1. NCCU Academic Hub
  2. Publications
  3. College of Commerce
  4. Theses
  5. Item

Publications-Theses

Article View/Open

Publication Export

Google ScholarTM

NCCU Library

Citation Infomation

Related Publications in TAIR

題名 監控相依品質變數比之位置的EWMA管制圖
EWMA Control Chart for Monitoring Location of Ratio of Correlated Quality Variables
作者 吳宥群
Wu, Yu-Chun
貢獻者 楊素芬
Yang, Su-Fen
吳宥群
Wu, Yu-Chun
關鍵詞 變數比的位置
二元分配管制圖
Wilcoxon排序和檢定
符號檢定
核密度估計方法
Control chart
Location of ratio
Bivariate distribution variables
Wilcoxon rank-sum test
Sign test
Kernel density estimation
日期 2023
上傳時間 2-Aug-2023 13:02:56 (UTC+8)
摘要 近年來，在許多產業中，兩相依品質變數比的位置之製程監控影響產出品質，故至關重要。然而文獻上現有的研究主要集中於假設二元常態分配下兩相依變數的分配之監控。在實際應用中，我們所收集的數據往往是未知或非常態分配。因此，本研究提出了無母數和自由分配的管制圖，以在不假設特定分配的情況下監控兩相依變數比的位置。
本研究介紹了三種EWMA管制圖：一種利用Wilcoxon排序和檢定方法，另一種利用符號檢定方法，第三種則採用核密度估計方法建管制圖以監控兩相依變數比的位置。我們評估了這三種管制圖的績效並與已知多元變數分配的EWMA位置管制圖進行比較。最後，以半導體產業的數據來說明所提出的比例位置管制圖的應用。
In recent years, monitoring the location of the ratio of two correlated variables has become crucial in many industries. However, existing research on monitoring the distribution of the ratio of bivariate variables has predominantly focused on the assumption of bivariate normal variables. In practical applications, the data we collect often exhibit unknown or non-normal distributions. Hence, we propose a set of control charts, allowing us to monitor the location of the ratio of two correlated variables without assuming a specific distribution.
In this study, we introduce three EWMA control charts: one utilizing the Wilcoxon rank-sum statistic, another using the sign statistic, and the third employing the kernel density estimation method. These charts are designed to monitor the location of the ratio of two correlated variables. We evaluate the out-of-control detecting performance of the proposed charts and compare them with the exact EWMA mean charts, assuming knowledge of the distributions. Additionally, we present real data from the semiconductor industry to demonstrate the application of the proposed charts.
參考文獻 [1] Abubakar, S. S., Khoo, M. B., Saha, S., & Teoh, W. L. (2022). Run sum control chart for monitoring the ratio of population means of a bivariate normal distribution. Communications in Statistics-Theory and Methods, 51(13), 4559-4588.
[2] 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.
[3] Azzalini, A., & Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3), 579-602.
[4] Azzalini, A., & Valle, A. D. (1996). The multivariate skew-normal distribution. Biometrika, 83(4), 715-726.
[5] Bell, R. C., Jones-Farmer, L. A., & Billor, N. (2014). A distribution-free multivariate phase I location control chart for subgrouped data from elliptical distributions. Technometrics, 56(4), 528-538.
[6] 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.
[7] Chakraborti, S., & Graham, M. A. (2019). Nonparametric (distribution-free) control charts: An updated overview and some results. Quality Engineering, 31(4), 523-544.
[8] Chen, Y. C. (2017). A tutorial on kernel density estimation and recent advances. Biostatistics & Epidemiology, 1(1), 161-187.
[9] Farokhnia, M., & Niaki, S. T. A. (2020). Principal component analysis-based control charts using support vector machines for multivariate non-normal distributions. Communications in Statistics-Simulation and Computation, 49(7), 1815-1838.
[10] Hotelling, H. (1947). Multivariate quality control. Techniques of statistical analysis.
[11] Jones-Farmer, L. A., Jordan, V., & Champ, C. W. (2009). Distribution-free phase I control charts for subgroup location. Journal of Quality Technology, 41(3), 304-316.
[12] Lee, R. Y., Holland, B. S., & Flueck, J. A. (1979). Distribution of a ratio of correlated gamma random variables. SIAM Journal on Applied Mathematics, 36(2), 304-320.
[13] Li, S. Y., Tang, L. C., & Ng, S. H. (2010). Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology, 42(2), 209-226.
[14] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387.
