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An effective construction method for multi-level uniform designs. (English) Zbl 1279.62162

Summary: Uniform designs are widely used in various applications. However, it is computationally intractable to construct uniform designs, even for moderate number of runs, factors and levels. We establish a linear relationship between average squared centered \(L_2\)-discrepancy and generalized wordlength pattern, and then based on it, we propose a general method for constructing uniform designs with arbitrary number of levels. The main idea is to choose a generalized minimum aberration design and then permute its levels. We propose a novel stochastic algorithm and obtain many new uniform designs that have smaller centered \(L_2\)-discrepancies than the existing ones.

MSC:

62K15 Factorial statistical designs
65C60 Computational problems in statistics (MSC2010)
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References:

[1] Fang, K. T.; Ge, G. N.; Liu, M. Q.; Qin, H., Construction of uniform designs via super-simple resolvable \(t\)-designs, Utilitas Mathematica, 66, 15-32 (2004) · Zbl 1061.62113
[2] Fang, K. T.; Li, R.; Sudjianto, A., Design and Modeling for Computer Experiments (2006), Chapman and Hall, CRC: Chapman and Hall, CRC London · Zbl 1093.62117
[3] Fang, K. T.; Lu, X.; Tang, Y.; Yin, J., Constructions of uniform designs by using resolvable packings and coverings, Discrete Mathematics, 19, 692-711 (2003)
[4] Fang, K. T.; Lu, X.; Winker, P., Lower bounds for centered and wrap-around \(l_2\)-discrepancies and construction of uniform design by threshold accepting, Journal of Complexity, 20, 268-272 (2003)
[5] Fang, K. T.; Ma, C. X.; Winker, P., Centered \(l_2\)-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs, Mathematics of Computation, 71, 275-296 (2002) · Zbl 0977.68091
[6] Fang, K. T.; Maringer, D.; Tang, Y.; Winker, P., Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels, Mathematics of Computation, 75, 859-878 (2006) · Zbl 1093.90031
[7] Fang, K. T.; Mukerjee, R., A connection between uniformity and aberration in regular fractions of two-level factorials, Biometrika, 87, 193-198 (2000) · Zbl 0974.62059
[8] Hickernell, F. J., A generalized discrepancy and quadrature error bound, Mathematics of Computation, 67, 299-322 (1998) · Zbl 0889.41025
[9] Hickernell, F. J.; Liu, M. Q., Uniform designs limit aliasing, Biometrika, 89, 893-904 (2002) · Zbl 1036.62060
[10] Ma, C. X.; Fang, K. F., A note on generalized aberration in factorial designs, Metrika, 53, 85-93 (2001) · Zbl 0990.62067
[11] Tang, Y.; Xu, H.; Lin, D. K.J., Uniform fractional factorial designs, The Annals of Statistics, 40, 891-907 (2012) · Zbl 1274.62505
[12] Wang, Y.; Fang, K. T., A note on uniform distribution and experimental design, Chinese Science Bulletin, 26, 485-489 (1981) · Zbl 0493.62068
[13] Xu, H.; Wu, C. F.J., Generalized minimum aberration for asymmetrical fractional factorial designs, The Annals of Statistics, 29, 1066-1077 (2001) · Zbl 1041.62067
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