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Connecting U-type designs before and after level permutations and expansions. (English) Zbl 1476.62168

Summary: In both physical and computer experiments, U-type designs, including Latin hypercube designs, are commonly used. Two major approaches for evaluating U-type designs are orthogonality and space-filling criteria. Level permutations and level expansions are powerful tools for generating good U-type designs under the above criteria in the literature. In this paper, we systematically study the theoretical properties of U-type designs before and after level permutations and expansions. We establish the relationships between the initial designs’ generalized word-length patterns (GWLP) and the generated designs’ orthogonal and space-filling properties. Our findings generalize the existing results and provide theoretical justifications for the current level permutation and expansion algorithms.

MSC:

62K15 Factorial statistical designs
62K99 Design of statistical experiments
05B15 Orthogonal arrays, Latin squares, Room squares
62K10 Statistical block designs
62K05 Optimal statistical designs
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[1] Bingham, D.; Sitter, RR; Tang, B., Orthogonal and nearly orthogonal designs for computer experiments, Biometrika, 96, 51-65 (2009) · Zbl 1162.62073 · doi:10.1093/biomet/asn057
[2] Ba, S.; Myers, WR; Brenneman, WA, Optimal sliced Latin hypercube designs, Technometrics, 57, 479-487 (2015) · doi:10.1080/00401706.2014.957867
[3] Fang, KT, The uniform design: application of number-theoretic methods in experimental design, Acta Math Appl Sinica English Ser, 3, 363-372 (1980) · Zbl 0473.62067
[4] Fang, KT; Li, R.; Sudjianto, A., Design and modeling for computer experiments (2006), New York: CRC Press, New York · Zbl 1093.62117
[5] Georgiou, SD, Supersaturated designs: a review of their construction and analysis, J Stat Plan Inference, 144, 92-109 (2014) · Zbl 1278.62125 · doi:10.1016/j.jspi.2012.09.014
[6] Gramacy, RB, Surrogates: Gaussian process modeling, design, and optimization for the applied sciences (2020), New York: CRC Press, New York · doi:10.1201/9780367815493
[7] Hickernell, FJ, A generalized discrepancy and quadrature error bound, Math Comput, 67, 299-322 (1998) · Zbl 0889.41025 · doi:10.1090/S0025-5718-98-00894-1
[8] Hedayat, AS; Sloane, NJA; Stufken, J., Orthogonal arrays: theory and applications (1999), New York: Springer, New York · Zbl 0935.05001 · doi:10.1007/978-1-4612-1478-6
[9] Jiang, B.; Ai, M., Construction of uniform U-designs, J Stat Plan Inference, 181, 1-10 (2017) · Zbl 1356.62102 · doi:10.1016/j.jspi.2016.08.003
[10] Johnson, ME; Moore, LM; Ylvisaker, D., Minimax and maximin distance designs, J Stat Plan Inference, 26, 131-148 (1990) · doi:10.1016/0378-3758(90)90122-B
[11] Joseph, VR; Hung, Y., Orthogonal-maximin Latin hypercube designs, Stat Sin, 18, 171-186 (2008) · Zbl 1137.62050
[12] Joseph, VR; Gul, E.; Ba, S., Maximum projection designs for computer experiments, Biometrika, 102, 371-380 (2015) · Zbl 1452.62593 · doi:10.1093/biomet/asv002
[13] Lin, CD; Mukerjee, R.; Tang, B., Construction of orthogonal and nearly orthogonal Latin hypercubes, Biometrika, 96, 243-247 (2009) · Zbl 1161.62044 · doi:10.1093/biomet/asn064
[14] Lin, CD; Tang, B.; Dean, A.; Morris, M.; Stufken, J.; Bingham, D., Latin hypercubes and space-filling designs, Handbook of design and analysis of experiments, 593-625 (2015), New York: Chapman and Hall/CRC, New York · Zbl 1352.62128
[15] McKay, MD; Beckman, RJ; Conover, WJ, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 239-245 (1979) · Zbl 0415.62011