[15] Lowry, C. A., Woodall, W. H., Champ, C. W., & Rigdon, S. E. (1992). A multivariate exponentially weighted moving average control chart. Technometrics, 34(1), 46-53.
[16] Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The annals of mathematical statistics, 50-60.
[17] Malela-Majika, J. C. (2021). New distribution-free memory-type control charts based on the Wilcoxon rank-sum statistic. Quality Technology & Quantitative Management, 18(2), 135-155.
[18] McCann, M., & Johnston, A. (2008). SECOM Data Set Center for Machine Learning and Intelligent Systems. University of California, Irvine, CA.
[19] Melo, M. S., Ho, L. L., & Medeiros, P. G. (2017). M ax D: an attribute control chart to monitor a bivariate process mean. The International Journal of Advanced Manufacturing Technology, 90, 489-498.
[20] Montgomery, D. C. (2020). Introduction to statistical quality control. John Wiley & Sons.
[21] 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.
[22] Parzen, E. (1962). On estimation of a probability density function and mode. The annals of mathematical statistics, 33(3), 1065-1076.
[23] Pignatiello Jr, J. J., & Runger, G. C. (1990). Comparisons of multivariate CUSUM charts. Journal of quality technology, 22(3), 173-186.
[24] Roberts, S. W. (2000). Control chart tests based on geometric moving averages. Technometrics, 42(1), 97-101.
[25] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. The annals of mathematical statistics, 832-837.
[26] Setiawan, Adi. (2013). Control Chart based on Kernel Density Estimation.
[27] 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.
[28] 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.
[29] Tran, K. P., Castagliola, P., & Celano, G. (2016). The performance of the Shewhart-RZ control chart in the presence of measurement error. International Journal of Production Research, 54(24), 7504-7522.
[30] Wang, S., & Reynolds Jr, M. R. (2013). A GLR control chart for monitoring the mean vector of a multivariate normal process. Journal of Quality Technology, 45(1), 18-33.
[31] Wilcoxon, F. (1945). Individual Comparisons by Ranking Methods. Biometrics Bulletin, 1(6), 80–83.
[32] Yang, S. F. (2016). An improved distribution-free EWMA mean chart. Communications in Statistics-Simulation and Computation, 45(4), 1410-1427.
[33] 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.
[34] Yang, S. F., Lin, Y. C., & Yeh, A. B. (2021). A Phase II depth‐based variable dimension EWMA control chart for monitoring process mean. Quality and Reliability Engineering International, 37(6), 2384-2398.
[35] Zhou, M., Zhou, Q., & Geng, W. (2016). A new nonparametric control chart for monitoring variability. Quality and Reliability Engineering International, 32(7), 2471-2479.
描述 碩士
國立政治大學
統計學系
110354001
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0110354001
資料類型 thesis
dc.contributor.advisor 楊素芬zh_TW
dc.contributor.advisor Yang, Su-Fenen_US
dc.contributor.author (Authors) 吳宥群zh_TW
dc.contributor.author (Authors) Wu, Yu-Chunen_US
dc.creator (作者) 吳宥群zh_TW
dc.creator (作者) Wu, Yu-Chunen_US
dc.date (日期) 2023en_US
dc.date.accessioned 2-Aug-2023 13:02:56 (UTC+8)-
dc.date.available 2-Aug-2023 13:02:56 (UTC+8)-
dc.date.issued (上傳時間) 2-Aug-2023 13:02:56 (UTC+8)-
dc.identifier (Other Identifiers) G0110354001en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/146301-
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) 110354001zh_TW
dc.description.abstract (摘要) 近年來，在許多產業中，兩相依品質變數比的位置之製程監控影響產出品質，故至關重要。然而文獻上現有的研究主要集中於假設二元常態分配下兩相依變數的分配之監控。在實際應用中，我們所收集的數據往往是未知或非常態分配。因此，本研究提出了無母數和自由分配的管制圖，以在不假設特定分配的情況下監控兩相依變數比的位置。
本研究介紹了三種EWMA管制圖：一種利用Wilcoxon排序和檢定方法，另一種利用符號檢定方法，第三種則採用核密度估計方法建管制圖以監控兩相依變數比的位置。我們評估了這三種管制圖的績效並與已知多元變數分配的EWMA位置管制圖進行比較。最後，以半導體產業的數據來說明所提出的比例位置管制圖的應用。		zh_TW
dc.description.abstract (摘要) In recent years, monitoring the location of the ratio of two correlated variables has become crucial in many industries. However, existing research on monitoring the distribution of the ratio of bivariate variables has predominantly focused on the assumption of bivariate normal variables. In practical applications, the data we collect often exhibit unknown or non-normal distributions. Hence, we propose a set of control charts, allowing us to monitor the location of the ratio of two correlated variables without assuming a specific distribution.
