Double设计在对称化L_2-偏差下的均匀
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摘要
二水平部分因子设计在工农业生产、科学实验等领域得到广泛应用。在构造二水平部分因子设计中，一种称为doubling的方法最近被采用，特别是在构造分辨度为Ⅳ的设计时，doubling是一个简单但很有用的方法(Chen and Cheng,2006;Xu and Cheng,2006)。假定X为二水平部分因子设计，其中两水平分别为+1和-1，则称为X的double设计。本文从均匀设计的角度来研究double设计。我们利用对称化L_2-偏差作为均匀性的测度(Ma et al,2002;Wang et al,2006)，给出了double设计在这种偏差下的几个下界。利用这些下界，我们可以衡量double设计的均匀性程度，进而发现double设计D(X)的均匀性与初始设计X的均匀性有着密切的联系。我们还给出了double设计D(X)的对称化L_2-偏差值与初始设计X的广义字长型之间的关系。利用上述关系，我们断定对于具有较小低阶混杂的初始设计X，其对应的double设计D(X)也应该具有较小的对称化L_2-偏差值，也就是说，D(X)应该具有较好的均匀性。
本文的主要结论以三个定理的形式给出：
定理1对X∈u(n；2~k)，我们有
[SD_2(D(X))]~2≥L_(SD_2)~c(2；n，k)，
这里，且S_(n,m,2)是n被2~m除的余数，其中C=(4/3)~(2k)-2(11/8)~(2k)+2~(k-1)。
定理2对X∈u(n；2~k)，我们有
[SD_2(D(X))]~2≥L_(SD_2)~r(2；n，k)，这里L(SD_2)~r(2；n，k)=C+(n-1)4~θ(1+3f)/(2n)+4~k/(2n)，λ=k(n-2)/[2(n-1)]，λ=θ+f，θ是λ的整数部分，f是λ的分数部分，其中C=(4/3)~(2k)-2(11/8)~(2k)+2~(k-1)。
定理3对X∈D(n；2~k)，我们有其中C=(4/3)~(2k)-2(11/8)~(2k)+2~(k-1)。
Two-level fractional factorial designs have been widely used in industry, agriculture and scientific experiments. For constructing two-level fractional factorial designs, a new method called doubling method has been recently used. In particular, in constructing those of resolution IV, doubling is a simple but very powerful method (Chen and Cheng, 2006; Xu and Cheng, 2006). Suppose X is a two-level fractional factorial design, in which two levels are denoted by+1 and - 1. The double of X is the following matrixIn this paper, we study double designs in view of uniformity. We use the symmetric L_2-discrepancy as the measure of uniformity (Ma et al, 2002; Wang et al, 2006), and provide some lower bounds of double designs in the sense of the symmetric L_2-discrepancy. Using these lower bounds, we can value uniformity of double designs. Furthermore, a close connection between uniformity of double design D(X) and that of the initial design X is also given. In addition, we also obtain results connecting the symmetric L_2-discrepancy of D(X) and the generalized wordlength pattern (Xu and Wu, 2002) of X. We conclude that if X has less aberration, then D(X) has lower symmetric L_2-discrepancy, i.e., D(X) has better uniformity.
Main results of this paper are given as follows:
Theorem 1 Let X∈U(n; 2~k), we have [SD_2(D(X))]~2≥L_(SD_2)~c(2; n, k),where L_(SD_2)~c(2; n,k) = C + 5~k/2~(k+1) + 1/(2n~2) and S_(n,m,2) is the residual of n(mod 2~m), C = (4/3)~(2k) - 2(11/8)~(2k) + 2~(k-1)
Theorem 2 Let X∈U(n; 2~k), we have [SD_2(D(X))]~2≥L_(SD_2)~r(2; n, k), where L_(SD_2)~r(2; n, k) = C + (n - 1)4~θ(1 + 3f)/(2n) + 4~k/(2n), λ= k(n - 2)/[2(n - 1)],λ=θ+ f,θis the largest integer contained in A, f =λ-θ,C = (4/3)~(2k) - 2(11/8)~(2k) + 2~(k-1).
Theorem 3 Let X∈V(n; 2~k), we havewhere C = (4/3)~(2k) - 2(11/8)~(2k) + 2~(k-1).
