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題名 追蹤品質變數分配變化的Phase II管制圖
Phase II Control Charts for Monitoring Abnormal Process Distributions
作者 潘禹翔
Pan, Yu-Hsiang
貢獻者 楊素芬
Yang, Su-Fen
潘禹翔
Pan, Yu-Hsiang
關鍵詞 管制圖
指數移動平均
平均連串長度
累積分配函數
control chart
exponentially weighted moving average
average run length
CDF (cumulative distribution function)
日期 2023
上傳時間 1-Sep-2023 14:56:35 (UTC+8)
摘要 近年來，用於監測單維品質變數整個分配是否有變化的管制圖有新的發展。目前的文獻多以CDF服從均勻分配據以建立管制圖，用來偵測整個分配是否發生變化，且部分研究考慮樣本數為一(n=1)的情況。實務上，在製程管制中樣本數為一的情況較少，於是我們的研究動機是延伸1991年Hackl和Ledolter的研究，考慮樣本數大於一並且品質變數不受限制下的Phase II管制圖，用以監測品質變數分配是否發生變化。
在本研究中，我們建立兩個Phase II管制圖(SEWMA-R ̅ 管制圖和SEWMA-T管制圖)以監測品質變數的分配變化，並使用平均連串長度(ARL)作為管制圖偵測能力指標。我們考慮四個不同的失控分配，分別為常態、雙指數、偏斜常態和伽瑪分配，數據分析結果顯示在偏移平均數時，SEWMA-R ̅ 管制圖的偵測能力在伽瑪分配下最佳；在偏移變異數時，SEWMA-T管制圖的偵測能力在伽瑪分配表現最佳。最後，我們將SEWMA-R ̅ 管制圖和SEWMA-T管制圖與兩種現有的管制圖進行比較，在伽瑪分配下，SEWMA-R ̅ 管制圖在偵測平均數的偏移方面表現出較好的檢測性能，SEWMA-T管制圖在監測平均數和變異數同時偏移時表現出較好的偵測能力。
In recent years, there have been new developments in univariate control charts for monitoring changes in distribution. Current literature mostly focuses on control charts based on the cumulative distribution function (CDF) following a uniform distribution and used to detect distributional changes. Some studies consider the case where the sample size is one (n=1). However, in practice, the case of sample size one is less common in process control. Hence, our research motivation is considering sample size greater than one and developing the Phase II distribution-free control charts for monitoring abnormal process distributions by extending Hackl and Ledolter’s (1991) research.
In this study, we establish two Phase II control charts, the SEWMA-R ̅ chart and the SEWMA-T chart, to monitor changes in distribution. We use the average run length (ARL) as the performance measurement indicator for the control charts. We consider four different out-of-control distributions including normal, double exponential, skew normal and gamma distribution. The data analyses results show that, when mean shifts only, the SEWMA-R ̅ chart with gamma distributed performs the best. When variance shifts only, the SEWMA-T chart with gamma distributed performs the best. Finally, we compare the SEWMA-R ̅ chart and SEWMA-T chart with two existing control charts. Considering the gamma distribution, the SEWMA-R ̅ chart shows better detection performance in detecting shifts in process mean, while the SEWMA-T control chart exhibits better detection ability in monitoring shifts in the both mean and variance.
參考文獻 Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.
Amin, R. W., & Searcy, A. J. (1991). A nonparametric exponentially weighted moving average control scheme. Communications in Statistics-Simulation and Computation, 20(4), 1049-1072.
Amin, R. W., Reynolds Jr, M. R., & Saad, B. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistics-Theory and Methods, 24(6), 1597-1623.
Abu-Shawiesh, M. O. (2008). A simple robust control chart based on MAD. Journal of Mathematics and Statistics, 4(2), 102.
Abbasi, S. A., & Miller, A. (2013). MDEWMA chart: an efficient and robust alternative to monitor process dispersion. Journal of Statistical Computation and Simulation, 83(2), 247-268.
Abid, M., Nazir, H. Z., Riaz, M., & Lin, Z. (2017a). An efficient nonparametric EWMA Wilcoxon signed‐rank chart for monitoring location. Quality and Reliability Engineering International, 33(3), 669-685.
Abid, M., Nazir, H. Z., Riaz, M., & Lin, Z. (2017b). Investigating the impact of ranked set sampling in nonparametric CUSUM control charts. Quality and Reliability Engineering International, 33(1), 203-214.
Abbas, N. (2018). Homogeneously weighted moving average control chart with an application in substrate manufacturing process. Computers & Industrial Engineering, 120, 460-470.
