無分配假設剖面資料製程監控方法之研究

Abstract

在品管上，一般而言，製程或產品之品質特性都是一個變數。然而對某些製程而言， 品質特性是由反應變數和一或多個解釋變數間之關係來界定。因此一個品質特性乃以一 個函數、一條曲線或是一個曲面之資料型式來呈現，稱之為profile (縱斷面或剖面)。本 計畫主旨在於研究探討如何有效地監控製程profiles。文獻上大都假設固定效應或採用參 數迴歸模型。本計畫延續99 年度計畫採用隨機效應無母數迴歸模型讓profiles 的函數形 式更具彈性，可應用的範圍更廣。在此模型下，Shiau et al. (2009) 利用主成份分析 (principal component analysis, PCA)來剖析profiles 的特性，然後由此提出監控方法。不 過，此作法要有其效用，卻隱藏著一個不太明顯的假設：無論是in-control 還是 out-of-control profiles，其主要的特性都要能被此K 個有效主成份所描述。因此當一個形 式與in-control 的形式大不相同的out-of-control profile 產生時，就不一定能被由in-control 資料建構出來的管制圖所偵測到。99 年度之研究計畫，在離散化profile 具多變量常態 分配的假設下，對Shiau et al. (2009) 所提監控方法提出修正方法使之得以適用所有的情 形。然而，在實際的應用中，profiles 資料往往不服從多變量常態分配。本計畫擬發展在 隨機效應無母數迴歸模型之下的distribution-free profile 監控方法，讓profile 監控技術 可應用到更一般的情況。本計畫為二年期計畫，第一年研究發展第二階段distribution-free 線上profile 監控方法；第二年研究發展第一階段distribution-free profile 監控方法。本計 畫所提之研究在學術上和實際應用上都有相當的貢獻。

In many practical situations, the quality of a process or product is characterized by a relationship (or profile) between a response variable and one or more independent variables instead of by the distribution of a single quality characteristic. Most research works in the literature assumed fixed-effect and/or parametric regression models to model profiles. The main objective of this project is to develop tools for profile monitoring for profiles of more flexible shapes and under more general and practical situations. Under a random-effect model and adopting nonparametric regression approach, Shiau et al. (2009) proposed some profile monitoring schemes. They proposed to first characterize the process via functional principal component analysis and then construct control charts accordingly. However, this method works only when all profiles, including out-of-control profiles, all can be well characterized by the functional space spanned by the chosen effective principal components. Thus when out-of-control profiles have quite different features from the in-control profiles, the method might fail. In the last NSC-supported project, we proposed and studied a new profile monitoring scheme that provides a remedy to the above problem under the assumption that the profiles follow a Gaussian process. In this two-year project proposal, we propose to further extend the scope of profile monitoring by relaxing the Gaussian assumption. In the first year, we will propose and study a new distribution-free profile monitoring scheme for Phase II online monitoring of random-effect profiles of flexible shape. In the second year, we propose to develop a distribution-free tool for Phase I analysis under the same random-effect nonparametric regression model. The outcomes of this project will definitely make contributions in both academia and industries.