冪次轉換在製程監控上面的應用｜國立中央大學博碩士論文系統

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研究生：	高世州 Shih-Chou Kao
論文名稱：	冪次轉換在製程監控上面的應用 Applications of Power Transformation on Process Monitoring
指導教授：	賀全慶 Chuanching Ho 何應欽 Ying-Chin Ho
口試委員:
學位類別：	博士 Doctor
系所名稱：	管理學院 - 工業管理研究所 Graduate Institute of Industrial Management
畢業學年度：	94
語文別：	英文
論文頁數：	74
中文關鍵詞：	位移參數、冪次轉換、常態分配、自然對數轉換、伽瑪分配、指數分配
外文關鍵詞：	location parameter, power transformation, normal distribution, exponential distribution, gamma distribution, natural logarithm transformation
相關次數：	點閱：19 下載：0
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針對製程品質特性屬於指數分配型態者，多位學者提出以冪次轉換的方法，將原本屬於指數分配型態的資料轉換成近似常態資料，以便應用於統計品管上。但是，這些學者針對不同的準則，建議出不同的冪次轉換，並無一定的定論。本論文以轉換後的近似分配與理論的常態分配差異最小為評估基準，提出兩種不同評估準則來衡量指數資料轉換後近似常態的程度，此兩方法分別取轉換後近似分配與理論分配的（1）絕對差總和為最小，與（2）離差平方總和為最小，為評估基準。這兩種方法，得到轉換冪次分別為a = 3.5142和a = 3.5454。後經評估發現，轉換冪次介於[3.4, 3.77]時，其將指數分配轉換為近似常態分配的效果，彼此之間並沒有顯著的差異。然而，本研究也發現當指數分配具有位置參數(location parameter)時，不論使用的是冪次轉換或其他轉換方式，皆應先消除位置參數的影響，才能提升轉換的效果，否則轉換出的分配將明顯偏離常態。本論文進一步將研究成果推廣應用到伽瑪(gamma)分配型態的資料，以轉換後近似分配與理論分配的絕對差總和為最小為評估基準，獲得在不同樣本數下的轉換冪次係數，成功地將非常態分配的製程變異數轉換成近似常態分配，以利於統計品管的應用。

This dissertation presents the two methods, minimizing the sum of the absolute differences and minimizing the sum of the squared differences, to transform the exponential data to normality. a = 3.5142 and a = 3.5454 are obtained based on the two methods, and hence they can be applied in transforming an exponential distribution for statistical process control (SPC). This interval is [3.4, 3.77], which implies that exponents falling in this interval have very similar results in transforming the exponentials. This dissertation also presents transformation exponents to transform the gamma data by minimizing the sum of the absolute differences between these two distinct cumulative probability functions in SPC applications. The individual charts plotted using the transformed data by the proposed method are superior to those obtained using the original exponential data, the data and charts using the probability control limits, in terms of better performance, appearance and ease of interpretation and implementation by practitioners. Results of this dissertation demonstrate that the location parameter of an exponential distribution has a great effect on normalizing the exponential, whether it is the power transformation or natural logarithm transformation method. As long as a location parameter exists, the transformed data will deviate from normality.

Table on contents
摘要 ……………………………………………………………………………….......Ⅰ
Abstract …………………………………………………………………………Ⅱ
Table on contents ……………………………………………………………Ⅲ
List of Figure ……………………………………………………………………Ⅴ
List of Table ……………………………………………………………………..Ⅶ
Chapter 1 Introduction ……………………………………………………1
1.1	Background  …………………………………………………………………1
1.2	Literature Review .……………………………………………………………2
1.2.1 Transforming Exponential Characteristics …………………..........…2
1.2.2 Transforming Variance .……………….........………………….……...3
1.3	Scope and Purpose  …………………………………………………………3
Chapter 2 Monitoring a Process of Exponentially Distributed Characteristics through Minimizing the Sum of the Squared Differences………………………….6
2.1 Introduction ………………………………………………………………….6
2.2 Optimal Exponent for Transformation ……………………………………..6
2.3 Operating Characteristic Curves …….……………………………………..9
2.4 The Interval for the Exponent of Transformation ………………….…....12
2.5 SPC Applications …………………………………………………………..14
2.6 Examples …………………………………………………………………...16
Chapter 3 Transforming the Exponential by Minimizing Sum of the Absolute Differences……………………………… ………………………21
3.1 Introduction ………………………………………………………………….21
3.2 Determining the Optimal Value of a ……………………………………..21
3.3 Comparisons …………………………………………………………………23
3.4 SPC Applications ………………………………………………………….27
Chapter 4 Process Monitoring of the Sample Variances through an Optimal Normalizing Transformation …………………………………………………. …. ….….34
4.1 Introduction ………………………………………………………………….34
4.2 Transformation Coefficients ……………………………………………...34
4.3 OC Curves …………………………………………………………………43
4.4 SPC Applications ………………………………………………………….49
4.5 Examples …………………………………………………………………….50
Chapter 5 Notes on Normalizing the Exponential …………...54
5.1 Introduction ………………………………………………………………….54
5.2 Derivations of Probability Density Functions ……………………………54
5.3 Results Analyses ……………………………………………………………..57
5.4 Examples …………………………………………………………………….64
Chapter 6 Conclusions and Future Research Suggestions …………………………69
6.1 Conclusions …………………………….…..……………………………….69
6.2 Future Research Suggestions ……………………………………………71
References ……………………………………………………………………72

                                

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