Designing Uniform Computer Sequential Experiments with Mixture Levels Using Lee Discrepancy
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摘要
Computer experiments are constructed to simulate the behavior of complex physical systems. Uniform designs have good performance in computer experiments from several aspects. In practical use, the experimenter needs to choose a small size uniform design at the beginning of an experiment due to a limit of time, budget, resources, and so on, and later conduct a follow up experiment to obtain precious information about the system, that is, a sequential experiment. The Lee distance has been widely used in coding theory and its corresponding discrepancy is an important measure for constructing uniform designs. This paper proves that all the follow up designs of a uniform design are uniform and at least two of them can be used as optimal follow up experimental designs. Thus, it is not necessary that the union of any two uniform designs yields a uniform sequential design. Therefore, this article presents a theoretical justification for choosing the best follow up design of a uniform design to construct a uniform sequential design that involves a mixture of ω≥ 1 factors with β_k ≥ 2, 1 ≤ k ≤ωlevels. For illustration of the usage of the proposed results, a closer look is given at using these results for the most extensively used six particular cases, three symmetric and three asymmetric designs, which are often met in practice.
Computer experiments are constructed to simulate the behavior of complex physical systems. Uniform designs have good performance in computer experiments from several aspects. In practical use, the experimenter needs to choose a small size uniform design at the beginning of an experiment due to a limit of time, budget, resources, and so on, and later conduct a follow up experiment to obtain precious information about the system, that is, a sequential experiment. The Lee distance has been widely used in coding theory and its corresponding discrepancy is an important measure for constructing uniform designs. This paper proves that all the follow up designs of a uniform design are uniform and at least two of them can be used as optimal follow up experimental designs. Thus, it is not necessary that the union of any two uniform designs yields a uniform sequential design. Therefore, this article presents a theoretical justification for choosing the best follow up design of a uniform design to construct a uniform sequential design that involves a mixture of ω ≥ 1 factors with β_k ≥ 2, 1 ≤ k ≤ ωlevels. For illustration of the usage of the proposed results, a closer look is given at using these results for the most extensively used six particular cases, three symmetric and three asymmetric designs, which are often met in practice.
Computer experiments are constructed to simulate the behavior of complex physical systems. Uniform designs have good performance in computer experiments from several aspects. In practical use, the experimenter needs to choose a small size uniform design at the beginning of an experiment due to a limit of time, budget, resources, and so on, and later conduct a follow up experiment to obtain precious information about the system, that is, a sequential experiment. The Lee distance has been widely used in coding theory and its corresponding discrepancy is an important measure for constructing uniform designs. This paper proves that all the follow up designs of a uniform design are uniform and at least two of them can be used as optimal follow up experimental designs. Thus, it is not necessary that the union of any two uniform designs yields a uniform sequential design. Therefore, this article presents a theoretical justification for choosing the best follow up design of a uniform design to construct a uniform sequential design that involves a mixture of ω ≥ 1 factors with β_k ≥ 2, 1 ≤ k ≤ ωlevels. For illustration of the usage of the proposed results, a closer look is given at using these results for the most extensively used six particular cases, three symmetric and three asymmetric designs, which are often met in practice.
引文
[1]Fang K T and Li R,Uniform design for computer experiments and its optimal properties,Int.J.Materials and Product Technology,2006,25(1/2/3):198-210.
[2]Simpson T W,Lin D K J,and Chen W,Sampling strategies for computer experiments:Design and analysis,Int.J.Relability and Applications,2001,2(3):209-240.
[3]Fang K T,The uniform design:Application of number-theoretic methods in experimental design,Acta Math.Appl.Sinica,1980,3:363-372.
[4]Wang Y and Fang K T,A note on uniform distribution and experimental design.Chin.Sci.Bull.,1981,26:485-489.
[5]Hickernell F J,A generalized discrepancy and quadrature error bound,Math.Comp.,1998,67:299-322.
[6]Hickernell F J,Lattice Rules:How Well Do They Measure Up?Random and Quasi-Random Point Sets,Eds.by Hellekalek P and Larcher G,Springer,New York,1998.
[7]Roth R M,Introduction to Coding Theory,Cambridge University Press,Cambridge,UK,2006.
[8]Zhou Y D,Ning J H,and Song X B,Lee discrepancy and its applications in experimental designs,Statist.Probab.Lett.,2008,78:1933-1942.
[9]Ji Y B,Alaerts G,Xu C J,et al.,Sequential uniform designs for fingerprints development of Ginkgo biloba extracts by capillary electrophoresis,J.Chromatography,2006,A1128:273-281.
