基于似然率检验的过程控制方法研究
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摘要
统计过程控制是保证过程处于统计受控状态的重要技术手段,也是保证产品质量特性持续稳定的有效方法。传统的过程控制方法主要包括休哈特控制图、累计和控制图和指数加权移动平均控制图等,这些控制方法在对过程进行控制时都必须事先估计过程分布参数,以便确定控制图的中心线和控制限。
由于在实际应用中,过程分布的参数值往往不是事先已知的。常用的做法是通过第一阶段的抽样数据来估计过程分布的参数值,然后再对第二阶段动态数据进行过程控制。但是通过这种参数估计方法一定会产生误差,即使第一阶段通过大量抽样来进行参数估计,该误差也是十分明显的,将导致过程运行长度的分布不稳定,大大降低了传统控制图的准确度。
针对上述传统控制方法存在的缺陷,本文将介绍一种基于似然率检验的过程控制方法,它应用似然率原理,借助未知参数的变点方程,采用统一的均值方差联合控制图,诠释了一种新的过程控制方法。它不需要第一阶段的大量抽样来确定过程分布参数,节约了成本,同时也弥补了传统控制方法的缺陷。
本文还深入分析了基于似然率检验的过程控制方法性能表现,以及确定变异来源的准确度等。另外,通过与累计和控制图和指数加权移动平均控制图进行横向对比研究,似然率检验的控制方法对于均值和方差偏移十分敏感,能够及时的发现过程的异常波动,而且该方法的表现还具有很大的优越性和稳定性。
Statistical process control (SPC) is not only an important technical method to ensure that the process is in a state of statistic,but also a method to maintain the stability of product quality characteristics.Conventionally,process control methods include Shewhart control chart, cumulative sum chart and exponentially weighted moving average chart, and so on.In order to determine the centerline and control limits,these control charts require the parameters of process distribution were estimated in advance.
Actually,the parameters are often not known beforehand,they are commonly estimated from a phaseⅠsample.Even if we use a large number of samples to estimate the parameters in phaseⅠ,the random errors are very obvious, and they will lead to uncertain run length distribution of the resulting charts.In order to solve these problems,this article introduces a process control method based on unknown likelihood ratio test.It applies the principle of likelihood ratio,with change-point formulation and a single chart to detect a shift in mean, in variance or in both.
This paper also analyzes the performance of the process control method based on likelihood ratio test.In addition,through a comparative study with cumulative sum chart and exponentially moving average chart,this control method is very sensitive to a shift in mean or variance,which can timely detect the abnormal fluctuations in the process.Moreover,comparing with the traditional control methods,it also has great advantages and stability.
由于在实际应用中,过程分布的参数值往往不是事先已知的。常用的做法是通过第一阶段的抽样数据来估计过程分布的参数值,然后再对第二阶段动态数据进行过程控制。但是通过这种参数估计方法一定会产生误差,即使第一阶段通过大量抽样来进行参数估计,该误差也是十分明显的,将导致过程运行长度的分布不稳定,大大降低了传统控制图的准确度。
针对上述传统控制方法存在的缺陷,本文将介绍一种基于似然率检验的过程控制方法,它应用似然率原理,借助未知参数的变点方程,采用统一的均值方差联合控制图,诠释了一种新的过程控制方法。它不需要第一阶段的大量抽样来确定过程分布参数,节约了成本,同时也弥补了传统控制方法的缺陷。
本文还深入分析了基于似然率检验的过程控制方法性能表现,以及确定变异来源的准确度等。另外,通过与累计和控制图和指数加权移动平均控制图进行横向对比研究,似然率检验的控制方法对于均值和方差偏移十分敏感,能够及时的发现过程的异常波动,而且该方法的表现还具有很大的优越性和稳定性。
Statistical process control (SPC) is not only an important technical method to ensure that the process is in a state of statistic,but also a method to maintain the stability of product quality characteristics.Conventionally,process control methods include Shewhart control chart, cumulative sum chart and exponentially weighted moving average chart, and so on.In order to determine the centerline and control limits,these control charts require the parameters of process distribution were estimated in advance.
Actually,the parameters are often not known beforehand,they are commonly estimated from a phaseⅠsample.Even if we use a large number of samples to estimate the parameters in phaseⅠ,the random errors are very obvious, and they will lead to uncertain run length distribution of the resulting charts.In order to solve these problems,this article introduces a process control method based on unknown likelihood ratio test.It applies the principle of likelihood ratio,with change-point formulation and a single chart to detect a shift in mean, in variance or in both.
This paper also analyzes the performance of the process control method based on likelihood ratio test.In addition,through a comparative study with cumulative sum chart and exponentially moving average chart,this control method is very sensitive to a shift in mean or variance,which can timely detect the abnormal fluctuations in the process.Moreover,comparing with the traditional control methods,it also has great advantages and stability.
