非線性隨機效應剖面資料之無母數監控方法

Nonparametric Monitoring Schemes for Nonlinear Profiles with Random Effects

鄭清仁(Ching-Ren Cheng)

指導教授 ： 洪志真

國立交通大學/理學院/統計學研究所/博士(2012年)

https://doi.org/10.6842/NCTU.2012.00992

摘要

隨著現代工業製程的進步，品質的特性經常是以應變量（response）與共變量（covariate）的函數關係呈現，也就是文獻裡所謂的剖面資料（profile）。因此，為了因應實際的需求，發展剖面資料的管制方法是必要的。近年來亦有多篇文獻探討該議題。此篇論文對於具隨機效應的剖面資料提供了完整的管制方法。首先，我們先考慮服從常態分配的剖面資料。為了提升管制方法的效率，我們利用主成分分析得到其主成分記分（principal component score），並利用該記分來發展管制圖。在此論文，我們對於常態分配的剖面資料分別探討在第一階段（Phase I）和第二階段（Phase II）的分析。在實際的應用裡，資料經常並非服從常態分配的假設。因此，在沒有假設資料分配的情況下，我們亦發展剖面資料的管制方法。為此，我們先對於多變量資料，發展無分配假設（distribution-free）的第一階段管制方法。接著，再對剖面資料發展無分配假設的第一及第二階段的管制方法。在第一階段的分析，我們利用型一錯誤（type-I error）及型二錯誤（type-II error）來當作衡量準則。而在第二階段的分析裡，我們利用平均運行步長（average run length）來衡量。透過模擬分析，我們所發展的管制方法對於各種製程的變化都能有效地偵測，包含平均位置的位移、資料散佈的位移或是函數形狀的改變。我們亦利用真實的資料來示範我們所提出的方法的適用性及效率。

關鍵字

管制圖 ； 無分配假設 ； 無母數 ； 剖面資料 ； 第一階段 ； 第二階段 ； 隨機效應 ； 主成分分析

並列摘要

As modern technology advances in many industrial processes, the quality characteristics are often gathered in the form of a relationship between the response variable and explanatory variable(s), which are often referred to as profiles in the literature. Therefore, developing schemes for monitoring various types of functional characteristics becomes necessary for practical use and has attracted many researchers in resent years. The purpose of this dissertation is to provide a comprehensive analysis for profiles with random effects. First, the case of the profiles following the Gaussian distribution is considered. To monitor the profiles efficiently, the principal component scores of profiles obtained from the principal component analysis are utilized to construct control charts. Both the Phase I analysis and Phase II monitoring for Gaussian profiles are discussed in this dissertation. Since the Gaussian assumption may be violated in many practical applications, we also develop a distribution-free control chart for profiles. To this end, we first develop a novel distribution-free Phase I control chart for multivariate data. Then, two distribution-free control charts for profile data are constructed for Phase I and Phase II applications, respectively. The type-I and type-II error rates are considered as the performance measures for Phase I analysis whereas the average run length is used for Phase II analysis. Our simulation studies indicate that the proposed control charts are efficient in detecting shifts in various kinds of aspects, including the mean, dispersion, and shape of the profile. Some real data analysis are also provided to demonstrate the applicability and effectiveness of the proposed control charts.

並列關鍵字

control chart ； distribution-free ； nonparametric ； profile ； Phase I ； Phase II ； random effect ； PCA

參考文獻

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