Asghari, S., Sadeghpour Gildeh, B., Ahmadi, J., & Mohtashami Borzadaran, G. (2018). Sign control chart based on ranked set sampling. Quality Technology & Quantitative Management, 15(5), 568-588.
Alevizakos, V., Koukouvinos, C., & Chatterjee, K. (2020). A nonparametric double generally weighted moving average signed‐rank control chart for monitoring process location. Quality and Reliability Engineering International, 36(7), 2441-2458.
Alevizakos, V., Chatterjee, K., & Koukouvinos, C. (2021). An extended nonparametric homogeneously weighted moving average sign control chart. Quality and Reliability Engineering International, 37(8), 3395-3416.
Bates, G. E. (1955). Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya scheme. The Annals of Mathematical Statistics, 705-720.
Bakir, S. T., & Reynolds, M. R. (1979). A nonparametric procedure for process control based on within-group ranking. Technometrics, 21(2), 175-183.
Bakir, S. T. (2004). A distribution-free Shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623.
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.
Bakir, S. T. (2012). A nonparametric shewhart-type quality control chart for monitoring broad changes in a process distribution. Journal of Quality and Reliability Engineering, 2012.
Chen, G., Cheng, S. W., & Xie, H. (2001). Monitoring process mean and variability with one EWMA chart. Journal of Quality Technology, 33(2), 223-233.
Chen, J. T., Gupta, A. K., & Nguyen, T. T. (2004). The density of the skew normal sample mean and its applications. Journal of Statistical Computation and Simulation, 74(7), 487-494.
Champ, C. W., Jones-Farmer, L. A., & Rigdon, S. E. (2005). Properties of the T 2 control chart when parameters are estimated. Technometrics, 47(4), 437-445.
Chakraborti, S., & Eryilmaz, S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Communications in Statistics—Simulation and Computation®, 36(2), 335-356.
Chen, J. H., Lu, S. L., & Sheu, S. H. (2020). A nonparametric generally weighted moving average sign chart based on repetitive sampling. Communications in Statistics-Simulation and Computation, 51(3), 1137-1156.
Das, N., & Bhattacharya, A. (2008). A new non-parametric control chart for controlling variability. Quality Technology & Quantitative Management, 5(4), 351-361.
Fligner, M. A., & Killeen, T. J. (1976). Distribution-free two-sample tests for scale. Journal of the American Statistical Association, 71(353), 210-213.
Feller, W. (1991). An introduction to probability theory and its applications, Volume 2 (Vol. 81). John Wiley & Sons.
Graham, M. A., Chakraborti, S., & Human, S. W. (2011). A nonparametric EWMA sign chart for location based on individual measurements. Quality Engineering, 23(3), 227-241.
Godase, D. G., Rakitzis, A. C., Mahadik, S. B., & Khoo, M. B. (2022). Deciles‐based EWMA‐type sign charts for process dispersion. Quality and Reliability Engineering International, 38(7), 3726-3740.
Hackl, P., & Ledolter, J. (1991). A control chart based on ranks. Journal of Quality Technology, 23(2), 117-124.
Hackl, P., & Ledolter, J. (1992). A new nonparametric quality control technique. Communications in Statistics-Simulation and Computation, 21(2), 423-443.
Hawkins, D. M., Qiu, P., & Kang, C. W. (2003). The changepoint model for statistical process control. Journal of quality technology, 35(4), 355-366.
Hou, S., & Yu, K. (2021). A non-parametric CUSUM control chart for process distribution change detection and change type diagnosis. International Journal of Production Research, 59(4), 1166-1186.
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John wiley & sons.
Jones, L. A. (2002). The statistical design of EWMA control charts with estimated parameters. Journal of Quality Technology, 34(3), 277-288.
Jensen, W. A., Jones-Farmer, L. A., Champ, C. W., & Woodall, W. H. (2006). Effects of Parameter Estimation on Control Chart Properties: A Literature Review. Journal of Quality Technology, 38(4), 349-364.
Jones-Farmer, L. A., & Champ, C. W. (2010). A distribution-free Phase I control chart for subgroup scale. Journal of Quality Technology, 42(4), 373-387.
Khilare, S. K., & Shirke, D. T. (2012). Nonparametric synthetic control charts for process variation. Quality and Reliability Engineering International, 28(2), 193-202.
Lucas, J.M., & Saccucci, M.S. (1990). Exponentially weighted moving average control schemes: properties and enhancements (with discussion). Technometrics, 32(1), 1-29.
Liu R.Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387.
