数据污染下的稳健设计及最优参数的置信区间估计
|
摘要
传统的稳健设计一般假定试验数据为正态分布且无污染,然而在实践中,由于缺乏足够的试验数据以及数据中存在污染等因素,导致用传统的点估计方法无法准确获得某设计点下真实过程的输出值。因此,提出了基于bootstrap重抽样技术估计数据污染下最优参数置信区间的稳健设计方法。首先,采用Hodges-Lehman估计量和Shamos估计量分别估计位置参数和尺度参数;其次,构建过程均值和方差双响应曲面模型,实现稳健设计;然后,利用分位数bootstrapping (Percentile Bootstrapping,PB)、偏差校正分位数bootstrapping (Bias-Corrected Percentile Bootstrapping,BCPB)和偏差校正及加速分位数bootstrapping (Bias-Corrected and accelerated Percentile Bootstrapping,BCaPB)方法分别估计最优参数的bootstrap置信区间;最后,引入欧式距离和广义方差分别度量不同置信区间抵抗污染值的稳健性。通过仿真表明,在解决数据污染的稳健设计中BCPB和BCaPB方法估计的精确度明显高于PB方法,同时BCaPB方法在抵抗污染值干扰方面优于BCPB方法。
Due to the fact that the traditional point estimates of optimal settings may not be properly estimated resulting from lack of sufficient experiment data under data contamination,a robust design method is proposed based on bootstrap technique for obtaining confidence interval of the optimum operating conditions.Firstly,Hodges-Lehman and Shamos methods were used to estimate the location parameters and the scale parameters respectively.Secondly,dual response surface model was built for process mean and variance,and the optimal settings were attained by robust optimization.Next,the bootstrap confidence intervals of the optimal settings were estimated according to percentile bootstrapping(PB),bias-corrected percentile bootstrapping(BCPB) and bias-corrected and accelerated percentile bootstrapping(BCaPB) methods respectively.Finally,by introducing Euclidean distance and generalized variance,the robustness of different confidence intervals for outlier-resistant were measured.Through Monte Carlo simulation,the results show that BCaPB method is superior to BCPB method when the data are contaminated,yet also more robust and accurate than the counterpart of PB method in outlier-resistant.
Due to the fact that the traditional point estimates of optimal settings may not be properly estimated resulting from lack of sufficient experiment data under data contamination,a robust design method is proposed based on bootstrap technique for obtaining confidence interval of the optimum operating conditions.Firstly,Hodges-Lehman and Shamos methods were used to estimate the location parameters and the scale parameters respectively.Secondly,dual response surface model was built for process mean and variance,and the optimal settings were attained by robust optimization.Next,the bootstrap confidence intervals of the optimal settings were estimated according to percentile bootstrapping(PB),bias-corrected percentile bootstrapping(BCPB) and bias-corrected and accelerated percentile bootstrapping(BCaPB) methods respectively.Finally,by introducing Euclidean distance and generalized variance,the robustness of different confidence intervals for outlier-resistant were measured.Through Monte Carlo simulation,the results show that BCaPB method is superior to BCPB method when the data are contaminated,yet also more robust and accurate than the counterpart of PB method in outlier-resistant.
引文
[1] Ouyang L,Zhou D,Park C,et al.Ensemble modelling technique for a micro-drilling process based on a two-stage bootstrap[J].Engineering Optimization,2018:1-17.DOI:10.1080/0305215X.2018.1472251.
[2] 韩云霞,马义中,欧阳林寒,等.基于两类因子波动下的稳健优化设计[J].工业工程与管理,2017,22 (6):25-31.
[3] Park C,Ouyang L,Byun J H,et al.Robust design under normal model departure[J].Computers & Industrial Engineering,2017,113:206-220.
[4] 顾晓光,马义中,刘健,等.基于单值数据过程能力指数的多元质量特性稳健参数设计[J].系统工程与电子技术,2017,39 (2):362-368.
[5] Vining G G,Myers R H.Combining Taguchi and response surface philosophies:A dual response approach[J].Journal of Quality Technology,1990,22 (1):38-45.
[6] Hund E,Massart D L,Smeyers-Verbeke J.Robust regression and outlier detection in the evaluation of robustness tests with different experimental design[J].Analytica Chimica Acta,2002,463 (1):53-73.
[7] Ripley B D.Robust statistics[M].Oxford:University of Oxford,2004.
