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題名 監控多維度品質變異數比的EWMA管制圖
EWMA Control Charts for Monitoring Multi-Dimensional Ratios of Process Variances
作者 林正和
Lin, Cheng-Ho
貢獻者 楊素芬
Yang, Su-Fen
林正和
Lin, Cheng-Ho
關鍵詞 多元統計製程控制
變異數比值
不偏估計式
資料深度
指數加權移動平均
無分佈假設
Multivariate statistical process control
Ratios of variances
Unbiased estimators
Data depth
Exponentially weighted moving average
Distribution-Free
日期 2024
上傳時間 4-Sep-2024 14:56:11 (UTC+8)
摘要 近年來，在工業製造或服務過程中，多變量品質變數的多維度變異數比的監控在某些實務製程中至關重要，但是，在統計製程管制 (SPC) 的研究中，尚無文獻探討。因此，本研究提出了新的管制圖以追蹤多個兩相依母體變異數之比值製程是否穩定的狀況。 在此研究中，我們分別考慮已知分佈為多元常態分配 (multivariate normal distribution) 、多元伽瑪分配 (multivariate gamma distribution) 和多元偏態分配 (multivariate skew normal distribution) 之下，以三個不同方法建立指數加權移動平均 (EWMA) 管制圖來追蹤多個相依母體變異數之比值。第一種方法是將兩個母體變異數線性組合的不偏估計量轉為T^2統計量，再分別根據樣本大小來建立EWMA-DT管制圖與近似的卡方管制圖，第二種方法則是使用資料深度 (data depth)的方法來建立EWMA-DU管制圖監控多個變異數之比值。第三種方法是以符號方法 (sign method) 來建立EWMA-DS管制圖監控多個變異數之比值。接著，以數值分析計算此三種管制圖之管制界線並考慮在失控的變動幅度相同下的平均連串長度來評估此三種管制圖的偵測能力。 最後，我們選出兩個偵測能力比較好的管制圖，並以牛奶與半導體的實際數據來說明它們的應用，並驗證其偵測能力。
In recent years, monitoring the multiple ratios of variances of multiple quality variables has become crucial in industrial manufacturing and service processes. However, there is a lack of research on this topic in statistical process control study. To address this gap, we propose three control charts for tracking the stability and variations of variance ratios for the multiple dependent process variables. In this study, we consider the multivariate normal, multivariate gamma and multivariate skew normal distributions, and develop three new exponentially weighted moving average (EWMA) control charts to monitor the multiple ratios of variances. The first method transforms the unbiased estimators of linear combinations of two population variances into T² statistic to construct the EWMA-DT control chart and the approximate chi-square control charts based on small or large sample sizes. The second method uses data depth procedure to create the EWMA-DU control chart. The third method employs the sign method to develop the EWMA-DS control chart. We evaluate the detection capabilities of these three control charts by calculating the average run length under the out-of-control processes. Finally, we select two better performance control charts to demonstrate their application and effectiveness in the real data using the milk and semiconductor processes respectively.
參考文獻 [1] Alt, F. B. (1982). Multivariate quality control: state of the art. In ASQC Quality Congress Transactions (pp. 886-893). Milwaukee, WI: American Society for Quality Control. [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] Bakir, S. T. (2004). A distribution-free Shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623. [6] Balakrishnan, N., Triantafyllou, I. S., & Koutras, M. V. (2010). A distribution-free control chart based on order statistics. Communications in Statistics—Theory and Methods, 39(20), 3652-3677. [7] 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. [8] 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. [9] Cho, E., & Cho, M. J. (2008). Variance of sample variance. Section on Survey Research Methods–JSM, 2, 1291-1293. [10] Crosier, R. B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics, 30(3), 291-303. [11] de Oliveira Moraes, D. A. (2014). A self-oriented control chart for multivariate process location. [12] Hotelling, H. (1949), "Multivariate Quality Control," in Techniques in Statistical Analysis, eds. C. Eisenhart, M. W. Hastay, and W. A. Wallis, New York: McGraw-Hill. [13] 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. [14] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387. [15] McCann, M., & Johnston, A. (2008). SECOM Data Set Center for Machine Learning and Intelligent Systems. University of California, Irvine, CA. [16] Nguyen, H. D., Tran, K. P., & Goh, T. N. (2020). Variable sampling interval control charts for monitoring the ratio of two normal variables. Journal of Testing and Evaluation, 48(3), 2505-2529. [17] Prabawani, N. A., & Mashuri, M. (2020, March). Performance of robust EWMA control chart for variability process using non-normal data. In Journal of Physics: Conference Series (Vol. 1511, No. 1, p. 012054). IOP Publishing. [18] Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technimetrics, 1, 239-250. [19] 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. [20] 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. [21] 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. [22] Yang, S. F., & Arnold, B. C. (2016). A new approach for monitoring process variance. Journal of Statistical Computation and Simulation, 86(14), 2749-2765. [23] 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. [24] Yang, S. F., Arnold, B. C., Liu, Y. L., Lu, M. C., & Lu, S. L. (2022). A new phase II EWMA dispersion control chart. Quality and Reliability Engineering International, 38(4), 1635-1658. [25] 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.