[10]Tong C,Refinement strategies for stratified sampling methods,Reliability Engineering and System Safety,2006,91:1257-1265.
[11]Durrieu G and Briollais L,Sequential design for microarray experiments,J.Amer.Statist.Association,2009,104:650-660.
[12]Loeppky J L,Moore L M,and Williams B J,Batch sequential designs for computer experiments,J.Statist.Plann.Inference,2010,140:1452-1464.
[13]Cheong K T W,Htay K,Tan R H C,et al.,Identifying combinatorial growth inhibitory effects of various plant extracts on leukemia cells through systematic experimental design,Amer.J.Plant Sci.,2012,3:1390-1398.
[14]Elsawah A M,Constructing optimal router bit life sequential experimental designs:New results with a case study,Commun.Stat.Simulat.Comput.,2017,http://dx.doi.org/10.1080/03610918.2017.1397164.
[15]Bullington K E,Hool J N,and Maghsoodloo S,A simple method for obtaining resolution IVdesigns for use with Taguchi orthogonal arrays,J.Qual.Technol.,1990,22(4):260-264.
[16]Li W and Lin D K J,Optimal foldover plans for two-level fractional factorial designs,Technometrics,2003,45:142-149.
[17]Box,G E P,Hunter W G,and Hunter J S,Statistics for Experiments,John Wiley and Sons,New York,1978.
[18]Montgomery D C and Runger G C,Foldover of 2k-presolution IV experimental designs,J.Qual.Technol.,1996,28:446-450.
[19]Li W,Lin D K J,and Ye K Q,Optimal foldover plans for non-regular orthogonal designs,Technometrics,2003,45:347-351.
[20]Li P F,Liu M Q,and Zhang R C,Choice of optimal initial designs in sequential experiments,Metrika,2005,61(2):127-135.
[21]Wang B,Robert G M,and John F B,A note on the selection of optimal foldover plans for 16-and 32-run fractional factorial designs,J.Stat.Plan.Inference,2010,140:1497-1500.
[22]Fang K T,Lin D K J,and Qin H,A note on optimal foldover design,Statist.Probab.Lett.,2003,62:245-250.
[23]Elsawah A M and Qin H,An efficient methodology for constructing optimal foldover designs in terms of mixture discrepancy,J.Korean Statist.Soc.,2016,45:77-88.
[24]Elsawah A M and Qin H,Optimum mechanism for breaking the confounding effects of mixed-level designs,Comput.Stat.,2017,32(2):781-802.
[25]Ou Z and Qin H,Optimal foldover plans of asymmetric factorials with minimum wrap-around L2-discrepancy,Stat.Papers,2017,DOI:10.1007/s00362-017-0892-x.
[26]Elsawah A M,A closer look at de-aliasing effects using an efficient foldover technique,Statistics,2017,51(3):532-557.
[27]Elsawah A M,A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs,Aust.N.Z.J.Stat.,2017,59(1):17-41.
[28]Elsawah A M,Choice of optimal second stage designs in two-stage experiments,Comput.Stat.,2018,33(2):933-965
[29]Elsawah A M and Qin H,A new strategy for optimal foldover two-level designs,Statist.Probab.Lett.,2015,103:116-126.
[30]Elsawah A M and Qin H,New lower bound for centered L2-discrepancy of four-level U-type designs,Statist.Probab.Lett.,2014,93:65-71.
[31]Elsawah A M and Qin H,A new look on optimal foldover plans in terms of uniformity criteria,Commun.Stat.Theory Methods,2017,46(4):1621-1635.
[32]Elsawah A M,Constructing optimal asymmetric combined designs via Lee discrepancy,Statist.Probab.Lett.,2016,118:24-31.
[33]Fang K T,Ke X,and Elsawah A M,Construction of uniform designs via an adjusted threshold accepting algorithm,J.Complexity,2017,43:28-37.
[2]Simpson T W,Lin D K J,and Chen W,Sampling strategies for computer experiments:Design and analysis,Int.J.Relability and Applications,2001,2(3):209-240.
[3]Fang K T,The uniform design:Application of number-theoretic methods in experimental design,Acta Math.Appl.Sinica,1980,3:363-372.
[4]Wang Y and Fang K T,A note on uniform distribution and experimental design.Chin.Sci.Bull.,1981,26:485-489.
[5]Hickernell F J,A generalized discrepancy and quadrature error bound,Math.Comp.,1998,67:299-322.
