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題名 無母數指數加權移動平均管制圖伴隨變動管制界限
A Nonparametric EWMA-Type Signed-Rank Control Chart with Time-Varying Control Limits
作者 鄭舜壕
貢獻者 黃子銘
鄭舜壕
關鍵詞 無母數管制圖
指數加權移動平均
平均連串長度
日期 2009
上傳時間 5-Sep-2013 15:11:19 (UTC+8)
摘要 根據Steiner (1999) 提出指數加權移動平均 (EWMA) 管制圖之管制界限應伴隨時間變動，相較於傳統以漸近管制界限建構的 X-bar EWMA 管制圖，具備類似於快速起始反應之功能。然而，無母數EWMA 管制圖相關文獻中，大多採用漸近管制界限，甚少提及變動管制界限對於製程初期偵測能力之影響，因此本研究依據Wilcoxon 符號排序統計量為基礎，建構無母數EWMA 管制圖，並定義變動管制界限之形式，進而探討在製程初期的監控效果。假設製程為常態、均勻或雙指數分配下，使用非齊一性馬可夫鏈及蒙地卡羅模擬，求得製程穩定或失控狀態下的平均連串長度。模擬結果顯示，當加權常數越小，若採用變動管制界限能有效提升對於製程初期異常之偵測能力，且在厚尾分配下(例如：雙指數分配) 效果更為明顯。
According to Steiner (1999), the control limits of exponentially weighted moving average (EWMA) control charts should vary with time, so that the charts would have properties similar to the fast initial response (FIR) feature, when compared with asymptotic X-bar EWMA charts. However, previous analyses of nonparametric EWMA control charts consider only asymptotic control limits and are not sensitive to the shifts in a process at early stages. In this thesis, a nonparametric control chart with time-varying control limits is constructed based on EWMA control chart built upon the Wilcoxon signed-rank statistics. When the underlying distribution is normal, uniform, or double exponential, the average run lengths in both in-control and out-of-control conditions are approximated using non-homogenous Markov chain and based on Monte Carlo simulations. Simulation results show that EWMA charts with varying control limits are more efficient to detect early process shifts when weighting constants are small, and the underlying distributions are heavy-tailed distribution (such as double exponential distribution).
參考文獻 [1] Abate, J. and Whitt, W. (1992), The Fourier Series Method for Inverting Transforms of Probability Distributions", Queueing systems, 10(1), 5-87.
[2] Alloway, J. and Raghavachari, M. (1991), Control Chart Based on the Hodges-Lehmann Estimator", Journal of Quality Technology, 23(4), 336-347.
[3] Amin, R. W., Reynolds, M. R. Jr., and Bakir, S. T. (1995), Nonparametric Quality Control Charts Based on the Sign Statistic", Communications in Statistics - Theory and Methods, 24(6), 1597-1623.
[4] Amin, R. W. and Searcy, A. J. (1991), A Nonparametric Exponentially Weighted Moving Average Control Scheme", Communications in Statistics - Simulation and Computation, 20(4), 1049-1072.
[5] Arnold, H. J. (1965), Small Sample Power of the one Sample Wilcoxon Test for Non-Normal Shift Alternatives", The Annals of Mathematical Statistics, 36(6), 1767-1778.
[6] Bakir, S. T. (2004), A Distribution Free Shewhart Quality Control Chart Based on Signed Ranks", Quality Engineering, 16(4), 613-623.
[7] Bakir, S. T. (2006), Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics", Communications in Statistics - Theory and Methods, 35(4), 743-757.
[8] Bakir, S. T., Chakraborti, S., and Van der Laan, P. (2001), Nonparametric Control Charts: An Overview and Some Results", Journal of Quality Technology, 33(3), 304-315.
[9] Bakir, S. T. and Reynolds, M. R. Jr. (1979), A Nonparametric Procedure for Process Control Based on Within-Group Ranking", Technometrics, 21(2), 175-183.
[10] Borror, C. M., Champ, C. W., and Rigdon, S. E. (1998), Poisson EWMA Control Charts", Journal of Quality Technology, 30(4), 352-361.
[11] Chakraborti, S. and Eryilmaz, S. (2007), A Nonparametric Shewhart-Type Signed-Rank Control Chart Based on Runs", Communications in Statistics: Simulation and Computation, 36(2), 335-356.
