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題名 監控相依品質變數比之變異數的 EWMA 管制圖
EWMA Control Chart for Monitoring Variance of Ratio of Correlated Quality Variables
作者 陳韋豫
Chen, Wei-Yu
貢獻者 楊素芬<br>蕭又新
Yang,Su-Fen
陳韋豫
Chen, Wei-Yu
關鍵詞 相依品質變數比
變異數管制圖
Ratio of correlated variables
Variance control chart
日期 2023
上傳時間 2-Aug-2023 13:04:20 (UTC+8)
摘要 在品質管制的領域中，我們常使用管制圖來監控製程以提升產出的品質。在眾多產業中，追蹤相依品質變數之間的比例變化相當重要。在過去，文獻上對於監控相依品質變數比的平均值或變異數的管制圖研究較少。因此，如何監控相依品質變數比的平均數或變異數的製程管制圖是值得探討的。
本研究提出三種監控相依品質變數比的變異數管制圖，分別以符號檢定(sign test)方法、Mood (1954)的Rank test與Siegel & Tukey (1960)檢定兩分配變異數是否相同的檢定方法運用於建立相依變數比的變異數管制圖。本文在考慮不同的二元分配之下評估所提出的管制圖的表現，並與文獻中的比例變異數管制圖進行比較。最後，以半導體資料說明我們所提出的三種相依品質變數比的變異數管制圖的應用。
In quality control, control charts are commonly used to monitor processes. In many industries, monitoring the proportions of correlated process variables is crucial. Currently, there has been less research on control charts for monitoring the mean or variance of ratio of correlated process variables.
This study proposes three control charts for monitoring the variance of ratio of two correlated process variables. These control charts combine the sign test method, the Rank test method for dispersion proposed by Mood (1954), and the test for differences in variability proposed by Siegel & Tukey (1960). Moreover, the performance of the proposed control charts is evaluated under different bivariate distributions and also compared with some existing control charts from the literature. Additionally, the application of the three proposed control charts for monitoring the variance of ratio between two correlated process variables is demonstrated using semiconductor data.
參考文獻 [1] Alt, F. B. (1985). Multivariate quality control. Encyclopedia of Statistical Science, 6, 110-122.
[2] Alt, F. B., Smith, N. D. (1988). Multivariate process control. In: Krishnaiah PR, Rao CR, eds. Handbook of Statistics. Elsevier; 331-351.
[3] Azzalini, A. and Valle, A. D. (1996). The multivariate skew normal distribution. Biometrika, 83, 715–726
[4] Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. J. R. Statist. Soc. B, 61, 579–602.
[5] Celano, G., Castagliola, P., Faraz, A., & Fichera, S. (2014). Statistical performance of a control chart for individual observations monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 30(8), 1361-1377.
[6] Celano, G., Castagliola, P. (2016). Design of a phase II control chart for monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 32(1):291–308.
[7] Das, N. (2008). Nonparametric control chart for controlling variability based on rank test. Economic Quality Control, 23 (2):227-242.
[8] Fan, J., Shu, L., Zhao, H., and H. Yeung. (2017). Monitoring multivariate process variability via eigenvalues. Computers & Industrial Engineering, 113:269–81.
[9] Jones-Farmer, L. A. and Champ, C. W. (2010). A distribution-free phase I control charts for subgroup scale. Journal of Quality Technology, 42, pp. 373–387.
[10] Lee, R. Y., Holland, B. S., and Flueck, J. A. (1979). Distribution of a ratio of correlated gamma random variables. SIAM J Appl Math, 36:304–320.
[11] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387.
[12] Mood, A. M. (1954). On the asymptotic efficiency of certain non-parametric two sample test. The Annals of Mathematical Statistics, 25:514-522.
[13] Memar, A. O., and Niaki, S. T. A. (2009). New control charts for monitoring covariance matrix with individual observations. Quality and Reliability Engineering International, 25 (7):821–38.