[6]Hickernell F J,Lattice Rules:How Well Do They Measure Up?Random and Quasi-Random Point Sets,Eds.by Hellekalek P and Larcher G,Springer,New York,1998.
[7]Roth R M,Introduction to Coding Theory,Cambridge University Press,Cambridge,UK,2006.
[8]Zhou Y D,Ning J H,and Song X B,Lee discrepancy and its applications in experimental designs,Statist.Probab.Lett.,2008,78:1933-1942.
[9]Ji Y B,Alaerts G,Xu C J,et al.,Sequential uniform designs for fingerprints development of Ginkgo biloba extracts by capillary electrophoresis,J.Chromatography,2006,A1128:273-281.
[10]Tong C,Refinement strategies for stratified sampling methods,Reliability Engineering and System Safety,2006,91:1257-1265.
[11]Durrieu G and Briollais L,Sequential design for microarray experiments,J.Amer.Statist.Association,2009,104:650-660.
[12]Loeppky J L,Moore L M,and Williams B J,Batch sequential designs for computer experiments,J.Statist.Plann.Inference,2010,140:1452-1464.
[13]Cheong K T W,Htay K,Tan R H C,et al.,Identifying combinatorial growth inhibitory effects of various plant extracts on leukemia cells through systematic experimental design,Amer.J.Plant Sci.,2012,3:1390-1398.
[14]Elsawah A M,Constructing optimal router bit life sequential experimental designs:New results with a case study,Commun.Stat.Simulat.Comput.,2017,http://dx.doi.org/10.1080/03610918.2017.1397164.
[15]Bullington K E,Hool J N,and Maghsoodloo S,A simple method for obtaining resolution IVdesigns for use with Taguchi orthogonal arrays,J.Qual.Technol.,1990,22(4):260-264.
[16]Li W and Lin D K J,Optimal foldover plans for two-level fractional factorial designs,Technometrics,2003,45:142-149.
[17]Box,G E P,Hunter W G,and Hunter J S,Statistics for Experiments,John Wiley and Sons,New York,1978.
[18]Montgomery D C and Runger G C,Foldover of 2k-presolution IV experimental designs,J.Qual.Technol.,1996,28:446-450.
[19]Li W,Lin D K J,and Ye K Q,Optimal foldover plans for non-regular orthogonal designs,Technometrics,2003,45:347-351.
[20]Li P F,Liu M Q,and Zhang R C,Choice of optimal initial designs in sequential experiments,Metrika,2005,61(2):127-135.
[21]Wang B,Robert G M,and John F B,A note on the selection of optimal foldover plans for 16-and 32-run fractional factorial designs,J.Stat.Plan.Inference,2010,140:1497-1500.
[22]Fang K T,Lin D K J,and Qin H,A note on optimal foldover design,Statist.Probab.Lett.,2003,62:245-250.
[23]Elsawah A M and Qin H,An efficient methodology for constructing optimal foldover designs in terms of mixture discrepancy,J.Korean Statist.Soc.,2016,45:77-88.
[24]Elsawah A M and Qin H,Optimum mechanism for breaking the confounding effects of mixed-level designs,Comput.Stat.,2017,32(2):781-802.
[25]Ou Z and Qin H,Optimal foldover plans of asymmetric factorials with minimum wrap-around L2-discrepancy,Stat.Papers,2017,DOI:10.1007/s00362-017-0892-x.
[26]Elsawah A M,A closer look at de-aliasing effects using an efficient foldover technique,Statistics,2017,51(3):532-557.
[27]Elsawah A M,A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs,Aust.N.Z.J.Stat.,2017,59(1):17-41.
[28]Elsawah A M,Choice of optimal second stage designs in two-stage experiments,Comput.Stat.,2018,33(2):933-965
[29]Elsawah A M and Qin H,A new strategy for optimal foldover two-level designs,Statist.Probab.Lett.,2015,103:116-126.
[30]Elsawah A M and Qin H,New lower bound for centered L2-discrepancy of four-level U-type designs,Statist.Probab.Lett.,2014,93:65-71.
[31]Elsawah A M and Qin H,A new look on optimal foldover plans in terms of uniformity criteria,Commun.Stat.Theory Methods,2017,46(4):1621-1635.
[32]Elsawah A M,Constructing optimal asymmetric combined designs via Lee discrepancy,Statist.Probab.Lett.,2016,118:24-31.
[33]Fang K T,Ke X,and Elsawah A M,Construction of uniform designs via an adjusted threshold accepting algorithm,J.Complexity,2017,43:28-37.