[12] Chakraborti, S. and Van de Wiel, M. A. (2008), A Nonparametric Control Chart Based on The Mann Whitney Statistic", Institute of Mathematical Statistics, 1, 156-172.
[13] Chandrasekaran, S., English, J. R., and Disney, R. L. (1995), Modeling and Analysis of EWMA Control Schemes with Variance Adjusted Control Limits", IIE Transactions, 27(3), 282-290.
[14] Crowder, S. V. (1987), A Simple Method for Studying Run Length Distributions of Exponentially Weighted Moving Average Charts", Technometrics, 29(4), 401-407.
[15] Hackl, P. and Ledolter, J. (1991), A Control Chart Based on Ranks", Journal of Quality Technology, 23(2), 117-124.
[16] Hackl, P. and Ledolter, J. (1992), A New Nonparametric Quality Control Technique", Communications in Statistics - Simulation and Computation, 21(2), 423-443.
[17] Knoth, S. (2003), EWMA Schemes with Nonhomogeneous Transition Kernels", Sequential Analysis, 22(3), 241-255.
[18] Knoth, S. (2005), Fast Initial Response Features for EWMA Control Charts", Statistical Papers, 46(1), 47-64.
[19] Lucas, J. M. and Crosier, R. B. (1982), Fast Initial Response for CUSUM Quality Control Schemes: Give Your CUSUM a Head Start", Technometrics, 24(3), 199-205.
[20] Lucas, J. M. and Saccucci, M. S. (1990), Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements", Technometrics, 32(1), 1-12.
[21] MacGregor, J. F. and Harris, T. J. (1993), The Exponentially Weighted Moving Variance", Journal of Quality Technology, 25, 106-106.
[22] McDonald, D. (1990), A CUSUM Procedure Based on Sequential Ranks", Naval Research Logistics, 37(5), 627-646.
[23] Montgomery, D. C. (1991), Introduction to Statistical Quality Control, New York : Wiley, 2 edition.
[24] Reynolds, M. R. Jr. (1975), A Sequential Signed Rank Test for Symmetry", The Annals of Statistics, 3(2), 382-400.
[25] Rhoads, T. R., Montgomery, D. C., and Mastrangelo, C. M. (1996), A Fast Initial Response Scheme for the Exponentially Weighted Moving Average Control Chart", Quality Engineering, 9(2), 317-327.
[26] Roberts, S. W. (1959), Control Chart Tests Based on Geometric Moving Averages", Technometrics, 3, 239-250.
[27] Robinson, P. B. and Ho, T. Y. (1978), Average Run Lengths of Geometric Moving Average Charts by Numerical Methods", Technometrics, 20(1), 85-93.
[28] Steiner, S. H. (1999), Exponentially Weighted Moving Average Control Charts with Time Varying Control Limits and Fast Initial Response", Journal of Quality Technology, 31, 75-86.
[29] Tarter, M. E. and Kronmal, R. A. (1976), An Introduction to The Implementation and Theory of Nonparametric Density Estimation", American Statistician, 30(3), 105-112.
[30] Woodall, W. H. (2000), Controversies and Contradictions in Statistical Process Control", Journal of Quality Technology, 32(4), 341-378.
[31] Woodall, W. H. and Montgomery, D. C. (1999), Research Issues and Ideas in Statistical Process Control", Journal of Quality Technology, 31, 376-386.
[32] Yourstone, S. A. and Zimmer, W. J. (2007), Non Normality and the Design of Control Charts for Averages", Decision Sciences, 23(5), 1099-1113.