引文
[1]郎志正,质量管理及其技术和方法,北京:中国标准出版社,2003.
[2]Quesenberry,C.P.SPC Q-Charts for Start-up Processes and Short or Long Runs.Journal of Quality Technology,1991,33,3.
[3]Hawkins,D.M.and Olwell,D.H.Cumulative Sum Charts and Charting for Quality Improbement.New York:Springer-Verlag,1998.
[4]Hawkins,D.M.Testing a Sequence of Observations for a Shift in Location.Journal of the American Statistical Association,1977,72:180~186.
[5]Worsley,K.J.On the likelihood Ratio Test for a Shift in Location of Normal Populations.Journal of the American Statistical Association,1979,74:365~367.
[6]Carlstein,E.G.,Muller,H.G.,and Siegmund,D.Change-Point Problems.Hayward,CA:Institute for Mathematical Statistics,1994.
[7]Lai,T.L.Sequential Analysis:Some Classical Problems and New Challenges.Statistica Sinica,2001,11:303~350.
[8]Pignatiello,J.J.Jr,and Samuel,T.R.Estimation of the Change Point of a Normal Process Mean in SPC Applications.Journal of Quality Technology,2001,33:82~95.
[9]Samuel,T.R.,Pignatiello,J.J.Jr.,and Calvin,J.A.Identifying the Time of a Step Change With Xbar Control Charts.Quality Engineering,1998,10:521~527.
[10]Hawkins,D.M.,Qiu,P.,and Kang,C.W.The Changepoint Model for Statistical Process Control.Journal of Quality Technology,2003,35:355~365.
[11]Hawkins,D.M.,and Zamba,K.D.A Change Point Model for a Shift in Variance.Journal of Quality Technology,2005,37:21~31.
[12]Hawkins,D.M.,and Zamba,K.D.Statistical Process Control for Shift in Mean or Variance Using a Changepoint Formulation.Technometrics,2005,47,2:164~173.
[13] Sullivan, J. H. & Woodall, W. H.A control chart for preliminary analysis of individual observations.Journal of Quality Technology,1996,28:265~278.
[14] Koning, A. J. and Does, R. J. M. M.CUSUM control charts for preliminary analysis of individual observations.Journal of Quality Technology,2000,32:122~132.
[15]代毅,王兆军,邹长亮,CUSUM control charts based on likelihood ratio for preliminary analysis.Science in China Series A:Mathematics Jan.,2007,50,1:47~62.
[16]Page,E.S.Continuous Inspection Schemes.Biometrics,1954,41.
[17] Stoumbos, Z. G., and Reynolds, Jr. M. R.Control charts applying a sequential test at fixed sampling intervals.Journal of Quality Technology,1997,29: 21~40.
[18] Stoumbos, Z. G., Reynolds, Jr., M. R.The SPRT control chart for the process mean with samples starting at fixed times.Nonlinear Analysis:Real World Applications,2001,2:1~34.
[19]赵宁,韩东,宗福季,CUSUM,GLR,GEWMA以及RFCuscore控制图监测稳定过程均值漂移的效果比较,应用概率统计,2005,21,4:403~411.
[20]张翠玲,基于似然率方法的语音证据评价,证据科学,2008,16,3:337~342.
[21]Roberts,S.W.Control Chart Tests Based on Geometric Moving Averages.Technometrics,1959,1.
[22]杨跃进,统计过程控制技术,北京:航空工业出版社,2003.
[23]罗国勋,质量管理与可靠性,北京:高等教育出版社,2005.
[24]马林,何桢,六西格玛管理,北京:中国人民大学出版社,2007.
[25]Montgomery,D.C.Introduction to Statistical Quality Control,3nd ed.New York:John Wiley&Sons,Inc,1996.
[26]Wadsworth,H.M.Stephens,K.S.and Godfrey,A.B.Adapted from Modern Methods for Quality Control and Improvement.John Wiley&Sons,1986.
[27]Lawley,D.M.A General Method for Approximation to the Distribution of Likelihood Ratio Criteria.Biometrika,1956,43:295~303.
[28]Kendall,M.G,and Stuart,A.The Advanced Theory of Statistics.New York:Hafner,1961,2.
[29]Sullivan,J.H,and Woodall,W.H.A Control Chart for Preliminary Analysis of Individual Observations.Journal of Quality Technology,1996,28,3.
[30]Hawkins,D.M.Cumulative Sum Control Charting:An Underutilized SPC Tool.Quality Engineering,1993,5.
[31]Crowder,S.V.A Simple Method for Studying Run-Length Distributions of Exponentially Weighted Moving Average Charts.Technometrics,1987,29,4:401~407.
[32]Lucas,J.M,and M.S.Saccucci.Exponentially Weighted Moving Average Control Schemes:Properties and Enhancements.Technometrics,1990,32,1:1~29.
