mercadier
  • Céline Helbert


One of the most used criterion for evaluating space-filling design in computer experiments is the minimal distance between pairs of points. The focus of this paper is to propose a normalized quality index that is based on the distribution of the minimal distance when points are drawn independently from the uniform distribution over the unit hypercube. Expressions of this index are explicitly given in terms of polynomials under any \(L_p\) distance. When the size of the design or the dimension of the space is large, approximations relying on extreme value theory are derived. Some illustrations of our index are presented on simulated data and on a real problem.


Minimal distance Maximin Space-filling design  Computer experiments Extreme value theory 



The authors are grateful to the Associate Editor and the anonymous referees for helpful suggestions which helped to greatly improve the initial text. Luc Pronzato deserves thanks for careful reading and interesting ideas on the occasion of a first version of the paper. Special thanks also go to John P. Nolan for his help on correcting our English.


  1. Auffray, Y., Barbillon, P., Marin, J.: Constrained maximin designs for computer experiments. Stat. Comput. 22, 703–712 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Carnell, R.: Latin Hypercube Sample. R package version 0.10 (2012).
  3. Damblin, G., Couplet, M., Iooss, B.: Numerical studies of space-filling designs: optimization of Latin Hypercube Samples and subprojection properties. J. Simul. 7, 276–289 (2013)CrossRefGoogle Scholar
  4. David, H.: Order Statistics. Wiley, New York (1980)Google Scholar
  5. Fang, K., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments. Chapman & Hall, London (2005)CrossRefzbMATHGoogle Scholar
  6. Franco, J., Dupuy, D., Roustant, O.: DiceDesign: Design of computer experiments. R package version 1.7 (2015).
  7. Husslage, B.: Maximin designs for computer experiments. PhD thesis Universiteit van Tilburg (2006)Google Scholar
  8. Jin, R., Chen, W., Sudjianto, A.: An efficient algorithm for constructing optimal design of computer experiments. J. Stat. Plan. Inf. 134, 268–287 (2005)Google Scholar
  9. Magand, S., Pidol, L., Chaudoye, F., Sinoquet, D., Wahl, F., Castagne, M., Lecointe, B.: Use of ethanol/diesel blend and advanced calibration methods to satisfy Euro 5 emission standards without a DPF. Oil Gas Sci. Technol. 66, 855–875 (2011)CrossRefGoogle Scholar
  10. Mckay, M.D., Beckman, R.J., Conover, W.J.: Constrained maximin designs for computer experiments. Technometrics 21, 239–245 (1979)MathSciNetzbMATHGoogle Scholar
  11. Niederreiter, H.: Low-discrepancy and low-dispersion sequences. J. Number Theory 30, 51–70 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Petelet, M., Iooss, B., Asserin, O., Loredo, A.: Latin hypercube sampling with inequality constraints. Adv. Stat. Anal. 3, 11–21 (2010)Google Scholar
  13. Philip, J.: The probability distribution of the distance between two random points in a box. TRITA MAT 10 (2007)Google Scholar
  14. Philip, J.: The distance between two random points in a 4- and 5-cube. J. Chemom. 21, 198–207 (2010)Google Scholar
  15. Pronzato, L., Muller, G.: Design of computer experiments: space-filling and beyond. Stat. Comput. 23, 681–701 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4, 409–442 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  18. Stinstra, E., den Hertog, D., Stehouwer, P., Vestjens, A.: Constrained maximin designs for computer experiments. Oper. Res. 57, 595–608 (2003)Google Scholar
  19. Van Dam, E., Rennen, G., Husslage, B.: Bounds for maximin latin hypercube designs. Oper. Res. 57, 595–608 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • François Wahl
    • 1
    • 2
    Email author
  • Cécile Mercadier
    • 1
  • Céline Helbert
    • 1
  1. 1.Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille JordanVilleurbanneFrance
  2. 2.IFPENSolaizeFrance

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