描述 碩士
國立政治大學
統計研究所
97354020
98
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0097354020
資料類型 thesis
dc.contributor.advisor 黃子銘zh_TW
dc.contributor.author (Authors) 鄭舜壕zh_TW
dc.creator (作者) 鄭舜壕zh_TW
dc.date (日期) 2009en_US
dc.date.accessioned 5-Sep-2013 15:11:19 (UTC+8)-
dc.date.available 5-Sep-2013 15:11:19 (UTC+8)-
dc.date.issued (上傳時間) 5-Sep-2013 15:11:19 (UTC+8)-
dc.identifier (Other Identifiers) G0097354020en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/60435-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計研究所zh_TW
dc.description (描述) 97354020zh_TW
dc.description (描述) 98zh_TW
dc.description.abstract (摘要) 根據Steiner (1999) 提出指數加權移動平均 (EWMA) 管制圖之管制界限應伴隨時間變動，相較於傳統以漸近管制界限建構的 X-bar EWMA 管制圖，具備類似於快速起始反應之功能。然而，無母數EWMA 管制圖相關文獻中，大多採用漸近管制界限，甚少提及變動管制界限對於製程初期偵測能力之影響，因此本研究依據Wilcoxon 符號排序統計量為基礎，建構無母數EWMA 管制圖，並定義變動管制界限之形式，進而探討在製程初期的監控效果。假設製程為常態、均勻或雙指數分配下，使用非齊一性馬可夫鏈及蒙地卡羅模擬，求得製程穩定或失控狀態下的平均連串長度。模擬結果顯示，當加權常數越小，若採用變動管制界限能有效提升對於製程初期異常之偵測能力，且在厚尾分配下(例如：雙指數分配) 效果更為明顯。zh_TW
dc.description.abstract (摘要) According to Steiner (1999), the control limits of exponentially weighted moving average (EWMA) control charts should vary with time, so that the charts would have properties similar to the fast initial response (FIR) feature, when compared with asymptotic X-bar EWMA charts. However, previous analyses of nonparametric EWMA control charts consider only asymptotic control limits and are not sensitive to the shifts in a process at early stages. In this thesis, a nonparametric control chart with time-varying control limits is constructed based on EWMA control chart built upon the Wilcoxon signed-rank statistics. When the underlying distribution is normal, uniform, or double exponential, the average run lengths in both in-control and out-of-control conditions are approximated using non-homogenous Markov chain and based on Monte Carlo simulations. Simulation results show that EWMA charts with varying control limits are more efficient to detect early process shifts when weighting constants are small, and the underlying distributions are heavy-tailed distribution (such as double exponential distribution).en_US
dc.description.tableofcontents 第一章 緒論 1
第二章 文獻探討與回顧 3
第一節 無母數管制圖 3
第二節 指數加權移動平均管制圖 4
第三節 平均連串長度 6
第三章 研究方法 8
第一節 WSR-EWMA 8
第二節 WSR-TEWMA 10
第三節 計算平均連串長度 15
第四章 分析與比較 19
第五章 結論與未來研究方向 26
參考文獻 28
附錄甲：Wilcoxon 符號排序 32
附錄乙：傅立葉級數 35		zh_TW
dc.format.extent 1149697 bytes-
dc.format.extent 1149697 bytes-
dc.format.mimetype application/pdf-
dc.format.mimetype application/pdf-
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0097354020en_US
dc.subject (關鍵詞) 無母數管制圖zh_TW
dc.subject (關鍵詞) 指數加權移動平均zh_TW
dc.subject (關鍵詞) 平均連串長度zh_TW
dc.title (題名) 無母數指數加權移動平均管制圖伴隨變動管制界限zh_TW
dc.title (題名) A Nonparametric EWMA-Type Signed-Rank Control Chart with Time-Varying Control Limitsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] Abate, J. and Whitt, W. (1992), The Fourier Series Method for Inverting Transforms of Probability Distributions", Queueing systems, 10(1), 5-87.
[2] Alloway, J. and Raghavachari, M. (1991), Control Chart Based on the Hodges-Lehmann Estimator", Journal of Quality Technology, 23(4), 336-347.
[3] Amin, R. W., Reynolds, M. R. Jr., and Bakir, S. T. (1995), Nonparametric Quality Control Charts Based on the Sign Statistic", Communications in Statistics - Theory and Methods, 24(6), 1597-1623.
[4] Amin, R. W. and Searcy, A. J. (1991), A Nonparametric Exponentially Weighted Moving Average Control Scheme", Communications in Statistics - Simulation and Computation, 20(4), 1049-1072.
[5] Arnold, H. J. (1965), Small Sample Power of the one Sample Wilcoxon Test for Non-Normal Shift Alternatives", The Annals of Mathematical Statistics, 36(6), 1767-1778.
[6] Bakir, S. T. (2004), A Distribution Free Shewhart Quality Control Chart Based on Signed Ranks", Quality Engineering, 16(4), 613-623.
[7] Bakir, S. T. (2006), Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics", Communications in Statistics - Theory and Methods, 35(4), 743-757.
