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題名 監控兩相依品質變數之變異數比之 EWMA 管制圖
EWMA Control Charts for Monitoring the Ratio of Variances of Two Correlated Quality Variables
作者 周鈺宸
Chou, Yu-Chen
貢獻者 楊素芬
Yang, Su-Fen
周鈺宸
Chou, Yu-Chen
關鍵詞 變異數比值
二元分配
管制圖
平均連串長度
Ratio of variances
bivariate distributions
control chart
average run length
日期 2024
上傳時間 4-Sep-2024 14:56:47 (UTC+8)
摘要 在統計製程管制的研究領域中，兩相依品質變數之變異數比值的追蹤 在某些實務製程中是重要的，但是文獻中尚未有見探討。因此，我們的研 究旨在探索兩相依品質變異數比的變化，以監控製程是否失控。在實務上， 製程穩定性分析、參數優化以及生產效率評估等應用，都需要追蹤變異數 比值。 在本研究中，我們分別提出兩種方法，建立不同的 EWMA 變異數比例 管制圖。第一種，我們提出使用兩相依品質變數之樣本變異數之差異之分 配建立 EWMA 變異數比例管制圖，以追蹤兩相依品質變數之母體變異數之 比。第二種方法考慮符號檢定方法(sign test method)，根據兩相依品質變 數之樣本變異數的差異是否大於其期望值，並定義指標變數分配以建立符 號管制圖。我們分別再以數值分析評估在二元常態、伽馬、偏常態母體分 配下，這兩種管制圖的管制界線與失控的偵測力。最後，我們以實際的半 導體數據驗證這兩種管制圖的應用與失控的偵測力。
In statistical process control, monitoring the ratio of variances of correlated quality variables is crucial for some practical processes. However, this topic has not been explored in literature. Our study aims to investigate the ratio of variances of two correlated quality variables to monitor whether the ratio of two variances process is out of control. In practice, monitoring the ratio of two variances is essential for process stability analysis, parameter optimization and production efficiency evaluation. In this research, we propose two methods to establish EWMA ratio of variances control charts without the specified distributions of quality variables. The first method uses the distribution of the difference in the sample variances of two correlated quality variables to construct an EWMA ratio of variances control chart to monitor the population ratio of variances of two correlated quality variables. The second method considers the sign method to construct a sign-based control chart, where an indicator variable distribution is defined based on whether the difference in sample variances of two correlated quality variables exceeds its expected value. We conduct numerical analyses to calculate the control limits and evaluate detection capabilities of these two EWMA control charts under the bivariate normal, gamma, and skew-normal population distributions. Finally, we validate the application and detection capabilities of these two proposed control charts by using real semiconductor data.
參考文獻 [1] Alt, F. B., & Smith, N. D. (1988). 17 multivariate process control. In P. R. Krishnaiah & L. N. Rao (Eds.), Handbook of statistics (Vol. 7, pp. 333-351). North-Holland. [2] Aitchison, J. (2005). A concise guide to compositional data analysis. In Compositional Data Analysis Workshop. [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] 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] Chowdhury, S., Mukherjee, A., & Chakraborti, S. (2014). A new distribution‐ free control chart for joint monitoring of unknown location and scale parameters of continuous distributions. Quality and Reliability Engineering International, 30(2), 191-204. [7] Costa, A. F. B., & Machado, M. A. G. (2011). A control chart based on sample ranges for monitoring the covariance matrix of the multivariate processes. Journal of Applied Statistics, 38(2), 233-245. [8] Crosier, R. B. (1988). Multivariate generalizations of cumulative sum quality- control schemes. Technometrics, 30(3), 291-303. [9] Fan, J., Shu, L., Yang, A., & Li, Y. (2021). Phase I analysis of high-dimensional covariance matrices based on sparse leading eigenvalues. Journal of Quality Technology, 53(4), 333-346. [10] Hoteling, H. (1947). Multivariate quality control, illustrated by the air testing of sample bombsights. Techniques of statistical analysis, 111-184. [11] Kenney, J. F. (1939). Mathematics of statistics. D. van Nostrand. [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] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387 [14] Li, B., Wang, K., & Yeh, A. B. (2013). Monitoring the