[14] Mason, R. L., Chou, Y. M., and Young, J. C. (2009). Monitoring variation in a multivariate process when the dimension is large relative to the sample size. Communications in Statistics - Theory and Methods, 38 (6):939–51.
[15] Mason, R. L., Chou, Y. M. and Young, J. C. (2010). Decomposition of scatter ratios used in monitoring multivariate process variability. Communications in Statistics - Theory and Methods, 39 (12):2128–45.
[16] Nguyen, H. D., Tran, K. P., & Heuchenne, C. (2019). Monitoring the ratio of two normal variables using variable sampling interval exponentially weighted moving average control charts. Quality and Reliability Engineering International, 35(1), 439-460.
[17] Roberts, S. W. (1959). Control chart tests based on geometric moving averages. Technometrics, 1:239–250.
[18] Shewhart, W. A. (1924). Some applications of statistical methods to the analysis of physical and engineering data. Bell System Technical Journal, 3(1), 43-87.
[19] Siegel, S. and Tukey, J. W. (1960). Nonparametric sum of ranks procedure for relative spread in unpaired samples. Journal of the American Statistical Association, 55:429-445.
[20] Tran, K. P., Castagliola, P., & Celano, G. (2016). Monitoring the ratio of two normal variables using run rules type control charts. International Journal of Production Research, 54(6), 1670-1688.
[21] Tran, K. P., & Knoth, S. (2018). Steady-state ARL analysis of ARL-unbiased EWMA-RZ control chart monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 34(3), 377-390.
[22] Yang, S. F., Lin, J. S., and Cheng, S. W. (2011). A new nonparametric EWMA sign control chart. Expert Systems with Applications, 38(5), 6239-6243.
[23] Yang, S. F., Arnold, B. C., Liu, Y., Lu, M. C., and Lu, S. L. (2021). A new phase II EWMA dispersion control chart. Quality and Reliability Engineering International, 38:1635–1658.
[24] Yang, S. F., and Arnold, B. C. (2016). A new approach for monitoring process variance. J Stat Comput Simul, 86:2749–2765.
[25] Yen, C. L. and Shiau, J. J. H. (2010). A multivariate control chart for detecting increases in process dispersion. Statistica Sinica, 20:1683-1707.
[26] Yen, C. L., Shiau, J. J. H., & Yeh, A. B. (2012). Effective control charts for monitoring multivariate process dispersion. Quality and Reliability Engineering International, 28(4), 409-426.
[27] Zou, C. and Tsung, F. (2010). Likelihood ratio-based distribution-free EWMA control charts. Journal of Quality Technology, 42 (2):174-196.
[28] Zombade, D. M., Ghute, V. B. (2014). Nonparametric control chart for variability using runs rules. The Experiment, 24(4):1683–1691.
描述 碩士
國立政治大學
統計學系
110354013
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0110354013
資料類型 thesis
dc.contributor.advisor 楊素芬<br>蕭又新zh_TW
dc.contributor.advisor Yang,Su-Fenen_US
dc.contributor.author (Authors) 陳韋豫zh_TW
dc.contributor.author (Authors) Chen, Wei-Yuen_US
dc.creator (作者) 陳韋豫zh_TW
dc.creator (作者) Chen, Wei-Yuen_US
dc.date (日期) 2023en_US
dc.date.accessioned 2-Aug-2023 13:04:20 (UTC+8)-
dc.date.available 2-Aug-2023 13:04:20 (UTC+8)-
dc.date.issued (上傳時間) 2-Aug-2023 13:04:20 (UTC+8)-
dc.identifier (Other Identifiers) G0110354013en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/146307-
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) 110354013zh_TW
dc.description.abstract (摘要) 在品質管制的領域中，我們常使用管制圖來監控製程以提升產出的品質。在眾多產業中，追蹤相依品質變數之間的比例變化相當重要。在過去，文獻上對於監控相依品質變數比的平均值或變異數的管制圖研究較少。因此，如何監控相依品質變數比的平均數或變異數的製程管制圖是值得探討的。
本研究提出三種監控相依品質變數比的變異數管制圖，分別以符號檢定(sign test)方法、Mood (1954)的Rank test與Siegel & Tukey (1960)檢定兩分配變異數是否相同的檢定方法運用於建立相依變數比的變異數管制圖。本文在考慮不同的二元分配之下評估所提出的管制圖的表現，並與文獻中的比例變異數管制圖進行比較。最後，以半導體資料說明我們所提出的三種相依品質變數比的變異數管制圖的應用。		zh_TW
dc.description.abstract (摘要) In quality control, control charts are commonly used to monitor processes. In many industries, monitoring the proportions of correlated process variables is crucial. Currently, there has been less research on control charts for monitoring the mean or variance of ratio of correlated process variables.