[8] Bakir, S. T., Chakraborti, S., and Van der Laan, P. (2001), Nonparametric Control Charts: An Overview and Some Results", Journal of Quality Technology, 33(3), 304-315.
[9] Bakir, S. T. and Reynolds, M. R. Jr. (1979), A Nonparametric Procedure for Process Control Based on Within-Group Ranking", Technometrics, 21(2), 175-183.
[10] Borror, C. M., Champ, C. W., and Rigdon, S. E. (1998), Poisson EWMA Control Charts", Journal of Quality Technology, 30(4), 352-361.
[11] Chakraborti, S. and Eryilmaz, S. (2007), A Nonparametric Shewhart-Type Signed-Rank Control Chart Based on Runs", Communications in Statistics: Simulation and Computation, 36(2), 335-356.
[12] Chakraborti, S. and Van de Wiel, M. A. (2008), A Nonparametric Control Chart Based on The Mann Whitney Statistic", Institute of Mathematical Statistics, 1, 156-172.
[13] Chandrasekaran, S., English, J. R., and Disney, R. L. (1995), Modeling and Analysis of EWMA Control Schemes with Variance Adjusted Control Limits", IIE Transactions, 27(3), 282-290.
[14] Crowder, S. V. (1987), A Simple Method for Studying Run Length Distributions of Exponentially Weighted Moving Average Charts", Technometrics, 29(4), 401-407.
[15] Hackl, P. and Ledolter, J. (1991), A Control Chart Based on Ranks", Journal of Quality Technology, 23(2), 117-124.
[16] Hackl, P. and Ledolter, J. (1992), A New Nonparametric Quality Control Technique", Communications in Statistics - Simulation and Computation, 21(2), 423-443.
[17] Knoth, S. (2003), EWMA Schemes with Nonhomogeneous Transition Kernels", Sequential Analysis, 22(3), 241-255.
[18] Knoth, S. (2005), Fast Initial Response Features for EWMA Control Charts", Statistical Papers, 46(1), 47-64.
[19] Lucas, J. M. and Crosier, R. B. (1982), Fast Initial Response for CUSUM Quality Control Schemes: Give Your CUSUM a Head Start", Technometrics, 24(3), 199-205.
[20] Lucas, J. M. and Saccucci, M. S. (1990), Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements", Technometrics, 32(1), 1-12.
[21] MacGregor, J. F. and Harris, T. J. (1993), The Exponentially Weighted Moving Variance", Journal of Quality Technology, 25, 106-106.
[22] McDonald, D. (1990), A CUSUM Procedure Based on Sequential Ranks", Naval Research Logistics, 37(5), 627-646.
[23] Montgomery, D. C. (1991), Introduction to Statistical Quality Control, New York : Wiley, 2 edition.
[24] Reynolds, M. R. Jr. (1975), A Sequential Signed Rank Test for Symmetry", The Annals of Statistics, 3(2), 382-400.
[25] Rhoads, T. R., Montgomery, D. C., and Mastrangelo, C. M. (1996), A Fast Initial Response Scheme for the Exponentially Weighted Moving Average Control Chart", Quality Engineering, 9(2), 317-327.
[26] Roberts, S. W. (1959), Control Chart Tests Based on Geometric Moving Averages", Technometrics, 3, 239-250.
[27] Robinson, P. B. and Ho, T. Y. (1978), Average Run Lengths of Geometric Moving Average Charts by Numerical Methods", Technometrics, 20(1), 85-93.
[28] Steiner, S. H. (1999), Exponentially Weighted Moving Average Control Charts with Time Varying Control Limits and Fast Initial Response", Journal of Quality Technology, 31, 75-86.
[29] Tarter, M. E. and Kronmal, R. A. (1976), An Introduction to The Implementation and Theory of Nonparametric Density Estimation", American Statistician, 30(3), 105-112.
[30] Woodall, W. H. (2000), Controversies and Contradictions in Statistical Process Control", Journal of Quality Technology, 32(4), 341-378.
[31] Woodall, W. H. and Montgomery, D. C. (1999), Research Issues and Ideas in Statistical Process Control", Journal of Quality Technology, 31, 376-386.
[32] Yourstone, S. A. and Zimmer, W. J. (2007), Non Normality and the Design of Control Charts for Averages", Decision Sciences, 23(5), 1099-1113.		zh_TW