本文的主要结论以三个定理的形式给出：
定理1对X∈u(n；2~k)，我们有
[SD_2(D(X))]~2≥L_(SD_2)~c(2；n，k)，
这里，且S_(n,m,2)是n被2~m除的余数，其中C=(4/3)~(2k)-2(11/8)~(2k)+2~(k-1)。
定理2对X∈u(n；2~k)，我们有
[SD_2(D(X))]~2≥L_(SD_2)~r(2；n，k)，这里L(SD_2)~r(2；n，k)=C+(n-1)4~θ(1+3f)/(2n)+4~k/(2n)，λ=k(n-2)/[2(n-1)]，λ=θ+f，θ是λ的整数部分，f是λ的分数部分，其中C=(4/3)~(2k)-2(11/8)~(2k)+2~(k-1)。
定理3对X∈D(n；2~k)，我们有其中C=(4/3)~(2k)-2(11/8)~(2k)+2~(k-1)。
Two-level fractional factorial designs have been widely used in industry, agriculture and scientific experiments. For constructing two-level fractional factorial designs, a new method called doubling method has been recently used. In particular, in constructing those of resolution IV, doubling is a simple but very powerful method (Chen and Cheng, 2006; Xu and Cheng, 2006). Suppose X is a two-level fractional factorial design, in which two levels are denoted by+1 and - 1. The double of X is the following matrixIn this paper, we study double designs in view of uniformity. We use the symmetric L_2-discrepancy as the measure of uniformity (Ma et al, 2002; Wang et al, 2006), and provide some lower bounds of double designs in the sense of the symmetric L_2-discrepancy. Using these lower bounds, we can value uniformity of double designs. Furthermore, a close connection between uniformity of double design D(X) and that of the initial design X is also given. In addition, we also obtain results connecting the symmetric L_2-discrepancy of D(X) and the generalized wordlength pattern (Xu and Wu, 2002) of X. We conclude that if X has less aberration, then D(X) has lower symmetric L_2-discrepancy, i.e., D(X) has better uniformity.
Main results of this paper are given as follows:
Theorem 1 Let X∈U(n; 2~k), we have [SD_2(D(X))]~2≥L_(SD_2)~c(2; n, k),where L_(SD_2)~c(2; n,k) = C + 5~k/2~(k+1) + 1/(2n~2) and S_(n,m,2) is the residual of n(mod 2~m), C = (4/3)~(2k) - 2(11/8)~(2k) + 2~(k-1)
Theorem 2 Let X∈U(n; 2~k), we have [SD_2(D(X))]~2≥L_(SD_2)~r(2; n, k), where L_(SD_2)~r(2; n, k) = C + (n - 1)4~θ(1 + 3f)/(2n) + 4~k/(2n), λ= k(n - 2)/[2(n - 1)],λ=θ+ f,θis the largest integer contained in A, f =λ-θ,C = (4/3)~(2k) - 2(11/8)~(2k) + 2~(k-1).
Theorem 3 Let X∈V(n; 2~k), we havewhere C = (4/3)~(2k) - 2(11/8)~(2k) + 2~(k-1).
引文
[1] 方开泰、马长兴．(2001)．正交与均匀试验设计．科学出版社．
[2] Box, G. E. P., Hunter, W. G. and Hunter, J. S. (1978). Statistics for experiments. Wiley, New York.
[3] Chen, H. G.and Cheng, C. S. (2006) Doubling and projection: a method of constructing two-level designs of resolution Ⅳ. Ann. Statist, 34:546-558.
[4] Fang, K. T., Li, R. Z. and Sudjianto, A. Design and Modeling for Computer Experiments. Chapman Hall/CRC
[5] Fang, K. T., Lin, D. K. J., Winker, P. and Zhang, Y. (2000) Uniform design: theory and applications. Technometrics, 42:237-248.
[6] Fang, K. T., Lu, X. and Winker, P. (2003) Lower bounds for centered and wraparound L_2-discrepancies and construction of uniform designs by threshold accepting. J. Complexity, 19:692-711.
[7] Fang, K. T., Ma, C. X. and Ma, C. X. (2002) Uniformity in fractional factorials. In: Fang, K. T., Hickernell, F. J., Niederreiter, H. (Eds). Monte Carlo and QuaSi-Monte Carlo Methods in Scientific Computing. Springer, 232-241.
[8] Fang, K. T. and Mukerjee, R. (2000) A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika, 87:173-198.
[9] Fang, K. T. and Qin, H. (2004) Uniformity pattern and related criteria for two-level factorials. Science in China Ser. A Mathematics, 48(1):1-11.
[10] Fang, K. T. and Wang, Y. (1994) Number-Theoretic Methods in Statistics. London:Chapman and Hall.
[11] Fedorov, V. V. (1972) Theory of Optimal Experiments. Academic Press New York and London
[12] Fries, A.and Hunter, W. G. (1980) Minimum aberration 2~(k-p) designs. Technometrics, 22 (4):601-608.
[13] Hamming, R. W. (1950) Error detecting and error correcting codes. Bell System Tech. J., 29:147-160.
[14] Hickernell, F. J. (1998a) A generalized discrepancy and quadrature error Bound. Mathematics of Computation, 67:299-322.
[15] Hickernell, F. J. (1998b) Lattic RULES: How Well Do They Measure Up? In:Hellekalek, P.,Larcher, G. (Eds). Random and Quasi-Random Point Sets, Lecture Notes in Statistics. New York:Spinger, 138:109-166.
[16] Lin, D. K. J. (1993) A new class of supersaturated designs. Technometrics,35:28-31.
[17] Ma, C. X. and Fang, K. T. (2001) A note on generalized aberration factorial designs. Metrika, 53:85-93.
[18] Mukerjee, R. and Wu, C. F. J. (1995) On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist., 23:2102-2115.