covariance matrix via penalized likelihood estimation. IIE Transactions, 45, 132-146. [15] Li, Z., Zou, C., Wang, Z., & Huwang, L. (2013). A multivariate sign chart for monitoring process shape parameters. Journal of Quality Technology, 45(2), 149-165. [16] Lowry, C. A., & Montgomery, D. C. (1995). A review of multivariate control charts. IIE Transactions (Institute of Industrial Engineers), 27(6), 800-810 [17] 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. [18] Maboudou-Tchao, E. M., & Agboto, V. (2013). Monitoring the covariance matrix with fewer observations than variables. Computational Statistics & Data Analysis, 64, 99-112. 108 [19] McCann, M., & Johnston, A. (2008). SECOM Data Set Center for Machine Learning and Intelligent Systems. University of California, Irvine, CA. [20] Melo, M. S., Ho, L. L., & Medeiros, P. G. (2017). Max D: an attribute control chart to monitor a bivariate process mean. The International Journal of Advanced Manufacturing Technology, 90, 489-498. [21] Mukherjee, A., & Chakraborti, S. (2012). A distribution‐free control chart for the joint monitoring of location and scale. Quality and Reliability Engineering International, 28(3), 335-352. [22] Nguyen, H. D., Nadi, A. A., Tran, K. P., Castagliola, P., Celano, G., & Tran, K. D. (2021). The effect of autocorrelation on the Shewhart-RZ control chart. arXiv preprint arXiv:2108.05239. [23] ÖKSOY, D., Boulos, E., & DAVID PYE, L. (1993). Statistical process control by the quotient of two correlated normal variables. Quality Engineering, 6(2), 179-194. [24] Pignatiello Jr, J. J., & Runger, G. C. (1990). Comparisons of multivariate CUSUM charts. Journal of Quality Technology, 22(3), 173-186. [25] Riaz, M., Zaman, B., Raji, I.A., Omar, M.H., Mehmood, R., Abbas, N. (2022). An Adaptive EWMA Control Chart Based on Principal Component Method to Monitor Process Mean Vector. Mathematics, 10(12), 2025. [26] Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 239-250. [27] Rose, C., & Smith, M. D. (2002). MathStatica: mathematical statistics with mathematica. In Compstat: Proceedings in Computational Statistics (pp. 437- 442). Physica-Verlag HD. [28] Ross, G. J., & Adams, N. M. (2012). Two nonparametric control charts for detecting arbitrary distribution changes. Journal of Quality Technology, 44(2), 102-116. [29] 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. Tran, [30] Shu, L., & Fan, J. (2018). A distribution‐free control chart for monitoring high‐ dimensional processes based on interpoint distances. Naval Research Logistics (NRL), 65(4), 317-330. [31] Susanty, A., Ulkhaq, M. M., & Amalia, D. (2018). Using multivariate control chart to maintain the quality of drinking water in accordance with standard. International Journal of Applied Science and Engineering, 15(2), 83-94. [32] 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. [33] 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. [34] Yang, S. F. (2016). An improved distribution-free EWMA mean chart. Communications in Statistics-Simulation and Computation, 45(4), 1410-1427. 33. [35] Yang, S. F., & Arnold, B. C. (2016). A new approach for monitoring process variance. Journal of Statistical Computation and Simulation, 86(14), 2749-2765. [36] 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 110 [37] Yang, S. F., Lin, Y. C., & Yeh, A. B. (2021). A Phase II depth‐based variable dimension EWMA control chart for monitoring process means. Quality and Reliability Engineering International, 37(6), 2384-2398. [38] Yang, S. F., Yeh, A. B., & Chou, C. C. (2023). A phase II multivariate EWMA chart for monitoring multi-dimensional ratios of process means with individual observations. Computers & Industrial Engineering, 183, 109490 [39] Yen, C. L., & Shiau, J. J. H. (2010). A multivariate control chart for detecting increases in process dispersion. Statistica Sinica, 1683-1707.
