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研究生:姚妮君
研究生(外文):Ni-Chun Yao
論文名稱:藉由近似線性的分段監控非線性剖面
論文名稱(外文):Monitoring Nonlinear Profiles by Piecewise Linear Approximation
指導教授:范書愷范書愷引用關係
學位類別:碩士
校院名稱:元智大學
系所名稱:工業工程與管理學系
學門:工程學門
學類:工業工程學類
論文種類:學術論文
論文出版年:2009
畢業學年度:97
語文別:英文
論文頁數:94
中文關鍵詞:非線性剖面變化點統計製程監控垂直密度剖面資料赤池資訊準則Schwarz資訊準則
外文關鍵詞:Nonlinear ProfileChange PointsStatistical Process Control (SPC)Vertical Density Profiles (VDP)Akaike’s Information Criterion (AIC)Schwarz Information Criterion (SIC)
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許多實際製程情況下,使用一個或多個解釋(可控)變數與反應變數之間的關係來描述製程或產品之品質特性是比較適當的。在許多情况下,反應變數Y和解釋變數X之間的關係可以由剖面(profile)來呈現。近幾年來,監控剖面資料在統計製程監控領域已經成為熱門研究議題。但實際上,文獻中對於非線性剖面製程監控並無太多著墨。

本研究將非線性剖面分割成近似線性的多重線段,並且分別監控每一個分段;也就是說,在Phase I的監控包含了建立模型並估計管制界線,而在Phase II監控時,利用所估計出的管制界線來監控製程。本研究提出了新的方法來決定剖面上的變化點數量及其位置,以便分割非線性剖面,並利用兩個準則來決定最佳變化點數量以避免過度配適情況發生,同時使用多個製程剖面資料來測試本研究提出的方法以說明其可行性。文中也使用平均串聯長度(average run length)來比較多變量 管制圖及EWMA3這兩種監控方法之優劣。本論文最後使用垂直密度剖面資料(vertical density profiles)為例,使用多變量 管制圖配合線性近似分段來達到非線性剖面資料監控的目的,在模擬實驗中分別改變不同模型參數,以模擬製程處於非管制狀態下,比較本研究方法與Williams et al. (2007)製程監控績效。
In many practical situations, the quality of a process or product is better characterized and summarized by the relationship between a response variable and one or more explanatory variable(s). In many cases, there is a relationship between the response variable Y and explanatory variables X that can be represented by a profile. In recent years, profile monitoring has been a popular and fertile field of research in statistical process control. In fact, little work has been done to address the monitoring of nonlinear profiles.

In order to monitor the whole nonlinear profiles, this thesis focuses on segmenting the entire nonlinear profile into several linear approximations and monitoring each segment separately. That is to say, included in Phase I analysis are model building and the parameter estimation required to construct the control limits. In Phase II, the control limits are used to monitor the process. Here, this thesis proposes a new method for determining the number and the location of change points. Two criterions are used to select the best number of change points to avoid over fitting. Some manufacturing profiles are presented to verify the proposed method. The ARL comparison between method and EWMA3 (2003) are demonstrated based on the premise that the nonlinear profile is fitted by the proposed change point model. Lastly, the vertical density profile data is used to evaluate the monitoring performance between the proposed method and Williams et al. (2007).
Table of Contents
摘 要 i
ABSTRCT ii
Table of Contents iii
List of Tables iv
List of Figures vi
Nomenclature vii
Chapter 1 Introduction 1
1.1 Background 1
1.2 Motivation 2
1.3 Organization of the Thesis 4
Chapter 2 Literature Review 6
2.1 Statistical Process Control (SPC) 6
2.2 Profile Statistical Process Control 10
2.2.1 Linear Profiles 12
2.2.2 Nonlinear Profiles 17
2.3 Change Point Determination 21
2.4 Akaike Information Criterion (AIC) 24
2.4.1 Kullback-Leibler Information 24
2.4.2 Akaike Information Criterion 26
2.4.3 The Least Squares Case 29
2.4.4 Second Order Information Criterion ( AICc ) 30
2.4.5 A small sample correction for Schwarz Information Criterion 32
Chapter 3 The Proposed Method for Monitoring Nonlinear Profiles 33
3.1 Method 33
Chapter 4 Further Experimental Results of Change Point Detection and Process Monitoring 50
4.1 Examples of Other Manufacturing Processes 50
4.2 Monitoring of Nonlinear Profiles Using Linear Approximation 57
4.2.1 Monitoring Method of Kim et al. (2003) 57
4.2.2 Monitoring fluorescence anisotropy data 59
4.3 Monitoring Nonlinear Profiles of Vertical Density Profiles (VDP) 69
4.3.1 Vertical Density Profiles (VDP) 69
4.3.2 The Method of Williams et al. (2007) 70
4.3.3 Phase I Monitoring 74
4.3.4 Phase II Monitoring 78
Chapter 5 Conclusion and Future Research 90
References 92

