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研究生:陳韋廷
研究生(外文):Wei-Ting Chen
論文名稱:多變量自相關程序之鑑別與統計監控
論文名稱(外文):Identification and Statistical Monitoring of Multivariate Autocorrelated Processes
指導教授:黃世宏
指導教授(外文):Shyh-Hong Hwang
學位類別:碩士
校院名稱:國立成功大學
系所名稱:化學工程學系碩博士班
學門:工程學門
學類:化學工程學類
論文種類:學術論文
論文出版年:2008
畢業學年度:96
語文別:中文
論文頁數:93
中文關鍵詞:多變量SPC自相關性主成份分析
外文關鍵詞:AutocorrelationMultivariate SPCPCA
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統計製程管制已被廣泛地應用於多變量工業製程的監控。隨著量測技術的發展,現代化工業製程中可量測變量的個數也大為增加,這些變量間常存在互相關性,使得基於單變量觀點之傳統管制圖,如Shewhart、CUSUM、EWMA等,無法適用於多變量製程的管制。Hotelling所提之T2統計量加入互相關性的考量,因而可將傳統單變量管制圖修正應用至多變量製程中。另一方面,變量個數的增加不僅將計算複雜化,亦容易降低管制圖的監控靈敏度。主成份分析(PCA)法利用變量重組來排除不重要成份,簡化後之主變量更適合計算並能夠增強監控效能。
由於受到製程動態的影響,量測變量常具有自相關性,這容易讓基於T2統計量之管制圖與PCA法產生誤判警報。本論文利用量測變量之預估誤差建立多變量管制圖,可有效降低自相關性來減少誤判之發生。變量預估誤差之建立須知隨機擾動模型,在此我們考慮轉移函數與時間序列兩種模型,前者較為化學工程師所熟悉,後者則較適合於模型之鑑別。本論文提出兩種監控策略:當擾動模型(轉移函數或時間序列)為已知時,直接利用該模型來建立變量預估誤差,然後以PCA法將變量預估誤差中不重要成份排除,最後結合多變量管制圖來監控製程;當擾動模型為未知時,則先以PCA法將原始量測變量中不重要成份排除,然後針對主變量以最小平方法鑑別出向量自迴歸或向量自迴歸移動平均之時間序列模型,並據以建立預估誤差為基之管制圖。
所提兩種監控策略皆能有效偵測多變量製程之異常發生,對於較大擾動偏移以使用Hotelling管制圖為佳,而對於較小擾動偏移則以MEWMA管制圖較為有效。此外,以MEWMA管制圖另外監控PCA法中較不重要之變量,可以偵測到這些變量發生異常之情況,這能改善PCA法的不足。
Abstract

Statistical process control has been widely applied to multivariate manufacture and process industries. With the advance in measurement technology, the number of measured variates has been increased substantially in modern industrial processes. Because these variates are often cross-correlated, conventional control charts based on a univariate viewpoint, such as Shewhart, CUSUM and EWMA, are not suited to multivariate process monitoring. The T2 statistic developed by Hotelling takes the cross correlation into account and hence is able to extend conventional univariate control charts to multivariate processes. On the other hand, an increase in the number of variates would complicate computation and reduce the monitoring sensitivity of control charts. Principal component analysis (PCA) utilizes variate recombination to eliminate unimportant components, resulting in the principal variates that facilitate computation and enhance the monitoring efficiency.
Autocorrelation occurs frequently in measured variates due to the influence of process dynamics, causing control charts based on the T2 statistic and PCA to deliver false alarms. In this thesis, the prediction errors of measured variates are employed to establish multivariate control charts. Such control charts can effectively reduce autocorrelation to avoid false alarms. The prediction errors of variates are formed based on stochastic disturbance models. Here, transfer function models and time series models are considered; the former are more familiar to chemical engineers, whereas the latter are more suited to model identification. This thesis presents two monitoring strategies. When the disturbance model (transfer function or time series) is known, calculate the prediction errors of variates directly using the model. Then, employ PCA to eliminate the unimportant components. Finally, establish a multivariate control chart for process monitoring. When the disturbance model is unknown, first employ PCA to eliminate the unimportant components and then use the least squares method to identify a vector autoregressive or a vector autoregressive moving average time series model according to the principal variates. Finally, establish a multivariate control chart based on the prediction errors.
Both the developed monitoring strategies can effectively detect an abnormal condition in a multivariate process. Hotelling’s control chart is preferred for a larger disturbance shift, whereas the MEWMA chart is more suited for a smaller disturbance shift. In addition, using an extra MEWMA chart to monitor the unimportant components revealed by PCA can detect an abnormal condition corresponding to those variates. This can improve the deficiency of PCA.
摘要
英文摘要
總目錄......................................................Ⅰ
圖目錄......................................................Ⅳ
表目錄......................................................Ⅹ
符號表......................................................XI
第一章 緒論..................................................1
1-1 簡介..................................................1
1-2 文獻回顧..............................................3
1-3 研究動機..............................................6
第二章 多變量統計製程管制....................................8
2-1 多變量常態分配........................................8
2-2 多變量管制圖..........................................9
2-2.1 Hotelling管制圖................................. 9
2-2.2 多變量CUSUM管制圖............................11
2-2.3 多變量 EWMA管制圖............................12
2-3 主成份分析之概念.....................................14
2-4 自相關程序數據.......................................15
2-4.1 時間序列模型...................................16
2-4.2 轉移函數模型................................... 18
第三章 理論推導.............................................19
3-1 監控自相關程序數據...................................19
3-1.1 以時間序列模型為基礎之預估誤差............... 19
3-1.2 以轉移函數模型為基礎之預估誤差................ 28
3-2 模型的鑑別...........................................34
3-2.1 VAR(1)之最小平方估計...........................34
3-2.2 VAR(m)之最小平方估計........................... 35
3-3 主成份分析的算法與應用...............................39
3-3.1 主成份分析之處理步驟...........................39
3-3.2 主成份分析應用於監控圖.........................43
3-3.3 結合主成份分析與MEWMA.................... 45
第四章 多變量製程監控之模擬研究.............................51
4-1 以模型為基礎之監控模擬............................... 51
4-2 基於模型鑑別之監控模擬...............................60
4-3 結合主成份分析之監控模擬.............................68
4-3.1策略1之監控模擬................................ 68
4-3.2 策略2之監控模擬............................... 77
第五章 結論與未來工作.......................................88
參考文獻....................................................90
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