In this study, we introduce three EWMA control charts: one utilizing the Wilcoxon rank-sum statistic, another using the sign statistic, and the third employing the kernel density estimation method. These charts are designed to monitor the location of the ratio of two correlated variables. We evaluate the out-of-control detecting performance of the proposed charts and compare them with the exact EWMA mean charts, assuming knowledge of the distributions. Additionally, we present real data from the semiconductor industry to demonstrate the application of the proposed charts.		en_US
dc.description.tableofcontents 1. Introduction 1
2. Wilcoxon Rank-Sum Statistic Based EWMA Chart for Monitoring Location of Ratio of Bivariate Variables 5
2.1 Review the EWMA-WRS chart for monitoring univariate process location 5
2.2 Design the EWMA-WRSMR chart for monitoring location of ratio of bivariate variables based on Wilcoxon rank-sum statistic 7
2.3 The procedure to determine the control limits and average run lengths of the EWMA-WRSMR chart 9
2.4 Ratios of two specified bivariate distributions 11
2.4.1 Ratio of bivariate gamma variables 11
2.4.2 Ratio of bivariate skew normal variables 13
2.4.3 An example of ratio of bivariate distribution 15
2.5 Detection performance of the EWMA-WRSMR chart 16
3. Sign Statistic Based EWMA Chart for Monitoring Location of Ratio of Bivariate Variables 25
3.1 Review the EWMA-SN chart for monitoring univariate process location 25
3.2 Design the sign based EWMA-SNMR chart for monitoring the location of ratio of bivariate variables 26
3.3 The procedure to determine the control limits and average run lengths of the EWMA-SNMR chart 28
3.4 Detection performance of the EWMA-SNMR chart 32
4. Kernel Density Estimation Based EWMA Chart for Monitoring Location of Ratio of Bivariate Variables 37
4.1 Review the EWMA-K chart for monitoring univariate process location 37
4.2 Design the EWMA-KMR chart for monitoring location of ratio of bivariate variables based on kernel density estimation 39
4.3 The procedure to determine the control limits and average run lengths of the EWMA-KMR chart 40
4.4 Detection performance of the EWMA-KMR chart 42
5. Performance Comparison 47
5.1 Performance Comparison Among the EWMA-WRSMR, EWMA-SNMR, and EWMA-KMR charts 47
5.2 Performance Comparison with the RZ-Shewhart Chart 55
6. A Real Example 57
7. Conclusions 68
References 69		zh_TW
dc.format.extent 1545209 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0110354001en_US
dc.subject (關鍵詞) 變數比的位置zh_TW
dc.subject (關鍵詞) 二元分配管制圖zh_TW
dc.subject (關鍵詞) Wilcoxon排序和檢定zh_TW
dc.subject (關鍵詞) 符號檢定zh_TW
dc.subject (關鍵詞) 核密度估計方法zh_TW
dc.subject (關鍵詞) Control charten_US
dc.subject (關鍵詞) Location of ratioen_US
dc.subject (關鍵詞) Bivariate distribution variablesen_US
dc.subject (關鍵詞) Wilcoxon rank-sum testen_US
dc.subject (關鍵詞) Sign testen_US
dc.subject (關鍵詞) Kernel density estimationen_US
dc.title (題名) 監控相依品質變數比之位置的EWMA管制圖zh_TW
dc.title (題名) EWMA Control Chart for Monitoring Location of Ratio of Correlated Quality Variablesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Abubakar, S. S., Khoo, M. B., Saha, S., & Teoh, W. L. (2022). Run sum control chart for monitoring the ratio of population means of a bivariate normal distribution. Communications in Statistics-Theory and Methods, 51(13), 4559-4588.
[2] 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.
[3] Azzalini, A., & Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3), 579-602.
[4] Azzalini, A., & Valle, A. D. (1996). The multivariate skew-normal distribution. Biometrika, 83(4), 715-726.
[5] Bell, R. C., Jones-Farmer, L. A., & Billor, N. (2014). A distribution-free multivariate phase I location control chart for subgrouped data from elliptical distributions. Technometrics, 56(4), 528-538.