Liu, L., Chen, B., Zhang, J., & Zi, X. (2015). Adaptive phase II nonparametric EWMA control chart with variable sampling interval. Quality and Reliability Engineering International, 31(1), 15-26.
Lu, S. L. (2015). An extended nonparametric exponentially weighted moving average sign control chart. Quality and Reliability Engineering International, 31(1), 3-13.
Liang, W., Mukherjee, A., Xiang, D., & Xu, Z. (2022). A new nonparametric adaptive EWMA procedures for monitoring location and scale shifts via weighted Cucconi statistic. Computers & Industrial Engineering, 170, 108321.
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Perdikis, T., Psarakis, S., Castagliola, P., & Maravelakis, P. E. (2021). An EWMA signed ranks control chart with reliable run length performances. Quality and Reliability Engineering International, 37(3), 1266-1284.
Quesenberry, C. P. (1993). The effect of sample size on estimated limits for X ̅ and X control charts. Journal of Quality Technology, 25(4), 237-247.
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Reynolds Jr, M. R., & Stoumbos, Z. G. (2001). Monitoring the process mean and variance using individual observations and variable sampling intervals. Journal of Quality Technology, 33(2), 181-205.
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Riaz, M. (2015). A sensitive non-parametric EWMA control chart. Journal of the Chinese Institute of Engineers, 38(2), 208-219.
Riaz, M., Abid, M., Shabbir, A., Nazir, H. Z., Abbas, Z., & Abbasi, S. A. (2021). A non‐parametric double homogeneously weighted moving average control chart under sign statistic. Quality and Reliability Engineering International, 37(4), 1544-1560.
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Shirke, D. T., & Barale, M. S. (2018). A nonparametric CUSUM chart for process dispersion. Quality and Reliability Engineering International, 34(5), 858-866.
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Yang, S. F., Lin, J. S., & Cheng, S. W. (2011). A new nonparametric EWMA sign control chart. Expert Systems with Applications, 38(5), 6239-6243.
Yang, S. F., Cheng, T. C., Hung, Y. C., & W. Cheng, S. (2012). A new chart for monitoring service process mean. Quality and Reliability Engineering International, 28(4), 377-386.
Yang, S. F. (2016). An improved distribution-free EWMA mean chart. Communications in Statistics-Simulation and Computation, 45(4), 1410-1427.
Yang, S. F., & Arnold, B. C. (2016). A new approach for monitoring process variance. Journal of Statistical Computation and Simulation, 86(14), 2749-2765.
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描述 碩士
國立政治大學
統計學系
110354015
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0110354015
資料類型 thesis
dc.contributor.advisor 楊素芬zh_TW
dc.contributor.advisor Yang, Su-Fenen_US
dc.contributor.author (Authors) 潘禹翔zh_TW
dc.contributor.author (Authors) Pan, Yu-Hsiangen_US
dc.creator (作者) 潘禹翔zh_TW
dc.creator (作者) Pan, Yu-Hsiangen_US
dc.date (日期) 2023en_US
dc.date.accessioned 1-Sep-2023 14:56:35 (UTC+8)-
dc.date.available 1-Sep-2023 14:56:35 (UTC+8)-
dc.date.issued (上傳時間) 1-Sep-2023 14:56:35 (UTC+8)-
dc.identifier (Other Identifiers) G0110354015en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/146901-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 110354015zh_TW
dc.description.abstract (摘要) 近年來，用於監測單維品質變數整個分配是否有變化的管制圖有新的發展。目前的文獻多以CDF服從均勻分配據以建立管制圖，用來偵測整個分配是否發生變化，且部分研究考慮樣本數為一(n=1)的情況。實務上，在製程管制中樣本數為一的情況較少，於是我們的研究動機是延伸1991年Hackl和Ledolter的研究，考慮樣本數大於一並且品質變數不受限制下的Phase II管制圖，用以監測品質變數分配是否發生變化。
在本研究中，我們建立兩個Phase II管制圖(SEWMA-R ̅ 管制圖和SEWMA-T管制圖)以監測品質變數的分配變化，並使用平均連串長度(ARL)作為管制圖偵測能力指標。我們考慮四個不同的失控分配，分別為常態、雙指數、偏斜常態和伽瑪分配，數據分析結果顯示在偏移平均數時，SEWMA-R ̅ 管制圖的偵測能力在伽瑪分配下最佳；在偏移變異數時，SEWMA-T管制圖的偵測能力在伽瑪分配表現最佳。最後，我們將SEWMA-R ̅ 管制圖和SEWMA-T管制圖與兩種現有的管制圖進行比較，在伽瑪分配下，SEWMA-R ̅ 管制圖在偵測平均數的偏移方面表現出較好的檢測性能，SEWMA-T管制圖在監測平均數和變異數同時偏移時表現出較好的偵測能力。		zh_TW
dc.description.abstract (摘要) In recent years, there have been new developments in univariate control charts for monitoring changes in distribution. Current literature mostly focuses on control charts based on the cumulative distribution function (CDF) following a uniform distribution and used to detect distributional changes. Some studies consider the case where the sample size is one (n=1). However, in practice, the case of sample size one is less common in process control. Hence, our research motivation is considering sample size greater than one and developing the Phase II distribution-free control charts for monitoring abnormal process distributions by extending Hackl and Ledolter’s (1991) research.