描述 碩士
國立政治大學
統計學系
111354021
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0111354021
資料類型 thesis
dc.contributor.advisor 楊素芬zh_TW
dc.contributor.advisor Yang, Su-Fenen_US
dc.contributor.author (Authors) 周鈺宸zh_TW
dc.contributor.author (Authors) Chou, Yu-Chenen_US
dc.creator (作者) 周鈺宸zh_TW
dc.creator (作者) Chou, Yu-Chenen_US
dc.date (日期) 2024en_US
dc.date.accessioned 4-Sep-2024 14:56:47 (UTC+8)-
dc.date.available 4-Sep-2024 14:56:47 (UTC+8)-
dc.date.issued (上傳時間) 4-Sep-2024 14:56:47 (UTC+8)-
dc.identifier (Other Identifiers) G0111354021en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/153366-
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) 111354021zh_TW
dc.description.abstract (摘要) 在統計製程管制的研究領域中，兩相依品質變數之變異數比值的追蹤 在某些實務製程中是重要的，但是文獻中尚未有見探討。因此，我們的研 究旨在探索兩相依品質變異數比的變化，以監控製程是否失控。在實務上， 製程穩定性分析、參數優化以及生產效率評估等應用，都需要追蹤變異數 比值。 在本研究中，我們分別提出兩種方法，建立不同的 EWMA 變異數比例 管制圖。第一種，我們提出使用兩相依品質變數之樣本變異數之差異之分 配建立 EWMA 變異數比例管制圖，以追蹤兩相依品質變數之母體變異數之 比。第二種方法考慮符號檢定方法(sign test method)，根據兩相依品質變 數之樣本變異數的差異是否大於其期望值，並定義指標變數分配以建立符 號管制圖。我們分別再以數值分析評估在二元常態、伽馬、偏常態母體分 配下，這兩種管制圖的管制界線與失控的偵測力。最後，我們以實際的半 導體數據驗證這兩種管制圖的應用與失控的偵測力。zh_TW
dc.description.abstract (摘要) In statistical process control, monitoring the ratio of variances of correlated quality variables is crucial for some practical processes. However, this topic has not been explored in literature. Our study aims to investigate the ratio of variances of two correlated quality variables to monitor whether the ratio of two variances process is out of control. In practice, monitoring the ratio of two variances is essential for process stability analysis, parameter optimization and production efficiency evaluation. In this research, we propose two methods to establish EWMA ratio of variances control charts without the specified distributions of quality variables. The first method uses the distribution of the difference in the sample variances of two correlated quality variables to construct an EWMA ratio of variances control chart to monitor the population ratio of variances of two correlated quality variables. The second method considers the sign method to construct a sign-based control chart, where an indicator variable distribution is defined based on whether the difference in sample variances of two correlated quality variables exceeds its expected value. We conduct numerical analyses to calculate the control limits and evaluate detection capabilities of these two EWMA control charts under the bivariate normal, gamma, and skew-normal population distributions. Finally, we validate the application and detection capabilities of these two proposed control charts by using real semiconductor data.en_US
dc.description.tableofcontents 1. Introduction 8 2. Distributions of Bivariate Normal and Non-normal Quality Variables 14 2.1 The distribution under a Bivariate Normal Process 2.2 The distribution under a Bivariate Gamma Process 2.3 The distribution under a Bivariate Skew Normal Process 3. New EWMA Control Charts for Monitoring Ratio of Variances Under Three Different Bivariate Distributions 17 3.1 The distribution of the difference of the sample variance 3.2 Design of the EWMA-U control chart 3.3 Design of the EWMA-D control chart 3.4 Design of the EWMA-UC control chart 3.5 Design of the EWMA-DC control chart 3.6 Determination of the control limits of the three EWMA charts using Monte Carlo simulation method 3.7 Detection performance of the proposed EWMA-U, EWMA-D, EWMA-UC and EWMA-DC charts 4. The Sign Based EWMA Control Chart Using the Distribution of the Differences in Sample Variances 60 4.1 Design of the Sign based EWMA-ISR chart 4.2 Design of the Sign based EWMA-DSR chart 4.3 Determination of the control limits of the two sign-based EWMA charts using Monte Carlo simulation method 4.4 Detection performance of the proposed EWMA-ISR and EWMA DSR charts 5. Out-of-Control Detection Performance Comparison of the Proposed Two EWMA Control Charts 78 6. Application of the Two Proposed EWMA Control Charts Using the Semiconductor Dataset 87 7. Conclusions 106 References 107zh_TW
dc.format.extent 3587803 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0111354021en_US
dc.subject (關鍵詞) 變異數比值zh_TW
dc.subject (關鍵詞) 二元分配zh_TW
dc.subject (關鍵詞) 管制圖zh_TW
dc.subject (關鍵詞) 平均連串長度zh_TW
dc.subject (關鍵詞) Ratio of variancesen_US
dc.subject (關鍵詞) bivariate distributionsen_US
dc.subject (關鍵詞) control charten_US
dc.subject (關鍵詞) average run lengthen_US
dc.title (題名) 監控兩相依品質變數之變異數比之 EWMA 管制圖zh_TW
