Abstract

Computer experiments are widely used to study complex real-world systems. Space-filling

designs, which aim to provide adequate coverage of the input domain, are commonly used to facilitate informative sampling and accurate surrogate modeling. Although maximin distance designs for

continuous variables have been extensively studied, general-purpose constructions for mixed input

types remain relatively limited. This paper develops interleaved lattice designs under the maximin

distance criterion for input spaces comprising continuous, ordinal, and binary variables. The proposed

framework accommodates a range of run sizes, combinations of variable types, and prespecified level

sets for ordinal variables. We develop three main construction algorithms, supplemented by seven

supporting procedures and corresponding theoretical results. These algorithms include an exhaustive search for low-dimensional problems and computationally tractable constructions for moderate-

and high-dimensional problems. In the numerical settings considered, the proposed designs attain

larger separation distances than the benchmark designs in most cases and yield lower or comparable

prediction errors under the specified Gaussian process emulators.

Key words and phrases: Maximin distance design, mixed-variable design, interleaved lattice

Information

Preprint No.SS-2025-0169
Manuscript IDSS-2025-0169
Complete AuthorsHui Lan, Xu He
Corresponding AuthorsXu He
Emailshexu@amss.ac.cn

References

  1. Ba, S., W. R. Myers, and W. A. Brenneman (2015). Optimal sliced Latin hypercube designs. Technometrics 57(4), 479–487.
  2. Conway, J. H. and N. J. A. Sloane (2013). Sphere packings, lattices and groups, Volume 290. Springer.
  3. Deng, X., Y. Hung, and C. D. Lin (2015). Design for computer experiments with qualitative and quantitative factors. Stat. Sin. 25(4), 1567–1581.
  4. Gaviano, M., D. E. Kvasov, D. Lera, and Y. D. Sergeyev (2003). Algorithm 829: Software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 29(4), 469–480.
  5. Gu, M., J. Palomo, and J. O. Berger (2019). RobustGaSP: Robust gaussian stochastic process emulation in R. The R Journal 11(1), 112–136.
  6. Haaland, B. and P. Z. G. Qian (2011). Accurate emulators for large-scale computer experiments. Ann. Statist. 39(6), 2974–3002.
  7. Haaland, B., W. Wang, and V. Maheshwari (2018). A framework for controlling sources of inaccuracy in Gaussian process emulation of deterministic computer experiments. SIAM/ASA J. Uncertain. Quantif. 6(2), 497–521.
  8. He, X. (2017). Interleaved lattice-based minimax distance designs. Biometrika 104(3), 713–725.
  9. He, X. (2019). Interleaved lattice-based maximin distance designs. Biometrika 106(2), 453–464.
  10. Johnson, M., L. Moore, and D. Ylvisaker (1990). Minimax and maximin distance designs. J. Stat. Plan. Infer. 26(2), 131–148.
  11. Joseph, V. R., E. Gul, and S. Ba (2015, 03).
  12. Maximum projection designs for computer experiments. Biometrika 102(2), 371–380.
  13. Joseph, V. R., E. Gul, and S. Ba (2020). Designing computer experiments with multiple types of factors: The MaxPro approach. J. Qual. Technol. 52(4), 343–354.
  14. Lekivetz, R. and B. Jones (2015). Fast flexible space-filling designs for nonrectangular regions. Qual. Reliab. Eng. Int. 31(5), 829–837.
  15. Li, W., M. Liu, and B. Tang (2021). A method of constructing maximin distance designs. Biometrika 108(4), 845–855.
  16. Morris, M. D. and T. J. Mitchell (1995). Exploratory designs for computational experiments. J. Stat. Plan. Infer. 43(3), 381–402.
  17. Mu, W. and S. Xiong (2017). On algorithmic construction of maximin distance designs. Commun. Stat.-Simul. Comput. 46(10), 7972–7985.
  18. Pelamatti, J., L. Brevault, M. Balesdent, E.-G. Talbi, and Y. Guerin (2021). Mixed variable gaussian process-based surrogate modeling techniques: Application to aerospace design. J. Aerosp. Inf. Syst. 18(11), 813–837.
  19. Qian, P. Z. G., H. Wu, and C. F. J. Wu (2008). Gaussian process models for computer experiments with qualitative and quantitative factors. Technometrics 50(3), 383–396.
  20. Santner, T. J., B. J. Williams, W. I. Notz, and B. J. Williams (2003). The design and analysis of computer experiments, Volume 1. Springer.
  21. Sobol’, I. M. (1993). Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1, 407.
  22. Stinstra, E., D. den Hertog, P. Stehouwer, and A. Vestjens (2003). Constrained maximin designs for computer experiments. Technometrics 45(4), 340–346.
  23. Sun, F., R. B. Gramacy, B. Haaland, S. Lu, and Y. Hwang (2019). Synthesizing simulation and field data of solar irradiance. Stat. Anal. Data Min. 12(4), 311–324.
  24. Trosset, M. W. et al. (1999). Approximate maximin distance designs. In Proceedings of the Section on Physical and Engineering Sciences, pp. 223–227.
  25. Wang, L., Q. Xiao, and H. Xu (2018). Optimal maximin L1-distance Latin hypercube designs based on good lattice point designs. Ann. Statist. 46(6B), 3741–3766.
  26. Wang, S. and R. Joseph (2025). SFDesign: Space-Filling Designs. R package version 0.1.2.
  27. Wu, C. J. and M. S. Hamada (2011). Experiments: planning, analysis, and optimization. John Wiley & Sons.
  28. Xiao, Q. and H. Xu (2017). Construction of maximin distance Latin squares and related Latin hypercube designs. Biometrika 104(2), 455–464.
  29. Yuan, R., Y. Yin, H. Xu, and M. Liu (2025). A construction method for maximin L1-distance Latin hypercube designs. Stat. Sin. 35(1), 1–24.
  30. Zhou, Q., P. Z. G. Qian, and S. Zhou (2011). A simple approach to emulation for computer models with qualitative and quantitative factors. Technometrics 53(3), 266–273.

Acknowledgments

The authors thank the three referees and the associate editor for their insightful comments. Xu He was supported by the National Natural Science Foundation of China (Grant

Nos. 12288201 and 12471247), the Strategic Priority Research Program of the Chinese

Academy of Sciences (Grant No. XDA0500400), and the CAS Project for Young Scientists

in Basic Research (Grant No. YSBR-149).

Supplementary Materials

The supplementary materials include proofs, a list of three-dimensional standard ILs, the

source code for the R package used to generate the proposed designs, and the scripts used

to reproduce the numerical studies.

Supplementary materials are available for download.