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研究生： 龍庭軒
Ting-Hsuan Long
論文名稱： A control chart based on copula-based Markov time series models
指導教授： 江村剛志
Takeshi Emura
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 統計研究所
Graduate Institute of Statistics
論文出版年： 2013
畢業學年度： 101
語文別： 英文
論文頁數： 58
中文關鍵詞： 平均串聯長度 、Copula 、相關性資料 、馬可夫鏈 、降低變異數
外文關鍵詞： Average run length, Copula, correlated data, Markov chain, variance reduction
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    • 不論在工業方面或是商業方面而言，統計製程管制(Statistical process control)皆是一個非常重要的品質管制工具。傳統的Shewhart管制圖僅是運用在獨立性的假設之下。然而，在現實生活中，存在著許多相關性假設的資料，因此，傳統的管制圖在現實生活中是不被接受且不實用的。在本文，我們主要的目的是以建立在copula之下的馬可夫鏈模型去衍伸我們的相關性假設資料。此外，我們提出了最大概似估計量的方法估計我們的未知參數，分別為管制上限(UCL)以及管制下限(LCL)。接著，我們使用蒙地卡羅模擬法做出平均串聯長度(Average run length)以用來表現管制圖的性質。最後，我們提出了降低變異數的方法去增加資料的準確性。

    Statistical process control is an important and convenient tool for business and industry. The traditional Shewhart control chart has been a popular tool for process control, which however is valid under the independence assumption of consecutive observations. In real world applications, there exist many types of dependent observations in which the traditional control charts cannot be used. In this paper, we apply a copula-based Markov chain to model the correlated observations. In particular, we proposed a maximum likelihood method to obtain the estimates of upper control limit (UCL) and lower control limit (LCL). It is shown by simulations that the proposed method provide more accurate estimates of the UCL and LCL than the existing procedure and traditional procedure. We also consider Monte Carlo simulations to compute the value of the average run length (ARL) of the proposed charts. Here, we suggest a variance reduction technique, called antithetic variables method to gain computational efficiency. Two datasets are analyzed for illustration.

    摘要 i Abstract ii 致謝詞 iii List of Figures vii List of Tables viii Chapter 1 Introduction 1 Chapter 2 Background 4 Chapter 3 Estimation of process parameters 7 3.1. Chen and Fan’s method 8 3.2. Joe’s method (Proposed method) 9 3.2.1. Likelihood inference 9 3.2.2. Computational algorithm (Newton-Raphson) 10 3.3. Simulation result 12 3.3.1. Simulation method 12 3.3.2. The result for MSE 13 Chapter 4 Control chart based on Copula 19 4.1. 3 -Control Chart 19 4.2. Average Run Length Calculation 20 4.2.1. Monte Carlo and Data generation 20 4.2.2. Variance Reduction 21 4.3. Simulation Result 22 Chapter 5 Applications 26 5.1. Data Background 26 5.2. Numerical Result 26 Chapter 6 Conclusion and Discussion 35 Appendix A 37 A.1 Log-copula density 37 A.2 Likelihood function and its derivatives 37 A.3 R Codes for Joe's Method 40 A.3.1 Description 40 A.3.2 Usage 40 A.3.3 Arguments 40 A.3.4 Definition and example 40 Reference 47

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