[6] 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.
[7] Chakraborti, S., & Graham, M. A. (2019). Nonparametric (distribution-free) control charts: An updated overview and some results. Quality Engineering, 31(4), 523-544.
[8] Chen, Y. C. (2017). A tutorial on kernel density estimation and recent advances. Biostatistics & Epidemiology, 1(1), 161-187.
[9] Farokhnia, M., & Niaki, S. T. A. (2020). Principal component analysis-based control charts using support vector machines for multivariate non-normal distributions. Communications in Statistics-Simulation and Computation, 49(7), 1815-1838.
[10] Hotelling, H. (1947). Multivariate quality control. Techniques of statistical analysis.
[11] Jones-Farmer, L. A., Jordan, V., & Champ, C. W. (2009). Distribution-free phase I control charts for subgroup location. Journal of Quality Technology, 41(3), 304-316.
[12] Lee, R. Y., Holland, B. S., & Flueck, J. A. (1979). Distribution of a ratio of correlated gamma random variables. SIAM Journal on Applied Mathematics, 36(2), 304-320.
[13] Li, S. Y., Tang, L. C., & Ng, S. H. (2010). Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology, 42(2), 209-226.
[14] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387.
[15] Lowry, C. A., Woodall, W. H., Champ, C. W., & Rigdon, S. E. (1992). A multivariate exponentially weighted moving average control chart. Technometrics, 34(1), 46-53.
[16] Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The annals of mathematical statistics, 50-60.
[17] Malela-Majika, J. C. (2021). New distribution-free memory-type control charts based on the Wilcoxon rank-sum statistic. Quality Technology & Quantitative Management, 18(2), 135-155.
[18] McCann, M., & Johnston, A. (2008). SECOM Data Set Center for Machine Learning and Intelligent Systems. University of California, Irvine, CA.
[19] Melo, M. S., Ho, L. L., & Medeiros, P. G. (2017). M ax D: an attribute control chart to monitor a bivariate process mean. The International Journal of Advanced Manufacturing Technology, 90, 489-498.
[20] Montgomery, D. C. (2020). Introduction to statistical quality control. John Wiley & Sons.
[21] 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.
[22] Parzen, E. (1962). On estimation of a probability density function and mode. The annals of mathematical statistics, 33(3), 1065-1076.
[23] Pignatiello Jr, J. J., & Runger, G. C. (1990). Comparisons of multivariate CUSUM charts. Journal of quality technology, 22(3), 173-186.
[24] Roberts, S. W. (2000). Control chart tests based on geometric moving averages. Technometrics, 42(1), 97-101.
[25] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. The annals of mathematical statistics, 832-837.
[26] Setiawan, Adi. (2013). Control Chart based on Kernel Density Estimation.
[27] 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.
[28] 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.
[29] Tran, K. P., Castagliola, P., & Celano, G. (2016). The performance of the Shewhart-RZ control chart in the presence of measurement error. International Journal of Production Research, 54(24), 7504-7522.
[30] Wang, S., & Reynolds Jr, M. R. (2013). A GLR control chart for monitoring the mean vector of a multivariate normal process. Journal of Quality Technology, 45(1), 18-33.
[31] Wilcoxon, F. (1945). Individual Comparisons by Ranking Methods. Biometrics Bulletin, 1(6), 80–83.
[32] Yang, S. F. (2016). An improved distribution-free EWMA mean chart. Communications in Statistics-Simulation and Computation, 45(4), 1410-1427.
[33] 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.
[34] Yang, S. F., Lin, Y. C., & Yeh, A. B. (2021). A Phase II depth‐based variable dimension EWMA control chart for monitoring process mean. Quality and Reliability Engineering International, 37(6), 2384-2398.
[35] Zhou, M., Zhou, Q., & Geng, W. (2016). A new nonparametric control chart for monitoring variability. Quality and Reliability Engineering International, 32(7), 2471-2479.		zh_TW