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題名 多項比例精確管制圖之研究
Study of exact control chart for monitoring multinomial distribution processes
作者 甘勝進
Gan, Sheng-Jin
貢獻者 楊素芬<br>陳立榜
Yang Su-Fen<br>Chen Li-Pang
甘勝進
Gan, Sheng-Jin
關鍵詞 多項分配過程
皮爾森卡方統計量
加權指數滑動平均
測量誤差
measurement error
Multinomial distribution process
Pearson’s Chi-square statistic
EWMA
日期 2024
上傳時間 4-Sep-2024 14:55:34 (UTC+8)
摘要 管制圖已經廣汎應用于製造業中的質量監控, 在過程質量發生改變時及時發出報警方面,它扮演著重要的角色. 現有管制圖主要側重單變量或多變量連續型過程分配.爲了處理離散型分配,特別是多項分配過程, 借助皮爾遜卡方統計量來構建管制圖可能是一個共同的選擇. 然而, 這種管制圖嚴重依賴大樣本, 當樣本容量較小或者中等時產生不可靠結果. 本論文中, 我們主要探索多項分配過程管制圖. 我們首先重新審視了皮爾森卡方統計量,并推導出了其任意樣本下的均值和方差. 然後, 建立精確的EWMA比例管制圖. 與現有基於符號的EWMA管制圖和多項CUSUM圖相比, 模擬結果證明了我們方法的檢測性能. 另外, 測量誤差對精確管制圖的影響也得到研究, 一些模擬表明測量誤差延緩失控信號的發出.
Control charts have been widely used for monitoring output quality in manufacturing. It plays an important role in triggering a signal in time when detecting a change in process quality. Most existing control charts focus on the univariate or multivariate process data with continuous distribution. To deal with discrete distributions, in particular, the multinomial distribution processes, Pearson’s Chi-square statistic might be a common approach to construct control charts. However, it depends heavily on large sample sizes, which can yield unreliable result when sample size is small or moderate. In this thesis, we primarily explore the process control chart for multinomial distribution data. We first review Pearson’s Chi-square statistic, and derive the exact mean and variance regardless of sample sizes. After that, the exact exponentially weighted moving average (EWMA) proportions chart is derived under small or large sample sizes. Compared with existing sign-based EWMA chart and multinomial CUSUM chart for monitoring the multinomial distribution processes, simulation study is conducted to assess the performance of our proposed chart. Moreover, affection of measurement error on the exact control chart is also investigated, some simulation results suggest that measurement error delay detecting in out-of -control processes.
參考文獻 Abbasi, S. A. (2010). On the performance of EWMA chart in the presence of two-component measurement error. Quality Engineering, 22(3), 199-213. Chandrasekaran, S., English, J. R., & Disney, R. L. (1995). Modeling and analysis of EWMA control schemes with variance-adjusted control limits. IIE Transactions, 27(3), 282-290. Chen, L. P., & Yang, S. F. (2023). A new p-control chart with measurement error correction. Quality and Reliability Engineering International, 39(1), 81-98. Cocchi, D., & Scagliarini, M. (2011). Effects of the two-component measurement error model on X control charts. Statistica, 71(3), 307-327. Crosier, R. B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics, 30(3), 291-303. Falk, M. (1999). A simple approach to the generation of uniformly distributed random variables with prescribed correlations. Communications in Statistics-Simulation and Computation, 28(3), 785-791. Gan, S., & Yang, S. F.(2020). A EWMA control chart for multinomial distribution with application to monitoring multivariate mean shift. Technical report, National ChengChi University. Gan, S., Yang, S. F., & Chen, L. P. (2023). A new EWMA control chart for monitoring multinomial proportions. Sustainability, 15(15), 11797. Huang, W., Reynolds Jr, M. R., & Wang, S. (2012). A binomial GLR control chart for monitoring a proportion. Journal of Quality Technology, 44(3), 192-208. Huang, W., Wang, S., & Reynolds Jr, M. R. (2013). A generalized likelihood ratio chart for monitoring Bernoulli processes. Quality and Reliability Engineering International, 29(5), 665-679. Lee, J., Peng, Y., Wang, N., & Reynolds Jr, M. R. (2017). A GLR control chart for monitoring a multinomial process. Quality and Reliability Engineering International, 33(8), 1773-1782. Li, J., Tsung, F., & Zou, C. (2014). Multivariate binomial/multinomial control chart. IIE Transactions, 46(5), 526-542. Linna, K. W., & Woodall, W. H. (2001). Effect of measurement error on Shewhart control charts. Journal of Quality Technology, 33(2), 213-222. Linna, K. W., Woodall, W. H., & Busby, K. L. (2001). The performance of multivariate control charts in the presence of measurement error. Journal of Quality Technology, 33(3), 349-355. Lucas, J. M., & Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, 32(1), 1-12. Maravelakis, P. E. (2012). Measurement error effect on the CUSUM control chart. Journal of Applied Statistics, 39(2), 323-336. Maravelakis, P., Panaretos, J., & Psarakis, S. (2004). EWMA chart and measurement error. Journal of Applied Statistics, 31(4), 445-455. Marcucci, M. (1985). Monitoring multinomial processes. Journal of Quality Technology, 17(2), 86-91. Montgomery, D. C. (2009). Introduction to Statistical Quality Control. John Wiley & Sons. Nelson, L.S.(1987). A chi-square control chart for several proportions. Journal of Quality Technology, 19(4), 229-231. Qiu, P. (2008). Distribution-free multivariate process control based on log-linear modeling. IIE Transactions, 40(7), 664-677. Qiu, P. (2013). Introduction to Statistical Process Control. CRC press. Reynolds, M. R., & Stoumbos, Z. G. (1998). The SPRT chart for monitoring a proportion. IIE Transactions, 30(6), 545-561. Reynolds, M. R., & Stoumbos, Z. G. (2001). Monitoring a proportion using CUSUM and SPRT control charts. In Frontiers in Statistical Quality Control 6 (pp.155-175). Physica, Heidelberg. Ryan, A. G., Wells, L. J., & Woodall, W. H. (2011). Methods for monitoring multiple proportions when inspecting continuously. Journal of Quality Technology, 43(3), 237-248. Woodall, W. H. (1997). Control charts based on attribute data: bibliography and review. Journal of Quality Technology, 29(2), 172-183.