描述 碩士
國立政治大學
統計學系
111354017
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0111354017
資料類型 thesis
dc.contributor.advisor 楊素芬zh_TW
dc.contributor.advisor Yang, Su-Fenen_US
dc.contributor.author (Authors) 林正和zh_TW
dc.contributor.author (Authors) Lin, Cheng-Hoen_US
dc.creator (作者) 林正和zh_TW
dc.creator (作者) Lin, Cheng-Hoen_US
dc.date (日期) 2024en_US
dc.date.accessioned 4-Sep-2024 14:56:11 (UTC+8)-
dc.date.available 4-Sep-2024 14:56:11 (UTC+8)-
dc.date.issued (上傳時間) 4-Sep-2024 14:56:11 (UTC+8)-
dc.identifier (Other Identifiers) G0111354017en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/153363-
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) 111354017zh_TW
dc.description.abstract (摘要) 近年來，在工業製造或服務過程中，多變量品質變數的多維度變異數比的監控在某些實務製程中至關重要，但是，在統計製程管制 (SPC) 的研究中，尚無文獻探討。因此，本研究提出了新的管制圖以追蹤多個兩相依母體變異數之比值製程是否穩定的狀況。 在此研究中，我們分別考慮已知分佈為多元常態分配 (multivariate normal distribution) 、多元伽瑪分配 (multivariate gamma distribution) 和多元偏態分配 (multivariate skew normal distribution) 之下，以三個不同方法建立指數加權移動平均 (EWMA) 管制圖來追蹤多個相依母體變異數之比值。第一種方法是將兩個母體變異數線性組合的不偏估計量轉為T^2統計量，再分別根據樣本大小來建立EWMA-DT管制圖與近似的卡方管制圖，第二種方法則是使用資料深度 (data depth)的方法來建立EWMA-DU管制圖監控多個變異數之比值。第三種方法是以符號方法 (sign method) 來建立EWMA-DS管制圖監控多個變異數之比值。接著，以數值分析計算此三種管制圖之管制界線並考慮在失控的變動幅度相同下的平均連串長度來評估此三種管制圖的偵測能力。 最後，我們選出兩個偵測能力比較好的管制圖，並以牛奶與半導體的實際數據來說明它們的應用，並驗證其偵測能力。zh_TW
dc.description.abstract (摘要) In recent years, monitoring the multiple ratios of variances of multiple quality variables has become crucial in industrial manufacturing and service processes. However, there is a lack of research on this topic in statistical process control study. To address this gap, we propose three control charts for tracking the stability and variations of variance ratios for the multiple dependent process variables. In this study, we consider the multivariate normal, multivariate gamma and multivariate skew normal distributions, and develop three new exponentially weighted moving average (EWMA) control charts to monitor the multiple ratios of variances. The first method transforms the unbiased estimators of linear combinations of two population variances into T² statistic to construct the EWMA-DT control chart and the approximate chi-square control charts based on small or large sample sizes. The second method uses data depth procedure to create the EWMA-DU control chart. The third method employs the sign method to develop the EWMA-DS control chart. We evaluate the detection capabilities of these three control charts by calculating the average run length under the out-of-control processes. Finally, we select two better performance control charts to demonstrate their application and effectiveness in the real data using the milk and semiconductor processes respectively.en_US
dc.description.tableofcontents 1. Introduction 1 1.1. Literature review 1 1.2. Study motivation and proposed research methods 3 2. The In-control Distributions of Estimators for Monitoring the Ratios of Variances of Multivariate Quality Variables 5 2.1. The distribution of estimator for monitoring the ratios of variances of three different distributed multivariate quality variables 5 2.1.1. Consider multivariate normal distribution 8 2.1.2. Consider multivariate gamma distribution 10 2.1.3. Consider multivariate skew normal distribution 14 2.2. The distributions of estimators with the difference of observations under three multivariate quality variables 18 2.3. The distributions of estimators for different sample sizes under three multivariate quality variables 23 3. Design the EWMA-DT Chart for Monitoring the Multiple Ratios of Two Variances of Multivariate quality variables 37 3.1. Distribution of the DT² statistic for design of the EWMA chart 37 3.2. The EWMA-DT chart under three multivariate distributions 45 3.2.1. The EWMA-DT chart 46 3.2.2. The average EWMA-DT chart 60 3.2.3. The asymptotic EWMA-DC chart 62 4. Depth-Based EWMA Chart for Monitoring the Ratios of Two Variances of Multivariate Quality Variables 67 4.1. Distribution of the depth-based statistics 67 4.2. Design the depth-based EWMA-DU chart 68 4.3. Design the sign-based EWMA-DS chart 69 5. The Out-of-control Distributions of the Different Estimators of Ratios of Variances under Multivariate Quality Variables 72 5.1. The distribution of the out-of-control estimator under the multivariate normal quality variables 74 5.1.1. The parameters of the out-of-control variances under the multivariate normal variables 76 5.1.2. The out-of-control ratios of variances under the multivariate normal distribution with components variances 79 5.2. The distribution of the out-of-control estimator under the multivariate gamma quality variables 80 5.2.1. The parameters of the out-of-control