In this study, we establish two Phase II control charts, the SEWMA-R ̅ chart and the SEWMA-T chart, to monitor changes in distribution. We use the average run length (ARL) as the performance measurement indicator for the control charts. We consider four different out-of-control distributions including normal, double exponential, skew normal and gamma distribution. The data analyses results show that, when mean shifts only, the SEWMA-R ̅ chart with gamma distributed performs the best. When variance shifts only, the SEWMA-T chart with gamma distributed performs the best. Finally, we compare the SEWMA-R ̅ chart and SEWMA-T chart with two existing control charts. Considering the gamma distribution, the SEWMA-R ̅ chart shows better detection performance in detecting shifts in process mean, while the SEWMA-T control chart exhibits better detection ability in monitoring shifts in the both mean and variance.		en_US
dc.description.tableofcontents 1 Introduction 1
2 Preliminaries of the Existing rEWMA chart and the Distribution of the Monitoring Statistics 5
2.1 The Existing rEWMA chart 5
2.2 The Distributions of the Monitoring Statistics R, R ̅ and T 7
2.2.1 The distribution of the monitoring statistic R 7
2.2.2 Derivation of the Bates distribution and its general form 8
2.2.3 The distribution of the monitoring statistic R ̅ 9
2.2.4 The distribution of the monitoring statistic T 9
3 Constructions of the Phase II SEWMA-R ̅ chart and SEWMA-T chart 11
3.1 A Phase II SEWMA-R ̅ chart 11
3.2 A Phase II SEWMA-T chart 12
4 Determination of the Control Limits and Detection Performances of the Proposed Two Charts under the Four Specified Distributions 13
4.1 Determination of the Control Limits 13
4.2 Control Limits Comparison for the Two Approximation Methods and Monte Carlo Simulation 20
4.3 Detection Performances of the two proposed charts under the four specified distributions 25
5 Conclusion remarks and Future works 68
References 69		zh_TW
dc.format.extent 7069714 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0110354015en_US
dc.subject (關鍵詞) 管制圖zh_TW
dc.subject (關鍵詞) 指數移動平均zh_TW
dc.subject (關鍵詞) 平均連串長度zh_TW
dc.subject (關鍵詞) 累積分配函數zh_TW
dc.subject (關鍵詞) control charten_US
dc.subject (關鍵詞) exponentially weighted moving averageen_US
dc.subject (關鍵詞) average run lengthen_US
dc.subject (關鍵詞) CDF (cumulative distribution function)en_US
dc.title (題名) 追蹤品質變數分配變化的Phase II管制圖zh_TW
dc.title (題名) Phase II Control Charts for Monitoring Abnormal Process Distributionsen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.
Amin, R. W., & Searcy, A. J. (1991). A nonparametric exponentially weighted moving average control scheme. Communications in Statistics-Simulation and Computation, 20(4), 1049-1072.
Amin, R. W., Reynolds Jr, M. R., & Saad, B. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistics-Theory and Methods, 24(6), 1597-1623.
Abu-Shawiesh, M. O. (2008). A simple robust control chart based on MAD. Journal of Mathematics and Statistics, 4(2), 102.
Abbasi, S. A., & Miller, A. (2013). MDEWMA chart: an efficient and robust alternative to monitor process dispersion. Journal of Statistical Computation and Simulation, 83(2), 247-268.
Abid, M., Nazir, H. Z., Riaz, M., & Lin, Z. (2017a). An efficient nonparametric EWMA Wilcoxon signed‐rank chart for monitoring location. Quality and Reliability Engineering International, 33(3), 669-685.
Abid, M., Nazir, H. Z., Riaz, M., & Lin, Z. (2017b). Investigating the impact of ranked set sampling in nonparametric CUSUM control charts. Quality and Reliability Engineering International, 33(1), 203-214.
