當管制下共變異矩陣需估計時多變量製程變異之監控

Abstract

一般在實際應用管制圖時，通常管制狀態下的製程參數為未知，而必須執行階段一分析以取得管制狀態下之資料來估計製程參數，並用以建構階段二線上監控之管制圖。針對多變量製程之管制下共變異矩陣未知的狀況，我們利用概似比檢定(LRT)統計量來建構一個單邊與兩個雙邊之監控共變異矩陣的管制圖，並以控制期望假警報率於某一預設水準為準則，提供蒙特卡羅模擬方法來建構出適當的管制界限。我們以統計模擬計算這些管制圖在各種共變異數矩陣改變下之期望警報率來衡量這些管制圖之績效表現，並透過一半導體實例與模擬例子來闡述三種管制圖的應用及績效。

When implementing a control chart, the in-control parameters of the process are usually unknown and need to be estimated from the in-control data obtained from phase I analysis, and then used to construct the control limits for phase II online monitoring. Assume the in-control covariance matrix is unknown, we establish a one-sided-test-based control chart and two two-sided-test-based control charts based on the likelihood ratio test (LRT) statistics for testing the covariance matrix of the quality characteristic vector of the current process. Considering the randomness of the estimated covariance matrix, we construct the control limits by controlling the expected false alarm rate at a prescribed level. Algorithms that are computationally feasible are developed for constructing such control limits via Monte Carlo simulation. The performance of these control charts are evaluated in terms of the detecting power of various changes in the covariance matrix through a simulation study. The applicability and effectiveness of the three control charts are illustrated via a semiconductor example and two simulated examples.