函數資料的異常製程偵測與診斷及變異分析之研究

A Study on Functional FDD and Functional ANOVA

吳侑峻(Yu-Chun Wu)

指導教授 ： 洪志真

國立交通大學/理學院/統計學研究所/博士(2015年)

https://doi.org/10.6842/NCTU.2015.00163

摘要

隨著科技進步，工業製程資料的型態也更為複雜，近年製程剖面資料的監控以及
異常偵測、診斷等問題逐漸受到關注。許多剖面資料有別於過去多變量製程資料，具
有高維度、高相關性等特性，導致傳統多變量方法無法直接應用。本研究利用特徵函
數來描述剖面資料特徵，藉此降低維度，並在特徵函數空間上定義能區別剖面資料之
特徵差異的統計量。本論文所提之方法可應用在處理多類別異常製程偵測與診斷的問
題，以及描述剖面資料的實驗之因子效應，主要研究成果有三部分。若收集到的剖面
資料來自一高斯隨機過程，我們提出將剖面資料投影到特徵函數空間上，再利用其
函數主成份分析（functional principal components analysis, FPCA）得到的分數向量及其
Hotelling’s T2 作為識別資料與歷史資料特徵相似與否的統計量，並以Hotelling’s T2 統
計量所對應之p-值做為診斷依據。由於剖面資料在實務上未必符合高斯假設，因此
我們亦提出無母數製程偵測與診斷的方法。當已有數類異常製程的歷史資料時，我
們結合函數部分最小平方法和邏輯斯迴歸模型來預測各類異常發生的機率，並藉此
診斷出異常原因。同時，我們以FPCA 分數向量的多變量符號-順序統計量（spatial
signed-rank, SSR）來比較新資料和歷史資料的特徵；若特徵相似度過低則判斷為製程
發生未知異常。SSR 的管制界限則以加權平滑拔靴法得之。剖面資料的實驗設計旨在
分析在不同因子水準下的應變量之剖面資料特徵是否相異，藉此找出影響應變量之重
要因子。我們利用剖面資料的特徵函數空間來描述因子效應，並提出以交叉驗證法選
取適合描述因子效應之特徵函數空間，以提高因子效應顯著性的檢定力。

關鍵字

剖面資料 ； 函數主成份分析 ； 函數部分最小平方法 ； 函數變異分析 ； 異常製程偵測與診斷 ； 空間符號-順序統計量

並列摘要

With recent advances in technology, industrial process data have become complicated, and
more and more researches have been devoted on process control, fault detection and diagnosis
for profile data. Many of the multivariate techniques are improper to analyze profile data because
of the high dimensionality and high correlation of profile data. The purpose of the dissertation
is to provide methods for dimension reduction and feature extraction for profiles. We describe
the curve characteristics of profiles by Karhunen–Loève expansion and construct characteristic
similarity metrics based on the eigenspace. The proposed approach is apply to multi-fault
detection and diagnosis (FDD) and functional analysis of variance (FANOVA). The first study
focuses on FDD when profiles follow a Gaussian stochastic process. The principal component
scores of profiles are utilized to construct Hotelling’s T2 statistic for assessing the characteristic
similarity between an incoming profile and historical profiles. We diagnose the potential fault
according to the p-value of Hotelling’s T2 statistics. Since the Gaussian assumption may be not
valid in practice, we propose a non-parametric FDD procedure for profiles. The first step of the
non-parametric FDD procedure is to build a logistic regression model via functional partial least
squares for forecasting the possible fault. Then spatial signed-rank (SSR) statistic of the principal
component scores are used to compare incoming profile and historical profiles. The cut-off
value of the SSR statistic is obtained by the weighted smoothed bootstrap method. Finally, the
functional analysis of variance with profile response is considered. A Hotelling’s generalized
ii
T2 statistic via principal component scores is proposed. A scheme is also constructed in order
to determine a proper eigenspace for describing factor effects.

並列關鍵字

profile ； functional principal components analysis ； functional partial least squares ； functional analysis of variance ； fault detection and diagnosis ； spatial signed-rank

參考文獻

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