[8] Serfling R.Asymptotic relative efficiency in estimation[M].Springer Berlin Heidelberg,2011.
[9] Park C,Cho B R.Development of robust design under contaminated and non-normal data[J].Quality Engineering,2003,15 (3):463-469.
[10] Lee S B,Park C,Cho B R.Development of a highly efficient and resistant robust design[J].International Journal of Production Research,2007,45 (1):157-167.
[11] Huber P J.Robust estimation of a location parameter[J].The Annals of Mathematical Statistics,1964,35 (1):73-101.
[12] Yohai V J,Stahel W A,Zamar R H.A procedure for robust estimation and inference in linear regression[M].Springer New York,1991.
[13] Park Y,Kim D,Kim S.Robust regression using data partitioning and M-estimation[J].Communications in Statistics Simulation and Computation,2012,41 (8):1282-1300.
[14] Park C,Leeds M.A highly efficient robust design under data contamination[J].Computers & Industrial Engineering,2016,93:131-142.
[15] Lehmann E L.Elements of large-sample theory[M].Springer Science and Business Media,2004.
[16] Lévy-Leduc C,Boistard H,Moulines E,et al.Large sample behaviour of some well-known robust estimators under long-range dependence[J].Statistics,2011,45 (1):59-71.
[17] Efron B.The jackknife,the bootstrap and other resampling plans[M].Society for industrial and applied mathematics,1982.
[18] Vining G G,Myers R H.Combining Taguchi and Response Surface Philosophies:A Dual Response Approach[J].Journal of Quality Technology,1990,22 (1):38-45.
[19] Johnson R A,Wichern D W.Applied multivariate statistical analysis[M].New Jersey:Prentice-Hall,2014.
[2] 韩云霞,马义中,欧阳林寒,等.基于两类因子波动下的稳健优化设计[J].工业工程与管理,2017,22 (6):25-31.
[3] Park C,Ouyang L,Byun J H,et al.Robust design under normal model departure[J].Computers & Industrial Engineering,2017,113:206-220.
[4] 顾晓光,马义中,刘健,等.基于单值数据过程能力指数的多元质量特性稳健参数设计[J].系统工程与电子技术,2017,39 (2):362-368.
[5] Vining G G,Myers R H.Combining Taguchi and response surface philosophies:A dual response approach[J].Journal of Quality Technology,1990,22 (1):38-45.
[6] Hund E,Massart D L,Smeyers-Verbeke J.Robust regression and outlier detection in the evaluation of robustness tests with different experimental design[J].Analytica Chimica Acta,2002,463 (1):53-73.
[7] Ripley B D.Robust statistics[M].Oxford:University of Oxford,2004.
[8] Serfling R.Asymptotic relative efficiency in estimation[M].Springer Berlin Heidelberg,2011.
[9] Park C,Cho B R.Development of robust design under contaminated and non-normal data[J].Quality Engineering,2003,15 (3):463-469.
[10] Lee S B,Park C,Cho B R.Development of a highly efficient and resistant robust design[J].International Journal of Production Research,2007,45 (1):157-167.
[11] Huber P J.Robust estimation of a location parameter[J].The Annals of Mathematical Statistics,1964,35 (1):73-101.
[12] Yohai V J,Stahel W A,Zamar R H.A procedure for robust estimation and inference in linear regression[M].Springer New York,1991.
[13] Park Y,Kim D,Kim S.Robust regression using data partitioning and M-estimation[J].Communications in Statistics Simulation and Computation,2012,41 (8):1282-1300.
[14] Park C,Leeds M.A highly efficient robust design under data contamination[J].Computers & Industrial Engineering,2016,93:131-142.
[15] Lehmann E L.Elements of large-sample theory[M].Springer Science and Business Media,2004.
[16] Lévy-Leduc C,Boistard H,Moulines E,et al.Large sample behaviour of some well-known robust estimators under long-range dependence[J].Statistics,2011,45 (1):59-71.
[17] Efron B.The jackknife,the bootstrap and other resampling plans[M].Society for industrial and applied mathematics,1982.
[18] Vining G G,Myers R H.Combining Taguchi and Response Surface Philosophies:A Dual Response Approach[J].Journal of Quality Technology,1990,22 (1):38-45.
[19] Johnson R A,Wichern D W.Applied multivariate statistical analysis[M].New Jersey:Prentice-Hall,2014.