Abbas, N. (2018). Homogeneously weighted moving average control chart with an application in substrate manufacturing process. Computers & Industrial Engineering, 120, 460-470.
Asghari, S., Sadeghpour Gildeh, B., Ahmadi, J., & Mohtashami Borzadaran, G. (2018). Sign control chart based on ranked set sampling. Quality Technology & Quantitative Management, 15(5), 568-588.
Alevizakos, V., Koukouvinos, C., & Chatterjee, K. (2020). A nonparametric double generally weighted moving average signed‐rank control chart for monitoring process location. Quality and Reliability Engineering International, 36(7), 2441-2458.
Alevizakos, V., Chatterjee, K., & Koukouvinos, C. (2021). An extended nonparametric homogeneously weighted moving average sign control chart. Quality and Reliability Engineering International, 37(8), 3395-3416.
Bates, G. E. (1955). Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya scheme. The Annals of Mathematical Statistics, 705-720.
Bakir, S. T., & Reynolds, M. R. (1979). A nonparametric procedure for process control based on within-group ranking. Technometrics, 21(2), 175-183.
Bakir, S. T. (2004). A distribution-free Shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623.
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.
Bakir, S. T. (2012). A nonparametric shewhart-type quality control chart for monitoring broad changes in a process distribution. Journal of Quality and Reliability Engineering, 2012.
Chen, G., Cheng, S. W., & Xie, H. (2001). Monitoring process mean and variability with one EWMA chart. Journal of Quality Technology, 33(2), 223-233.
Chen, J. T., Gupta, A. K., & Nguyen, T. T. (2004). The density of the skew normal sample mean and its applications. Journal of Statistical Computation and Simulation, 74(7), 487-494.
Champ, C. W., Jones-Farmer, L. A., & Rigdon, S. E. (2005). Properties of the T 2 control chart when parameters are estimated. Technometrics, 47(4), 437-445.
Chakraborti, S., & Eryilmaz, S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Communications in Statistics—Simulation and Computation®, 36(2), 335-356.
Chen, J. H., Lu, S. L., & Sheu, S. H. (2020). A nonparametric generally weighted moving average sign chart based on repetitive sampling. Communications in Statistics-Simulation and Computation, 51(3), 1137-1156.
Das, N., & Bhattacharya, A. (2008). A new non-parametric control chart for controlling variability. Quality Technology & Quantitative Management, 5(4), 351-361.
Fligner, M. A., & Killeen, T. J. (1976). Distribution-free two-sample tests for scale. Journal of the American Statistical Association, 71(353), 210-213.
Feller, W. (1991). An introduction to probability theory and its applications, Volume 2 (Vol. 81). John Wiley & Sons.
Graham, M. A., Chakraborti, S., & Human, S. W. (2011). A nonparametric EWMA sign chart for location based on individual measurements. Quality Engineering, 23(3), 227-241.
Godase, D. G., Rakitzis, A. C., Mahadik, S. B., & Khoo, M. B. (2022). Deciles‐based EWMA‐type sign charts for process dispersion. Quality and Reliability Engineering International, 38(7), 3726-3740.
Hackl, P., & Ledolter, J. (1991). A control chart based on ranks. Journal of Quality Technology, 23(2), 117-124.
Hackl, P., & Ledolter, J. (1992). A new nonparametric quality control technique. Communications in Statistics-Simulation and Computation, 21(2), 423-443.
Hawkins, D. M., Qiu, P., & Kang, C. W. (2003). The changepoint model for statistical process control. Journal of quality technology, 35(4), 355-366.
Hou, S., & Yu, K. (2021). A non-parametric CUSUM control chart for process distribution change detection and change type diagnosis. International Journal of Production Research, 59(4), 1166-1186.
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John wiley & sons.
Jones, L. A. (2002). The statistical design of EWMA control charts with estimated parameters. Journal of Quality Technology, 34(3), 277-288.
Jensen, W. A., Jones-Farmer, L. A., Champ, C. W., & Woodall, W. H. (2006). Effects of Parameter Estimation on Control Chart Properties: A Literature Review. Journal of Quality Technology, 38(4), 349-364.
Jones-Farmer, L. A., & Champ, C. W. (2010). A distribution-free Phase I control chart for subgroup scale. Journal of Quality Technology, 42(4), 373-387.
Khilare, S. K., & Shirke, D. T. (2012). Nonparametric synthetic control charts for process variation. Quality and Reliability Engineering International, 28(2), 193-202.
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