dc.title (題名) EWMA Control Charts for Monitoring the Ratio of Variances of Two Correlated Quality Variablesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Alt, F. B., & Smith, N. D. (1988). 17 multivariate process control. In P. R. Krishnaiah & L. N. Rao (Eds.), Handbook of statistics (Vol. 7, pp. 333-351). North-Holland. [2] Aitchison, J. (2005). A concise guide to compositional data analysis. In Compositional Data Analysis Workshop. [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] 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] Chowdhury, S., Mukherjee, A., & Chakraborti, S. (2014). A new distribution‐ free control chart for joint monitoring of unknown location and scale parameters of continuous distributions. Quality and Reliability Engineering International, 30(2), 191-204. [7] Costa, A. F. B., & Machado, M. A. G. (2011). A control chart based on sample ranges for monitoring the covariance matrix of the multivariate processes. Journal of Applied Statistics, 38(2), 233-245. [8] Crosier, R. B. (1988). Multivariate generalizations of cumulative sum quality- control schemes. Technometrics, 30(3), 291-303. [9] Fan, J., Shu, L., Yang, A., & Li, Y. (2021). Phase I analysis of high-dimensional covariance matrices based on sparse leading eigenvalues. Journal of Quality Technology, 53(4), 333-346. [10] Hoteling, H. (1947). Multivariate quality control, illustrated by the air testing of sample bombsights. Techniques of statistical analysis, 111-184. [11] Kenney, J. F. (1939). Mathematics of statistics. D. van Nostrand. [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] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387 [14] Li, B., Wang, K., & Yeh, A. B. (2013). Monitoring the covariance matrix via penalized likelihood estimation. IIE Transactions, 45, 132-146. [15] Li, Z., Zou, C., Wang, Z., & Huwang, L. (2013). A multivariate sign chart for monitoring process shape parameters. Journal of Quality Technology, 45(2), 149-165. [16] Lowry, C. A., & Montgomery, D. C. (1995). A review of multivariate control charts. IIE Transactions (Institute of Industrial Engineers), 27(6), 800-810 [17] 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. [18] Maboudou-Tchao, E. M., & Agboto, V. (2013). Monitoring the covariance matrix with fewer observations than variables. Computational Statistics & Data Analysis, 64, 99-112. 108 [19] McCann, M., & Johnston, A. (2008). SECOM Data Set Center for Machine Learning and Intelligent Systems. University of California, Irvine, CA. [20] Melo, M. S., Ho, L. L., & Medeiros, P. G. (2017). Max D: an attribute control chart to monitor a bivariate process mean. The International Journal of Advanced Manufacturing Technology, 90, 489-498. [21] Mukherjee, A., & Chakraborti, S. (2012). A distribution‐free control chart for the joint monitoring of location and scale. Quality and Reliability Engineering International, 28(3), 335-352. [22] Nguyen, H. D., Nadi, A. A., Tran, K. P., Castagliola, P., Celano, G., & Tran, K. D. (2021). The effect of autocorrelation on the Shewhart-RZ control chart. arXiv preprint arXiv:2108.05239. [23] ÖKSOY, D., Boulos, E., & DAVID PYE, L. (1993). Statistical process control by the quotient of two correlated normal variables. Quality Engineering, 6(2), 179-194. [24] Pignatiello Jr, J. J., & Runger, G. C. (1990). Comparisons of multivariate CUSUM charts. Journal of Quality Technology, 22(3), 173-186. [25] Riaz, M., Zaman, B., Raji, I.A., Omar, M.H., Mehmood, R., Abbas, N. (2022). An Adaptive EWMA Control Chart Based on Principal Component Method to Monitor Process Mean Vector. Mathematics, 10(12), 2025. [26] Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 239-250. [27] Rose, C., & Smith, M. D. (2002). MathStatica: mathematical statistics with mathematica. In Compstat: Proceedings in Computational Statistics (pp. 437- 442). Physica-Verlag HD. [28] Ross, G. J., & Adams, N. M. (2012). Two nonparametric control charts for detecting arbitrary distribution changes. Journal of Quality Technology, 44(2), 102-116. [29] 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. Tran, [30] Shu, L., & Fan, J. (2018). A distribution‐free control chart for monitoring high‐ dimensional processes based on interpoint distances. Naval Research Logistics (NRL), 65(4), 317-330. [31] Susanty, A., Ulkhaq, M. M., & Amalia, D. (2018). Using multivariate control chart to maintain the quality of drinking water in accordance with standard. International Journal of Applied Science and Engineering, 15(2), 83-94. [32] 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. [33] 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. [34] Yang, S. F. (2016). An improved distribution-free EWMA mean chart. Communications in Statistics-Simulation and Computation, 45(4), 1410-1427. 33. [35] Yang, S. F., & Arnold, B. C. (2016). A new approach for monitoring process variance. Journal of Statistical Computation and Simulation, 86(14), 2749-2765. [36] 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 110 [37] Yang, S. F., Lin, Y. C., & Yeh, A. B. (2021). A Phase II depth‐based variable dimension EWMA control chart for monitoring process means. Quality and Reliability Engineering International, 37(6), 2384-2398. [38] Yang, S. F., Yeh, A. B., & Chou, C. C. (2023). A phase II multivariate EWMA chart for monitoring multi-dimensional ratios of process means with individual observations. Computers & Industrial Engineering, 183, 109490 [39] Yen, C. L., & Shiau, J. J. H. (2010). A multivariate control chart for detecting increases in process dispersion. Statistica Sinica, 1683-1707.zh_TW