[2]Quesenberry,C.P.SPC Q-Charts for Start-up Processes and Short or Long Runs.Journal of Quality Technology,1991,33,3.
[3]Hawkins,D.M.and Olwell,D.H.Cumulative Sum Charts and Charting for Quality Improbement.New York:Springer-Verlag,1998.
[4]Hawkins,D.M.Testing a Sequence of Observations for a Shift in Location.Journal of the American Statistical Association,1977,72:180~186.
[5]Worsley,K.J.On the likelihood Ratio Test for a Shift in Location of Normal Populations.Journal of the American Statistical Association,1979,74:365~367.
[6]Carlstein,E.G.,Muller,H.G.,and Siegmund,D.Change-Point Problems.Hayward,CA:Institute for Mathematical Statistics,1994.
[7]Lai,T.L.Sequential Analysis:Some Classical Problems and New Challenges.Statistica Sinica,2001,11:303~350.
[8]Pignatiello,J.J.Jr,and Samuel,T.R.Estimation of the Change Point of a Normal Process Mean in SPC Applications.Journal of Quality Technology,2001,33:82~95.
[9]Samuel,T.R.,Pignatiello,J.J.Jr.,and Calvin,J.A.Identifying the Time of a Step Change With Xbar Control Charts.Quality Engineering,1998,10:521~527.
[10]Hawkins,D.M.,Qiu,P.,and Kang,C.W.The Changepoint Model for Statistical Process Control.Journal of Quality Technology,2003,35:355~365.
[11]Hawkins,D.M.,and Zamba,K.D.A Change Point Model for a Shift in Variance.Journal of Quality Technology,2005,37:21~31.
[12]Hawkins,D.M.,and Zamba,K.D.Statistical Process Control for Shift in Mean or Variance Using a Changepoint Formulation.Technometrics,2005,47,2:164~173.
[13] Sullivan, J. H. & Woodall, W. H.A control chart for preliminary analysis of individual observations.Journal of Quality Technology,1996,28:265~278.
[14] Koning, A. J. and Does, R. J. M. M.CUSUM control charts for preliminary analysis of individual observations.Journal of Quality Technology,2000,32:122~132.
[15]代毅,王兆军,邹长亮,CUSUM control charts based on likelihood ratio for preliminary analysis.Science in China Series A:Mathematics Jan.,2007,50,1:47~62.
[16]Page,E.S.Continuous Inspection Schemes.Biometrics,1954,41.
[17] Stoumbos, Z. G., and Reynolds, Jr. M. R.Control charts applying a sequential test at fixed sampling intervals.Journal of Quality Technology,1997,29: 21~40.
[18] Stoumbos, Z. G., Reynolds, Jr., M. R.The SPRT control chart for the process mean with samples starting at fixed times.Nonlinear Analysis:Real World Applications,2001,2:1~34.
[19]赵宁,韩东,宗福季,CUSUM,GLR,GEWMA以及RFCuscore控制图监测稳定过程均值漂移的效果比较,应用概率统计,2005,21,4:403~411.
[20]张翠玲,基于似然率方法的语音证据评价,证据科学,2008,16,3:337~342.
[21]Roberts,S.W.Control Chart Tests Based on Geometric Moving Averages.Technometrics,1959,1.
[22]杨跃进,统计过程控制技术,北京:航空工业出版社,2003.
[23]罗国勋,质量管理与可靠性,北京:高等教育出版社,2005.
[24]马林,何桢,六西格玛管理,北京:中国人民大学出版社,2007.
[25]Montgomery,D.C.Introduction to Statistical Quality Control,3nd ed.New York:John Wiley&Sons,Inc,1996.
[26]Wadsworth,H.M.Stephens,K.S.and Godfrey,A.B.Adapted from Modern Methods for Quality Control and Improvement.John Wiley&Sons,1986.
[27]Lawley,D.M.A General Method for Approximation to the Distribution of Likelihood Ratio Criteria.Biometrika,1956,43:295~303.
[28]Kendall,M.G,and Stuart,A.The Advanced Theory of Statistics.New York:Hafner,1961,2.
[29]Sullivan,J.H,and Woodall,W.H.A Control Chart for Preliminary Analysis of Individual Observations.Journal of Quality Technology,1996,28,3.
[30]Hawkins,D.M.Cumulative Sum Control Charting:An Underutilized SPC Tool.Quality Engineering,1993,5.
[31]Crowder,S.V.A Simple Method for Studying Run-Length Distributions of Exponentially Weighted Moving Average Charts.Technometrics,1987,29,4:401~407.
[32]Lucas,J.M,and M.S.Saccucci.Exponentially Weighted Moving Average Control Schemes:Properties and Enhancements.Technometrics,1990,32,1:1~29.