This study proposes three control charts for monitoring the variance of ratio of two correlated process variables. These control charts combine the sign test method, the Rank test method for dispersion proposed by Mood (1954), and the test for differences in variability proposed by Siegel & Tukey (1960). Moreover, the performance of the proposed control charts is evaluated under different bivariate distributions and also compared with some existing control charts from the literature. Additionally, the application of the three proposed control charts for monitoring the variance of ratio between two correlated process variables is demonstrated using semiconductor data.		en_US
dc.description.tableofcontents 1. Introduction 1
2. Distribution of Ratio of Bivariate Non-normal quality variables 5
2-1 The distribution of bivariate skew normal variables 5
2-2 The distribution of ratio of bivariate gamma variables 7
2-3 The same means and variances of the bivariate skew normal and bivariate gamma distributions 8
3. Sign Based EWMA Variance of Ratio Control Chart 14
3-1 Review of the Sign based variance control chart 14
3-2 Construction of the sign based EWMA variance of ratio control chart 14
3-3 Calculation of the average run length of the sign based EVRS control chart 17
3-4 Detection performance of the Sign based EWMA variance of ratio control chart 23
4. Mood Based EWMA Variance of Ratio Control Chart 30
4-1 Review of the Mood test 30
4-2 The Mood based EWMA variance of ratio control chart 30
4-3 Calculation of the average run length of the Mood based EVRM control chart 33
4-4 Detection performance of the Mood based EWMA variance of ratio control chart 37
5. Tukey Based EWMA Variance of Ratio Control Chart 42
5-1 Review of the Tukey test 42
5-2 The Tukey based EWMA variance of ratio control chart 42
5-3 Calculation of the average run length of the Tukey based EVRT control chart 45
5-4 Detection performance of the Tukey based EWMA variance ratio control chart 49
6. Performance Comparison 54
7. Real Numerical Example 58
8. Conclusions 69
Reference 70		zh_TW
dc.format.extent 1581868 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0110354013en_US
dc.subject (關鍵詞) 相依品質變數比zh_TW
dc.subject (關鍵詞) 變異數管制圖zh_TW
dc.subject (關鍵詞) Ratio of correlated variablesen_US
dc.subject (關鍵詞) Variance control charten_US
dc.title (題名) 監控相依品質變數比之變異數的 EWMA 管制圖zh_TW
dc.title (題名) EWMA Control Chart for Monitoring Variance of Ratio of Correlated Quality Variablesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Alt, F. B. (1985). Multivariate quality control. Encyclopedia of Statistical Science, 6, 110-122.
[2] Alt, F. B., Smith, N. D. (1988). Multivariate process control. In: Krishnaiah PR, Rao CR, eds. Handbook of Statistics. Elsevier; 331-351.
[3] Azzalini, A. and Valle, A. D. (1996). The multivariate skew normal distribution. Biometrika, 83, 715–726
[4] Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. J. R. Statist. Soc. B, 61, 579–602.
[5] Celano, G., Castagliola, P., Faraz, A., & Fichera, S. (2014). Statistical performance of a control chart for individual observations monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 30(8), 1361-1377.
