近年來，統計製程管制(SPC)方法已經廣泛地應用在工業界上，在SPC 的應用上， 製程品質的好壞大多可藉由重要品質特徵的分布是發生變化來決定。然而有些產品或製 程的品質好壞藉由反應變數與解釋變數間是否滿足的某個函數關係來表示會更為適 當，此種的資料型態即稱為輪廓資料，而輪廓監控就是用來監控此函數關係是否發生改 變，若函數關係改變，則此製程的品質已發生失控。目前對於簡單線性輪廓監控的研究 上，皆假設函數關係中的隨機誤差項滿足常態分配的假設，然而在某些應用上，混合常 態分配(mixture normal distribution)的假設要比常態分配更符合實際的狀況。當隨機誤差 項為一混合常態分配時，欲監控隨機誤差項的參數是否發生改變，首先必須適當的區分 出一輪廓資料中的隨機誤差值是來自於哪一個常態分配。因此，我們考慮當隨機誤差項 服從混合常態分配的情形，並於第二階段(Phase II)已有部分歷史資料下，欲利用EM 演 算法來估計穩定狀態下的參數，以及將任一簡單線性輪廓資料的殘差作分類，在我們的 計畫構想中，若殘差資料可以適當的被區分出來自於哪一個常態分配，則我們可以將所 有的殘差值標準化，再進一步使用EWMA管制圖來監控此函數關係以及隨機誤差項的 參數是否發生變化。 Statistical process control (SPC) has been successfully applied in a variety of industries. In most SPC applications, the quality of a process can be adequately represented by the distribution of some important quality characteristics. However, in some applications the quality of a process or product is better characterized and summarized by a functional relationship between a response variable and one or more explanatory variables. A collection of this kind of data points is called a profile. Profile monitoring is used to understand and to check the stability of this relationship or curve over time. If the relationship has changed, the quality of process or product is out-of-control. The normality assumption for the error term is used in the existing simple linear profile monitoring models. However, in certain applications, the mixture normal assumption for the error term may be more appropriate in real situations. Therefore, a process with mixture simple linear profiles is considered in this article. When the underlying distribution of error term is mixture normal distributed, it is essential to find the proper classification of the error terms. Hence, after fitting the simple linear regression, we consider to find the classification of the residuals and the in-control parameters using the EM-algorithm for Phase II monitoring. Then, we can standardize all the residuals and apply the EWMA control charts to monitor the stability of the functional relationship and the parameters of the error term.
