第二階段二維剖面監控之研究

Abstract

近年來以統計觀點來監控製程或產品剖面資料的研究已有廣泛發展，大多數 的研究皆針對一維剖面提出第一階段或第二階段監控策略。本篇文章是針對二維 剖面並納入剖面與剖面之間的變異，運用PCA 及MPCA 提出第二階段的監控策 略。對於運用PCA 方面，將二維剖面矩陣的行向量拉成一維向量，並利用主成 分分析的方法來分析此一維向量的共變異結構，進而對主成分投影量 (principalcomponent scores) 提出監控策略。對於運用MPCA 方面，先利用Ye (2005) 提出的演算法來找尋分別為行方向和列方向的兩個基底矩陣，然後將新進的二維剖面同時對這兩個基底矩陣投影而獲得座標矩陣 (coordinatematrix) ，進而對座標矩陣提出監控策略。兩方法皆採用常見的Hotelling T 2管制圖來確認製程穩定性。評估兩種方法之績效的方式則是利用平均連串長度來比較偵測能力；整體而言，MPCA 表現較PCA 來得好，且在執行時間方面，MPCA 較不會受到維度增大的影響。

The study of monitoring process or product profiles by means of statistics already has extensive development. Most research works focus on one-dimensional profile monitoring schemes of Phase I or Phase II. This thesis is directed towards two-dimensional profiles with profile-to-profile variation. We utilize principal component analysis (PCA) and multilinear principal component analysis (MPCA) to propose monitoring schemes of Phase II. Since two-dimensional profile data are represented as matrices, we vectorize these matrix data for PCA to analyze the covariance structure of the resulting one-dimensional vectors. With that, we propose a control chart based on principal component scores. For using MPCA, we first utilize the algorithm proposed by Ye (2005) to search for two basis matrices, one for columns and one for rows, and then project an incoming two-dimensional profile onto the two basis matrices simultaneously to obtain a score matrix. We construct a monitoring scheme based on this score matrix. Both control charts use the Hotelling’s T 2 statistic to monitor the stability of the process.The performances of the two methods are evaluated and compared in terms of the average run length. Generally, MPCA performs better than PCA. Moreover, for the execution time, the MPCA method is not affected as much as the PCA method when the size of the profile matrix increases.