[16] Morris, MD; Mitchell, TJ, Exploratory designs for computational experiments, J Stat Plan Inference, 43, 381-402 (1995) · Zbl 0813.62065 · doi:10.1016/0378-3758(94)00035-T
[17] Owen, AB, Controlling correlations in Latin hypercube samples, J Am Stat Assoc, 89, 1517-1522 (1994) · Zbl 0813.65060 · doi:10.1080/01621459.1994.10476891
[18] Pronzato, L.; Müller, WG, Design of computer experiments: space filling and beyond, Stat Comput, 22, 681-701 (2012) · Zbl 1252.62080 · doi:10.1007/s11222-011-9242-3
[19] Santner, TJ; Williams, BJ; Notz, WI, The design and analysis of computer experiments (2018), New York: Springer, New York · Zbl 1405.62110 · doi:10.1007/978-1-4939-8847-1
[20] Sun, F.; Tang, B., A general rotation method for orthogonal Latin hypercubes, Biometrika, 104, 465-472 (2017) · Zbl 1506.62355
[21] Tang, B., Orthogonal array-based Latin hypercubes, J Am Stat Assoc, 88, 1392-1397 (1993) · Zbl 0792.62066 · doi:10.1080/01621459.1993.10476423
[22] Tang, B., Selecting Latin hypercubes using correlation criteria, Stat Sin, 8, 965-1397 (1998) · Zbl 0905.62065
[23] Tang, Y.; Xu, H.; Lin, DKJ, Uniform fractional factorial designs, Ann Stat, 40, 891-907 (2012) · Zbl 1274.62505
[24] Tang, Y.; Xu, H., An effective construction method for multi-level uniform designs, J Stat Plan Inference, 143, 1583-1589 (2013) · Zbl 1279.62162 · doi:10.1016/j.jspi.2013.04.009
[25] Wang, L.; Xiao, Q.; Xu, H., Optimal maximin \(L_1\)-distance Latin hypercube designs based on good lattice point designs, Ann Stat, 46, 3741-3766 (2018) · Zbl 1411.62238
[26] Wang, Y.; Yang, J.; Xu, H., On the connection between maximin distance designs and orthogonal designs, Biometrika, 105, 471-477 (2018) · Zbl 07072426 · doi:10.1093/biomet/asy005
[27] Wang, Y.; Sun, F.; Xu, H., On design orthogonality, maximin distance and projection uniformity for computer experiments, J Am Stat Assoc (2020) · Zbl 1506.62356 · doi:10.1080/01621459.2020.1782221
[28] Xiao, Q.; Xu, H., Construction of maximin distance Latin squares and related Latin hypercube designs, Biometrika, 104, 455-464 (2017) · Zbl 1506.62357 · doi:10.1093/biomet/asx006
[29] Xiao, Q.; Xu, H., Construction of maximin distance designs via level permutation and expansion, Stat Sin, 28, 1395-1414 (2018) · Zbl 1394.62109
[30] Xiao, Q.; Wang, L.; Xu, H., Application of kriging models for a drug combination experiment on lung cancer, Stat Med, 38, 236-246 (2019) · doi:10.1002/sim.7971
[31] Xu, G.; Zhang, J.; Tang, Y., Level permutation method for constructing uniform designs under the wrap-around \({L}_2\)-discrepancy, J Complex, 30, 46-53 (2014) · Zbl 1295.05073 · doi:10.1016/j.jco.2013.09.003
[32] Xu, H., Minimum moment aberration for nonregular designs and supersaturated designs, Stat Sin, 13, 691-708 (2003) · Zbl 1028.62063
[33] Xu, H.; Wu, CFJ, Generalized minimum aberration for asymmetrical fractional factorial designs, Ann Stat, 29, 549-560 (2001) · Zbl 1012.62083
[34] Yamada, S.; Matsui, T., Optimality of mixed-level supersaturated designs, J Stat Plan Inference, 104, 459-468 (2002) · Zbl 0992.62069 · doi:10.1016/S0378-3758(01)00248-8
[35] Ye, KQ, Orthogonal column Latin hypercubes and their application in computer experiments, J Am Stat Assoc, 93, 1430-1439 (1998) · Zbl 1064.62553 · doi:10.1080/01621459.1998.10473803
[36] Zhou, YD; Fang, KT; Ning, J., Mixture discrepancy for quasi-random point sets, J Complex, 29, 283-301 (2013) · Zbl 1282.65018 · doi:10.1016/j.jco.2012.11.006
[37] Zhou, YD; Xu, H., Space-filling fractional factorial designs, J Am Stat Assoc, 109, 1134-1144 (2014) · Zbl 1368.62232 · doi:10.1080/01621459.2013.873367
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