[19] Mukerjee, R. and Wu, C. F. J. (2006) A Modern Theory of Factorial Designs. Springer.
[20] Tang, B.and D, L. Y. (1999) Minimum G_2-aberration for nonregular fractional designs. Ann. Statist., 27:1914-1926
[21] Wang, Y. and Fang, K. T. (1981) A note on uniform distribution and experimental design. Chin. Sci. Bull., 26:485-489.
[22] Wang, Z. H., Qin, H. and Chatterjee, K. (2007) A lower bound for the symmetric L_2-discrepancy and its application. Communications in Statistics, to appear.
[23] Wu, C. F. J. and Hamada, M. (2000) Experiments Planning, Analysis,and Parameter Design Optimization. A Wiely-interscience Publication
[24] Xu, H. Q. (2003) Minimum moment aberration for nonregular designs and supersaturated designs. Statistica Sinica, 13:691-708.
[25] Xu, H. Q. and Cheng, C. S. (2006) A complementary design theory for doubling. Ann.Statist., to appear.
[26] Zhang, A. J., Fang, K. T., Li, R. Z. and Sudjianto, A. (2005) Majorization framework for balanced lattice designs. Ann. Statist., 33 (6):2837-2853.
[2] Box, G. E. P., Hunter, W. G. and Hunter, J. S. (1978). Statistics for experiments. Wiley, New York.
[3] Chen, H. G.and Cheng, C. S. (2006) Doubling and projection: a method of constructing two-level designs of resolution Ⅳ. Ann. Statist, 34:546-558.
[4] Fang, K. T., Li, R. Z. and Sudjianto, A. Design and Modeling for Computer Experiments. Chapman Hall/CRC
[5] Fang, K. T., Lin, D. K. J., Winker, P. and Zhang, Y. (2000) Uniform design: theory and applications. Technometrics, 42:237-248.
[6] Fang, K. T., Lu, X. and Winker, P. (2003) Lower bounds for centered and wraparound L_2-discrepancies and construction of uniform designs by threshold accepting. J. Complexity, 19:692-711.
[7] Fang, K. T., Ma, C. X. and Ma, C. X. (2002) Uniformity in fractional factorials. In: Fang, K. T., Hickernell, F. J., Niederreiter, H. (Eds). Monte Carlo and QuaSi-Monte Carlo Methods in Scientific Computing. Springer, 232-241.
[8] Fang, K. T. and Mukerjee, R. (2000) A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika, 87:173-198.
[9] Fang, K. T. and Qin, H. (2004) Uniformity pattern and related criteria for two-level factorials. Science in China Ser. A Mathematics, 48(1):1-11.
[10] Fang, K. T. and Wang, Y. (1994) Number-Theoretic Methods in Statistics. London:Chapman and Hall.
[11] Fedorov, V. V. (1972) Theory of Optimal Experiments. Academic Press New York and London
[12] Fries, A.and Hunter, W. G. (1980) Minimum aberration 2~(k-p) designs. Technometrics, 22 (4):601-608.
[13] Hamming, R. W. (1950) Error detecting and error correcting codes. Bell System Tech. J., 29:147-160.
[14] Hickernell, F. J. (1998a) A generalized discrepancy and quadrature error Bound. Mathematics of Computation, 67:299-322.
[15] Hickernell, F. J. (1998b) Lattic RULES: How Well Do They Measure Up? In:Hellekalek, P.,Larcher, G. (Eds). Random and Quasi-Random Point Sets, Lecture Notes in Statistics. New York:Spinger, 138:109-166.
[16] Lin, D. K. J. (1993) A new class of supersaturated designs. Technometrics,35:28-31.
[17] Ma, C. X. and Fang, K. T. (2001) A note on generalized aberration factorial designs. Metrika, 53:85-93.
[18] Mukerjee, R. and Wu, C. F. J. (1995) On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist., 23:2102-2115.
[19] Mukerjee, R. and Wu, C. F. J. (2006) A Modern Theory of Factorial Designs. Springer.
[20] Tang, B.and D, L. Y. (1999) Minimum G_2-aberration for nonregular fractional designs. Ann. Statist., 27:1914-1926
[21] Wang, Y. and Fang, K. T. (1981) A note on uniform distribution and experimental design. Chin. Sci. Bull., 26:485-489.
[22] Wang, Z. H., Qin, H. and Chatterjee, K. (2007) A lower bound for the symmetric L_2-discrepancy and its application. Communications in Statistics, to appear.
[23] Wu, C. F. J. and Hamada, M. (2000) Experiments Planning, Analysis,and Parameter Design Optimization. A Wiely-interscience Publication
[24] Xu, H. Q. (2003) Minimum moment aberration for nonregular designs and supersaturated designs. Statistica Sinica, 13:691-708.
[25] Xu, H. Q. and Cheng, C. S. (2006) A complementary design theory for doubling. Ann.Statist., to appear.
[26] Zhang, A. J., Fang, K. T., Li, R. Z. and Sudjianto, A. (2005) Majorization framework for balanced lattice designs. Ann. Statist., 33 (6):2837-2853.