List of Tables
Table 3.1 Fluorescence anisotropy measurements 37
Table 3.2 Slope different with possible point and assigned level 39
Table 3.3 The RSS of each suggested point in the first iteration 40
Table 3.4 Results of the proposed method 42
Table 3.5 Results of Jones and Dey’s approach 43
Table 3.6 The locations of the points 48
Table 4.1 Results of reflow process 50
Table 4.2 Results of vacuum heat treatment 51
Table 4.3 Results of vacuum brazing aluminum 52
Table 4.4 Results of windscreen manufacturing 53
Table 4.5 Results of nocolok brazing aluminum 55
Table 4.6 The locations of the change points 56
Table 4.7 The locations of the change points 56
Table 4.8 Control limits of each segment 61
Table 4.9 a0 shift from a0 to a0+ (segment 1) 62
Table 4.10 a1 shift from a1 to a1 + (segment 1) 62
Table 4.11 shift from to (segment 1) 63
Table 4.12 a0 shift from a0 to a0+ (segment 2) 63
Table 4.13 a1 shift from a1 to a1 + (segment 2) 64
Table 4.14 shift from to (segment 2) 64
Table 4.15 a0 shift from a0 to a0+ (segment 3) 65
Table 4.16 a1 shift from a1 to a1 + (segment 3) 65
Table 4.17 shift from to (segment 3) 66
Table 4.18 a0 shift from a0 to a0+ (segment 4) 66
Table 4.19 a1 shift from a1 to a1 + (segment 4) 67
Table 4.20 shift from to (segment 4) 67
Table 4.21 Estimated parameter values from Williams et al. (2007) 71
Table 4.22 statistic for the VDP data 74
Table 4.23 Result of Phase I analysis 76
Table 4.24 a1 shift from a1 to a1 + 0.5 78
Table 4.25 a2 shift from a2 to a2 + 0.5 79
Table 4.26 b1 shift from b1 to b1 + 0.5 80
Table 4.27 b2 shift from b2 to b2 + 0.5 81
Table 4.28 c shift from c to c + 0.5 82
Table 4.29 d shift from d to d + 0.5 83
Table 4.30 a1 shift from a1 to a1 + 86
Table 4.31 a2 shift from a2 to a2 + 86
Table 4.32 b1 shift from b1 to b1 + 87
Table 4.33 b2 shift from b2 to b2 + 87
Table 4.34 c shift from c to c + 88
Table 4.35 d shift from d to d + 88

List of Figures
Figure 2.1 A typical control chart. 8
Figure 2.3 Milligrams of Aspartame Dissolved per Liter of Water. 11
Figure 3.1 The outline of the proposed method 36
Figure 3.2 The positions of the fluorescence anisotropy data 37
Figure 3.2 (a) Two linear regression models and one change point 44
Figure 3.2 (b) Three linear regression models and two change points 44
Figure 3.2 (c) Four linear regression models and three change points 45
Figure 3.2 (d) Five linear regression models and four change points 45
Figure 3.5 Flow chart of the proposed method 46
Figure 3.3 Line with three change points fit to the fluorescence anisotropy data of the proposed approach 47
Figure 3.4 Line with three change points fit to the fluorescence anisotropy data of Jones and Dey 47
Figure 4.1 Seven change points fit to the profile of reflow process 51
Figure 4.2 Twelve change points fit to the profile of vacuum heat treatment 52
Figure 4.3 Seven change points fit to the profile of aluminum vacuum brazing 53
Figure 4.4 Eleven change points fit to the profile of windscreen manufacturing 54
Figure 4.5 Sixteen change points fit to the profile of nocolok brazing aluminum 54
Figure 4.6 VDP of 24 particleboards 69
Figure 4.7 “Bathtub” function fit to VDP data 72
Figure 4.8 Eleven change points fit to the average VDP data 75
Figure 4.9 The out-of-control boards indicated by Williams et al. (with red line) 77
Figure 4.10 The out-of-control boards indicated by the proposed method (with red line) 77
Figure 4.11 a1 shift from a1 to a1 + 0.5 79
Figure 4.12 a2 shift from a2 to a2 + 0.5 80
Figure 4.13 b1 shift from b1 to b1 + 0.5 81
Figure 4.14 b2 shift from b2 to b2 + 0.5 82
Figure 4.15 c shift from c to c + 0.5 83
Figure 4.16 d shift from d to d + 0.5 84
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