Abstract
Discrepancy is a kind of important measure used in experimental designs. Among various existing discrepancies, the discrete discrepancy, centered L 2-(CD 2) and wrap-around L 2-discrepancy (WD 2) have been well justified and widely used. In this paper, using the second-order polynomials of indicator functions for these three discrepancies, we investigate the close relationships between them and the generalized wordlength pattern, and provide some conditions under which a design having one of these minimum discrepancies is equivalent to having generalized minimum aberration (GMA). These results provide further justifications for the criterion of GMA in terms of uniformity. In addition, the expressions of the discrepancies in the quadratic forms of the indicator functions are useful for us to find optimal designs under any of them.
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References
Box GEP, Hunter JS (1961) The 2k-p fractional factorial designs. Technometrics 3:311–352, 449–458
Cheng SW, Ye KQ (2004) Geometric isomorphism and minimum aberration for factorial designs with quantitative factors. Ann Stat 32: 2168–2185
Dey A, Mukerjee R (1999) Fractional factorial plans. Wiley, New York
Fang KT, Ge GN, Liu MQ (2002) Uniform supersaturated design and its construction. Sci China Ser A 47: 1080–1088
Fang KT, Li R, Sudjianto A (2006) Design and modeling for computer experiments. Chapman & Hall, Boca Raton
Fang KT, Lin DKJ, Liu MQ (2003) Optimal mixed-level supersaturated design. Metrika 58: 279–291
Fang KT, Lin DKJ, Winker P, Zhang Y (2000) Uniform design: theory and application. Technometrics 42: 237–248
Fang KT, Mukerjee R (2000) A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika 87: 193–198
Fang KT, Qin H (2003) A note on construction of nearly uniform designs with large number of runs. Stat Probab Lett 61: 215–224
Fontana R, Pistone G, Rogantin MP (2000) Classification of two-level factorial fractions. J Stat Plan Inference 87: 149–172
Fries A, Hunter WG (1980) Minimum aberration 2k-p designs. Technometrics 22: 601–608
Hickernell FJ (1998) A generalized discrepancy and quadrature error bound. Math Comp 67: 299–322
Hickernell FJ (1999) Goodness-of-fit statistics, discrepancies and robust designs. Stat Probab Lett 44: 73–78
Hickernell FJ, Liu MQ (2002) Uniform designs limit aliasing. Biometrika 89: 893–904
Liu MQ (2002) Using discrepancy to evaluate fractional factorial designs. In: Fang KT, Hickernell FJ, Niederreiter H (eds) Monte carlo and quasi-monte carlo methods 2000. Springer, Berlin, pp 357–368
Liu MQ, Fang KT, Hickernell FJ (2006) Connections among different criteria for asymmetrical fractional factorial designs. Stat Sin 16: 1285–1297
Liu MQ, Hickernell FJ (2002) E(s 2)-optimality and minimum discrepancy in 2-level supersaturated designs. Stat Sin 12: 931–939
Liu MQ, Hickernell FJ (2006) The relationship between discrepancies defined on a domain and on its subset. Metrika 63: 317–327
Liu MQ, Qin H, Xie MY (2005) Discrete discrepancy and its application in experimental design. In: Fan JQ, Li G (eds) Contemporary multivariate analysis and experimental designs. World Scientific Publishing, Singapore, pp 227–241
Ma CX, Fang KT (2001) A note on generalized aberration in factorial designs. Metrika 53: 85–93
Ma CX, Fang KT, Lin DKJ (2003) A note on uniformity and orthogonality. J Stat Plan Inference 113: 323–334
Pistone G, Rogantin MP (2008) Indicator function and complex coding for mixed fractional factorial designs. J Stat Plan Inference 138: 787–802
Qin H, Ai MY (2007) A note on connection between uniformity and generalized minimum aberration. Stat Papers 48: 491–502
Qin H, Fang KT (2004) Discrete discrepancy in factorial designs. Metrika 60: 59–72
Qin H, Li D (2006) Connection between uniformity and orthogonality for symmetrical factorial designs. J Stat Plan Inference 136: 2770–2782
Qin H, Zou N, Chatterjee K (2009) Connection between uniformity and minimum moment aberration. Metrika 70: 79–88
Sun FS, Liu MQ, Hao WR (2009) An algorithmic approach to finding factorial designs with generalized minimum aberration. J Complex 25: 75–84
Tang B, Deng LY (1999) Minimum G 2-aberration for nonregular fractional factorial designs. Ann Stat 27: 1914–1926
Xu H, Wu CFJ (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann Stat 29: 1066–1077
Ye KQ (2003) Indicator function and its application in two-level factorial designs. Ann Stat 31: 984–994
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Sun, F., Chen, J. & Liu, MQ. Connections between uniformity and aberration in general multi-level factorials. Metrika 73, 305–315 (2011). https://doi.org/10.1007/s00184-009-0279-7
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DOI: https://doi.org/10.1007/s00184-009-0279-7