偵測二維度資料之線性趨勢管制圖

Abstract

現今的統計製程管制圖在很多領域都被廣泛的運用，但是目前大部分的管制圖方法為偵測製程參數是否有偏移，而對於製程參數呈現線性的改變或是資料出現線性趨勢卻仍在管制圖的管制界限內的情形較少著墨。本篇文章利用Anderson (2003)所提出的線性模型和檢定方法為基礎，提出五種偵測線性趨勢資料的方法。在線性模型的共變異數矩陣未知的情形下，我們利用檢定斜率係數或全部的迴歸係數是否為0來偵測資料是否有線性趨勢；而在共變異數矩陣已知的情形下，我們也提出了利用檢定斜率係數或迴歸係數是否為0來偵測資料是否具有線性趨勢；最後的方法為在製程管制狀態下的分配與製程失控的分配其共變異數矩陣不相同的情形下，檢定迴歸係數是否為0來達到偵測資料是否具有線性趨勢的目的。之後利用平均連串長度來比較我們的方法和Hotelling’s 管制圖和MEWMA管制圖方法的偵測能力；整體而言，我們提出的檢定迴歸係數之方法表現得比檢定斜率係數的方法好；除此之外，我們提出的檢定迴歸係數之方法表現得比Hotelling’s 管制圖和MEWMA管制圖方法好。

Nowadays, control charts have been widely used in many fields, but most of the control charts are used to detect shifts of the process parameters. It is seldom to consider developing control charts to monitor the situation when the process parameters change linearly in two-dimensional data or data exhibit the linear trend within the control limit. We utilize the linear model and the associated tests given in Anderson (2003) to propose five methods for detecting the linear trend under various conditions. We construct control charts based on testing whether the slope coefficients or the entire regression coefficients are zero to detect the linear trend for the cases when the covariance matrix is known and unknown, respectively. Finally, when the covariance matrix of the in-control process is different from that of the out-of-control process, we construct a control chart based on testing whether the regression coefficients are zero to detect the linear trend. Then we compare the effectiveness of our methods with Hotelling’s control chart and MEWMA control chart in terms of average run length via simulation. The results of the simulation study indicate that the proposed methods for testing the regression coefficients perform better than testing the slope coefficients, and the proposed methods for testing the regression coefficients perform better than the existing control charts under comparism.