Optimal maximin \(L_{1}\)-distance Latin hypercube designs based on good lattice point designs. (English) Zbl 1411.62238

Authors’ abstract: Maximin distance Latin hypercube designs are commonly used for computer experiments, but the construction of such designs is challenging. We construct a series of maximin Latin hypercube designs via Williams transformations of good lattice point designs. Some constructed designs are optimal under the maximin \(L_{1}\)-distance criterion, while others are asymptotically optimal. Moreover, these designs are also shown to have small pairwise correlations between columns.


62K10 Statistical block designs
62K05 Optimal statistical designs
05B15 Orthogonal arrays, Latin squares, Room squares
Full Text: DOI Euclid


[1] Ba, S., Myers, W. R. and Brenneman, W. A. (2015). Optimal sliced Latin hypercube designs. Technometrics57 479–487.
[2] Bailey, R. A. (1982). The decomposition of treatment degrees of freedom in quantitative factorial experiments. J. Roy. Statist. Soc. Ser. B44 63–70. · Zbl 0481.62062
[3] Butler, N. A. (2001). Optimal and orthogonal Latin hypercube designs for computer experiments. Biometrika88 847–857. · Zbl 0985.62058
[4] Chen, R.-B., Hsieh, D.-N., Hung, Y. and Wang, W. (2013). Optimizing Latin hypercube designs by particle swarm. Stat. Comput.23 663–676. · Zbl 1322.90005
[5] Cioppa, T. M. and Lucas, T. W. (2007). Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics49 45–55.
[6] Edmondson, R. N. (1993). Systematic row-and-column designs balanced for low order polynomial interactions between rows and columns. J. R. Stat. Soc. Ser. B. Stat. Methodol.55 707–723. · Zbl 0794.62044
[7] Fang, K.-T., Li, R. and Sudjianto, A. (2006). Design and Modeling for Computer Experiments. Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1093.62117
[8] Fang, K. T. and Wang, Y. (1994). Number-Theoretic Methods in Statistics. Chapman & Hall, London. · Zbl 0925.65263
[9] Georgiou, S. D. and Efthimiou, I. (2014). Some classes of orthogonal Latin hypercube designs. Statist. Sinica24 101–120. · Zbl 06290088
[10] He, Y., Cheng, C. S. and Tang, B. (2018). Strong orthogonal arrays of strength two plus. Ann. Statist.46 457–468. · Zbl 1395.62234
[11] He, Y. and Tang, B. (2013). Strong orthogonal arrays and associated Latin hypercubes for computer experiments. Biometrika100 254–260. · Zbl 1284.62487
[12] He, Y. and Tang, B. (2014). A characterization of strong orthogonal arrays of strength three. Ann. Statist.42 1347–1360. · Zbl 1306.62188
[13] Johnson, M. E., Moore, L. M. and Ylvisaker, D. (1990). Minimax and maximin distance designs. J. Statist. Plann. Inference26 131–148.
[14] Joseph, V. R. and Hung, Y. (2008). Orthogonal-maximin Latin hypercube designs. Statist. Sinica18 171–186. · Zbl 1137.62050
[15] Korobov, N. M. (1959). The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR124 1207–1210. · Zbl 0089.04201
[16] Lin, C. D., Mukerjee, R. and Tang, B. (2009). Construction of orthogonal and nearly orthogonal Latin hypercubes. Biometrika96 243–247. · Zbl 1161.62044
[17] Lin, C. D. and Tang, B. (2015). Latin hypercubes and space-filling designs. In Handbook of Design and Analysis of Experiments (A. Dean, M. Morris, J. Stufken and D. Bingham, eds.) 593–625. Chapman & Hall/CRC, London. · Zbl 1352.62128
[18] Moon, H., Dean, A. and Santner, T. (2011). Algorithms for generating maximin Latin hypercube and orthogonal designs. J. Statist. Plann. Inference5 81–98. · Zbl 05902637
[19] Morris, M. D. (1991). Factorial plans for preliminary computational experiments. Technometrics33 161–174.
[20] Morris, M. D. and Mitchell, T. J. (1995). Exploratory designs for computational experiments. J. Statist. Plann. Inference43 381–402. · Zbl 0813.62065
[21] Morris, M. D. and Moore, L. M. (2015). Design of computer experiments: Introduction and background. In Handbook of Design and Analysis of Experiments (A. Dean, M. Morris, J. Stufken and D. Bingham, eds.) 577–591. CRC Press, Boca Raton, FL. · Zbl 1369.62194
[22] Sacks, J., Schiller, S. B. and Welch, W. J. (1989). Designs for computer experiments. Technometrics31 41–47.
[23] Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments. Springer, New York. · Zbl 1041.62068
[24] Steinberg, D. M. and Lin, D. KJ. (2006). A construction method for orthogonal Latin hypercube designs. Biometrika93 279–288. · Zbl 1153.62349
[25] Sun, F. S., Liu, M. Q. and Lin, D. K. J. (2009). Construction of orthogonal Latin hypercube designs. Biometrika96 971–974. · Zbl 1178.62083
[26] Sun, F. S. and Tang, B. (2017). A general rotation method for orthogonal Latin hypercubes. Biometrika104 465–472.
[27] Tang, B. (1993). Orthogonal array-based Latin hypercubes. J. Amer. Statist. Assoc.88 1392–1397. · Zbl 0792.62066
[28] Williams, E. J. (1949). Experimental designs balanced for the estimation of residual effects of treatments. Aust. J. Sci. Res.2 149–168.
[29] Xiao, Q. and Xu, H. (2017). Construction of maximin distance Latin squares and related Latin hypercube designs. Biometrika104 455–464.
[30] Xiao, Q. and Xu, H. (2018). Construction of maximin distance designs via level permutation and expansion. Statist. Sinica. 28 1395–1414. · Zbl 1394.62109
[31] Xu, H. (1999). Universally optimal designs for computer experiments. Statist. Sinica9 1083–1088. · Zbl 0940.62068
[32] Yang, J. Y. and Liu, M. Q. (2012). Construction of orthogonal and nearly orthogonal Latin hypercube designs from orthogonal designs. Statist. Sinica22 433–442. · Zbl 06013167
[33] Ye, K. Q. (1998). Orthogonal column Latin hypercubes and their application in computer experiments. J. Amer. Statist. Assoc.93 1430–1439. · Zbl 1064.62553
[34] Zhou, Y. and Xu, H. (2015). Space-filling properties of good lattice point sets. Biometrika102 959–966. · Zbl 1372.62036
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