[6] Celano, G., Castagliola, P. (2016). Design of a phase II control chart for monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 32(1):291–308.
[7] Das, N. (2008). Nonparametric control chart for controlling variability based on rank test. Economic Quality Control, 23 (2):227-242.
[8] Fan, J., Shu, L., Zhao, H., and H. Yeung. (2017). Monitoring multivariate process variability via eigenvalues. Computers & Industrial Engineering, 113:269–81.
[9] Jones-Farmer, L. A. and Champ, C. W. (2010). A distribution-free phase I control charts for subgroup scale. Journal of Quality Technology, 42, pp. 373–387.
[10] Lee, R. Y., Holland, B. S., and Flueck, J. A. (1979). Distribution of a ratio of correlated gamma random variables. SIAM J Appl Math, 36:304–320.
[11] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387.
[12] Mood, A. M. (1954). On the asymptotic efficiency of certain non-parametric two sample test. The Annals of Mathematical Statistics, 25:514-522.
[13] Memar, A. O., and Niaki, S. T. A. (2009). New control charts for monitoring covariance matrix with individual observations. Quality and Reliability Engineering International, 25 (7):821–38.
[14] Mason, R. L., Chou, Y. M., and Young, J. C. (2009). Monitoring variation in a multivariate process when the dimension is large relative to the sample size. Communications in Statistics - Theory and Methods, 38 (6):939–51.
[15] Mason, R. L., Chou, Y. M. and Young, J. C. (2010). Decomposition of scatter ratios used in monitoring multivariate process variability. Communications in Statistics - Theory and Methods, 39 (12):2128–45.
[16] Nguyen, H. D., Tran, K. P., & Heuchenne, C. (2019). Monitoring the ratio of two normal variables using variable sampling interval exponentially weighted moving average control charts. Quality and Reliability Engineering International, 35(1), 439-460.
[17] Roberts, S. W. (1959). Control chart tests based on geometric moving averages. Technometrics, 1:239–250.
[18] Shewhart, W. A. (1924). Some applications of statistical methods to the analysis of physical and engineering data. Bell System Technical Journal, 3(1), 43-87.
[19] Siegel, S. and Tukey, J. W. (1960). Nonparametric sum of ranks procedure for relative spread in unpaired samples. Journal of the American Statistical Association, 55:429-445.
[20] Tran, K. P., Castagliola, P., & Celano, G. (2016). Monitoring the ratio of two normal variables using run rules type control charts. International Journal of Production Research, 54(6), 1670-1688.
[21] Tran, K. P., & Knoth, S. (2018). Steady-state ARL analysis of ARL-unbiased EWMA-RZ control chart monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 34(3), 377-390.
[22] Yang, S. F., Lin, J. S., and Cheng, S. W. (2011). A new nonparametric EWMA sign control chart. Expert Systems with Applications, 38(5), 6239-6243.
[23] Yang, S. F., Arnold, B. C., Liu, Y., Lu, M. C., and Lu, S. L. (2021). A new phase II EWMA dispersion control chart. Quality and Reliability Engineering International, 38:1635–1658.
[24] Yang, S. F., and Arnold, B. C. (2016). A new approach for monitoring process variance. J Stat Comput Simul, 86:2749–2765.
[25] Yen, C. L. and Shiau, J. J. H. (2010). A multivariate control chart for detecting increases in process dispersion. Statistica Sinica, 20:1683-1707.
[26] Yen, C. L., Shiau, J. J. H., & Yeh, A. B. (2012). Effective control charts for monitoring multivariate process dispersion. Quality and Reliability Engineering International, 28(4), 409-426.
[27] Zou, C. and Tsung, F. (2010). Likelihood ratio-based distribution-free EWMA control charts. Journal of Quality Technology, 42 (2):174-196.
[28] Zombade, D. M., Ghute, V. B. (2014). Nonparametric control chart for variability using runs rules. The Experiment, 24(4):1683–1691.		zh_TW