描述 博士
國立政治大學
統計學系
108354502
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0108354502
資料類型 thesis
dc.contributor.advisor 楊素芬<br>陳立榜zh_TW
dc.contributor.advisor Yang Su-Fen<br>Chen Li-Pangen_US
dc.contributor.author (Authors) 甘勝進zh_TW
dc.contributor.author (Authors) Gan, Sheng-Jinen_US
dc.creator (作者) 甘勝進zh_TW
dc.creator (作者) Gan, Sheng-Jinen_US
dc.date (日期) 2024en_US
dc.date.accessioned 4-Sep-2024 14:55:34 (UTC+8)-
dc.date.available 4-Sep-2024 14:55:34 (UTC+8)-
dc.date.issued (上傳時間) 4-Sep-2024 14:55:34 (UTC+8)-
dc.identifier (Other Identifiers) G0108354502en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/153360-
dc.description (描述) 博士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) 108354502zh_TW
dc.description.abstract (摘要) 管制圖已經廣汎應用于製造業中的質量監控, 在過程質量發生改變時及時發出報警方面,它扮演著重要的角色. 現有管制圖主要側重單變量或多變量連續型過程分配.爲了處理離散型分配,特別是多項分配過程, 借助皮爾遜卡方統計量來構建管制圖可能是一個共同的選擇. 然而, 這種管制圖嚴重依賴大樣本, 當樣本容量較小或者中等時產生不可靠結果. 本論文中, 我們主要探索多項分配過程管制圖. 我們首先重新審視了皮爾森卡方統計量,并推導出了其任意樣本下的均值和方差. 然後, 建立精確的EWMA比例管制圖. 與現有基於符號的EWMA管制圖和多項CUSUM圖相比, 模擬結果證明了我們方法的檢測性能. 另外, 測量誤差對精確管制圖的影響也得到研究, 一些模擬表明測量誤差延緩失控信號的發出.zh_TW
dc.description.abstract (摘要) Control charts have been widely used for monitoring output quality in manufacturing. It plays an important role in triggering a signal in time when detecting a change in process quality. Most existing control charts focus on the univariate or multivariate process data with continuous distribution. To deal with discrete distributions, in particular, the multinomial distribution processes, Pearson’s Chi-square statistic might be a common approach to construct control charts. However, it depends heavily on large sample sizes, which can yield unreliable result when sample size is small or moderate. In this thesis, we primarily explore the process control chart for multinomial distribution data. We first review Pearson’s Chi-square statistic, and derive the exact mean and variance regardless of sample sizes. After that, the exact exponentially weighted moving average (EWMA) proportions chart is derived under small or large sample sizes. Compared with existing sign-based EWMA chart and multinomial CUSUM chart for monitoring the multinomial distribution processes, simulation study is conducted to assess the performance of our proposed chart. Moreover, affection of measurement error on the exact control chart is also investigated, some simulation results suggest that measurement error delay detecting in out-of -control processes.en_US
dc.description.tableofcontents Chapter 1. Introduction 1 Chapter 2. The exact chart for monitoring multinomial distribution processes 4 2.1 An EWMA chart based on Pearson statistic for monitoring a multinomial proportions processes 4 2.2. Simulation comparison between the proposed exact control chart and the asymptotic control chart for monitoring multinomial distribution processes 8 2.3. Performance comparison with the existing multinomial CUSUM chart 18 2.4. Performance comparison with the existing sign-based EWMA control chart 24 2.5. An application of monitoring mean or median of a multivariate processes using the proposed exact chart 30 2.6. A real data analysis 34 Chapter 3. Effect of measurement error on the proposed exact chart for monitoring multinomial distribution processes 39 Chapter 4. Conclusions and future study 51 References 52 Appendix A (Derivation of the variance of the in-control Pearson’s Chi-square statistic) 54 Appendix B (Derivation of the variance of the out-of-control Pearson’s Chi-square statistic) 57zh_TW
dc.format.extent 617282 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0108354502en_US
dc.subject (關鍵詞) 多項分配過程zh_TW
dc.subject (關鍵詞) 皮爾森卡方統計量zh_TW
dc.subject (關鍵詞) 加權指數滑動平均zh_TW
dc.subject (關鍵詞) 測量誤差zh_TW
dc.subject (關鍵詞) measurement erroren_US
dc.subject (關鍵詞) Multinomial distribution processen_US
dc.subject (關鍵詞) Pearson’s Chi-square statisticen_US
