## 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.

### MSC:

 62K10 Statistical block designs 62K05 Optimal statistical designs 05B15 Orthogonal arrays, Latin squares, Room squares
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### References:

 [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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