二維隨機效應剖面資料之監控方法

Abstract

在科技高度發展之下，工業的生產線亦越來越複雜，對這些複雜的製程發展適當的監控方法在品質管制上是重要的議題。近年來剖面資料監控方法之研究已成為統計品質管制(SPC)上大幅成長且相當熱門之研究領域，而現今許多工業製程已經複雜到必須監控曲面型的二維剖面資料，然而目前似乎尚無著墨於二維剖面資料監控之文獻。本研究考慮具隨機效應之二維剖面資料，採用無母數迴歸模型讓剖面的函數形式更具彈性，可應用的範圍更廣。在此模型下，我們利用二階(second order)多線性主成份分析(multilinear principal component analysis, 簡稱MPCA)來剖析管制中二維剖面資料的特性，然後由此提出監控方法。我們利用多線性主成份得到二維剖面資料的多線性主成份分數，並利用該分數來發展管制圖。第二階段中，在實際作業時管制圖的平均連串長度（ARL）績效表現會隨著不同應用中第一階段分析所估計的製程參數不同而有所變異，稱作「從業人員之間」的變異，因此可將ARL視為一隨機變數；所以我們探討二維剖面資料在第二階段下根據管制中ARL的分配來修正監控統計量的管制界限。在第一階段中，若是資料來自常態分配，我們根據常態特色發展一套管制方法；若是資料並非來自常態分配，我們亦發展無分配假設的第一階段管制方法；並利用偽陽性率與偽陰性率來當作衡量準則。最後使用真實的二維剖面資料來示範我們所提出的方法之適用性。

As high-tech advances rapidly, industrial production lines are getting more and more complicated. Developing suitable monitoring schemes for complicated processes as such has drawn much attention in the area of quality control. Among them, profile monitoring has been a growing and promising area of research in statistical process control (SPC) in recent years. Most research work in the literature focused on one-dimensional profiles. As technologies are advancing, two-dimensional profiles have become key quality characteristics for more and more processes; however, no monitoring schemes have been developed in the literature at the present time. Consider nonlinear two-dimensional profiles with random effects. Under a random-effect model and adopting the nonparametric regression approach, we propose using second-order multilinear principal component analysis (MPCA) to develop profile monitoring schemes. The multilinear principal component scores of two-dimensional profiles obtained from the multilinear principal component analysis are utilized to construct control charts. In Phase II, the average run length (ARL) performance of a control chart varies as the process parameter estimates from the Phase I analysis vary in each application. Consequently, the ARL becomes a random variable due to the so-called “practitioner-to-practitioner” variation. We develop an algorithm to construct a control limit with which practitioners would have a high “confidence” that the actual in-control ARL will exceed the nominal in-control ARL level. In Phase I, if two-dimensional profiles come from a normal distribution, we develop a control chart based on the distribution of the in-control ARL. If two-dimensional profiles violate the normal assumption, we develop a distribution-free control chart. The false-positive rate and false-negative rate are considered as the performance measures for Phase I analysis. Some real data analyses are provided to demonstrate the applicability of the proposed control charts.