dc.subject (關鍵詞) EWMAen_US
dc.title (題名) 多項比例精確管制圖之研究zh_TW
dc.title (題名) Study of exact control chart for monitoring multinomial distribution processesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Abbasi, S. A. (2010). On the performance of EWMA chart in the presence of two-component measurement error. Quality Engineering, 22(3), 199-213. Chandrasekaran, S., English, J. R., & Disney, R. L. (1995). Modeling and analysis of EWMA control schemes with variance-adjusted control limits. IIE Transactions, 27(3), 282-290. Chen, L. P., & Yang, S. F. (2023). A new p-control chart with measurement error correction. Quality and Reliability Engineering International, 39(1), 81-98. Cocchi, D., & Scagliarini, M. (2011). Effects of the two-component measurement error model on X control charts. Statistica, 71(3), 307-327. Crosier, R. B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics, 30(3), 291-303. Falk, M. (1999). A simple approach to the generation of uniformly distributed random variables with prescribed correlations. Communications in Statistics-Simulation and Computation, 28(3), 785-791. Gan, S., & Yang, S. F.(2020). A EWMA control chart for multinomial distribution with application to monitoring multivariate mean shift. Technical report, National ChengChi University. Gan, S., Yang, S. F., & Chen, L. P. (2023). A new EWMA control chart for monitoring multinomial proportions. Sustainability, 15(15), 11797. Huang, W., Reynolds Jr, M. R., & Wang, S. (2012). A binomial GLR control chart for monitoring a proportion. Journal of Quality Technology, 44(3), 192-208. Huang, W., Wang, S., & Reynolds Jr, M. R. (2013). A generalized likelihood ratio chart for monitoring Bernoulli processes. Quality and Reliability Engineering International, 29(5), 665-679. Lee, J., Peng, Y., Wang, N., & Reynolds Jr, M. R. (2017). A GLR control chart for monitoring a multinomial process. Quality and Reliability Engineering International, 33(8), 1773-1782. Li, J., Tsung, F., & Zou, C. (2014). Multivariate binomial/multinomial control chart. IIE Transactions, 46(5), 526-542. Linna, K. W., & Woodall, W. H. (2001). Effect of measurement error on Shewhart control charts. Journal of Quality Technology, 33(2), 213-222. Linna, K. W., Woodall, W. H., & Busby, K. L. (2001). The performance of multivariate control charts in the presence of measurement error. Journal of Quality Technology, 33(3), 349-355. Lucas, J. M., & Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, 32(1), 1-12. Maravelakis, P. E. (2012). Measurement error effect on the CUSUM control chart. Journal of Applied Statistics, 39(2), 323-336. Maravelakis, P., Panaretos, J., & Psarakis, S. (2004). EWMA chart and measurement error. Journal of Applied Statistics, 31(4), 445-455. Marcucci, M. (1985). Monitoring multinomial processes. Journal of Quality Technology, 17(2), 86-91. Montgomery, D. C. (2009). Introduction to Statistical Quality Control. John Wiley & Sons. Nelson, L.S.(1987). A chi-square control chart for several proportions. Journal of Quality Technology, 19(4), 229-231. Qiu, P. (2008). Distribution-free multivariate process control based on log-linear modeling. IIE Transactions, 40(7), 664-677. Qiu, P. (2013). Introduction to Statistical Process Control. CRC press. Reynolds, M. R., & Stoumbos, Z. G. (1998). The SPRT chart for monitoring a proportion. IIE Transactions, 30(6), 545-561. Reynolds, M. R., & Stoumbos, Z. G. (2001). Monitoring a proportion using CUSUM and SPRT control charts. In Frontiers in Statistical Quality Control 6 (pp.155-175). Physica, Heidelberg. Ryan, A. G., Wells, L. J., & Woodall, W. H. (2011). Methods for monitoring multiple proportions when inspecting continuously. Journal of Quality Technology, 43(3), 237-248. Woodall, W. H. (1997). Control charts based on attribute data: bibliography and review. Journal of Quality Technology, 29(2), 172-183.zh_TW