variances under the multivariate gamma variables 83 5.2.2. The out-of-control ratios of variances under the multivariate gamma distribution with components variances 87 5.3. The distribution of the out-of-control estimator under the multivariate skew normal quality variables 88 5.3.1. The parameters of the out-of-control variances under the multivariate skew normal variables 92 5.3.2. The out-of-control ratios of variances under the multivariate skew normal distribution with components variances 96 5.4. Summary 97 6. Out-of-Control Detection Performance Comparisons of the Proposed EWMA Ratios of Variances Control Charts 98 6.1. Detection performance comparison of the proposed EWMA-DT chart under the three different multivariate distributions 98 6.2. Detection performance comparison of the proposed EWMA-DU chart under the three different multivariate distributions 119 6.3. Detection performance comparison of the proposed EWMA-DS chart under the three different multivariate distributions 142 6.4. Detection performance comparison of the proposed EWMA-DT chart and the EWMA-DU chart under the three different multivariate distributions 144 7. The Application of the Proposed EWMA-DT and EWMA-DU Charts Using Real Data Sets 145 7.1. A real example with semiconductor data set 145 7.2. A real example with milk data set 154 8. Conclusion and Future Outlook 165 Reference 167zh_TW
dc.format.extent 17459422 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0111354017en_US
dc.subject (關鍵詞) 多元統計製程控制zh_TW
dc.subject (關鍵詞) 變異數比值zh_TW
dc.subject (關鍵詞) 不偏估計式zh_TW
dc.subject (關鍵詞) 資料深度zh_TW
dc.subject (關鍵詞) 指數加權移動平均zh_TW
dc.subject (關鍵詞) 無分佈假設zh_TW
dc.subject (關鍵詞) Multivariate statistical process controlen_US
dc.subject (關鍵詞) Ratios of variancesen_US
dc.subject (關鍵詞) Unbiased estimatorsen_US
dc.subject (關鍵詞) Data depthen_US
dc.subject (關鍵詞) Exponentially weighted moving averageen_US
dc.subject (關鍵詞) Distribution-Freeen_US
dc.title (題名) 監控多維度品質變異數比的EWMA管制圖zh_TW
dc.title (題名) EWMA Control Charts for Monitoring Multi-Dimensional Ratios of Process Variancesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Alt, F. B. (1982). Multivariate quality control: state of the art. In ASQC Quality Congress Transactions (pp. 886-893). Milwaukee, WI: American Society for Quality Control. [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] Bakir, S. T. (2004). A distribution-free Shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623. [6] Balakrishnan, N., Triantafyllou, I. S., & Koutras, M. V. (2010). A distribution-free control chart based on order statistics. Communications in Statistics—Theory and Methods, 39(20), 3652-3677. [7] 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. [8] 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. [9] Cho, E., & Cho, M. J. (2008). Variance of sample variance. Section on Survey Research Methods–JSM, 2, 1291-1293. [10] Crosier, R. B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics, 30(3), 291-303. [11] de Oliveira Moraes, D. A. (2014). A self-oriented control chart for multivariate process location. [12] Hotelling, H. (1949), "Multivariate Quality Control," in Techniques in Statistical Analysis, eds. C. Eisenhart, M. W. Hastay, and W. A. Wallis, New York: McGraw-Hill. [13] 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. [14] Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387. [15] McCann, M., & Johnston, A. (2008). SECOM Data Set Center for Machine Learning and Intelligent Systems. University of California, Irvine, CA. [16] Nguyen, H. D., Tran, K. P., & Goh, T. N. (2020). Variable sampling interval control charts for monitoring the ratio of two normal variables. Journal of Testing and Evaluation, 48(3), 2505-2529. [17] Prabawani, N. A., & Mashuri, M. (2020, March). Performance of robust EWMA control chart for variability process using non-normal data. In Journal of Physics: Conference Series (Vol. 1511, No. 1, p. 012054). IOP Publishing. [18] Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technimetrics, 1, 239-250. [19] 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. [20] 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. [21] 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. [22] Yang, S. F., & Arnold, B. C. (2016). A new approach for monitoring process variance. Journal of Statistical Computation and Simulation, 86(14), 2749-2765. [23] 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. [24] Yang, S. F., Arnold, B. C., Liu, Y. L., Lu, M. C., & Lu, S. L. (2022). A new phase II EWMA dispersion control chart. Quality and Reliability Engineering International, 38(4), 1635-1658. [25] 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.zh_TW