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研究生：	錢柏維 Chien, Bo-Wei
論文名稱：	各種指數加權移動平均多變量管制圖偵測能力之比較研究 A Comparative Study on the Detecting Performance for Various EWMA-type Multivariate Control Charts
指導教授：	潘浙楠 Pan, Jeh-Nan
學位類別：	碩士 Master
系所名稱：	管理學院 - 統計學系 Department of Statistics
論文出版年：	2013
畢業學年度：	101
語文別：	英文
論文頁數：	47
中文關鍵詞：	同時監控平均與變異指數加權移動平均多變量管制圖、指數加權移動平均、平均連串長度
外文關鍵詞：	EWMA-type multivariate control chart for simultaneously monitoring mean and variability, Multivariate exponentially weighted moving average (MEWMA), Average run length (ARL)
相關次數：	點閱：162 下載：0
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在實際使用管制圖監控製程時，通常呈管制狀態下的製程參數為未知，因此在試用第一階段欲取得穩定狀態下之資料以估計製程參數，並以此估計量做為製程參數。由於多變量管制圖在監控製程之第二階段會受到第一階段樣本數大小的影響，故已有多位學者探討第一階段下多變量管制圖樣本數大小之決定與管制界線之制定。然而前大多數文獻中只著重於偵測平均數研究，尚未有學者針對各種同時監控平均與變異之指數加權移動平均(EWMA)之多變量管制圖進行比較研究。由於大多數同時監控平均與變異之指數加權移動平均多變量管制圖均建立在製程參數已知下。因此，本研究乃針對Chen et al. (2005)所提出的Max-MEWMA管制圖之缺點進行修正，並提出參數未知時，以統計估計量修正Max-MEWMA管制圖的RMax-MEWMA管制圖，再與其他學者所提出之指數加權移動平均多變量管制圖進行比較與分析。藉由模擬結果證實，當第一階段樣本的組數(m)較小時，本研究所提出之修正後的多變量(RMax-MEWMA)管制圖其平均連串長度在穩定與失控狀態下在大多數情況均優於現有的管制圖。

When implementing a multivariate control chart, in-control parameters of the process are usually unknown in practice. Thus, we need to estimate in-control parameters using in-control data set from Phase I analysis and replace these parameters with their estimates. Moreover, the detecting performance of multivariate control charts in Phase II is affected by number of samples in Phase I. Several authors have dealt with this problem for process mean, but most previous studies did not compare the effect of parameter estimates on the detecting performance of various EWMA-type multivariate control charts in Phase II for simultaneously monitoring mean and variability, and the test statistics of these EWMA-type multivariate control charts are established when parameters are known. In this paper, we propose a RMax-MEWMA control chart in which the unknown in-control parameters are replaced by their estimates based on Chen’s Max-MEWMA control chart. Then, the detecting performances of our RMax-MEWMA and other EWMA-type multivariate control charts are compared. The simulation results show that RMax-MEWMA chart outperforms the other EWMA-type multivariate control charts in terms of in-control and out-of-control ARLs when number of samples is small.

Contents
1.	Introduction: Motivation and Research Objectives	1
2.	Mathematical notations and definitions	3
3.	Review of the pertinent multivariate control charts	5
3.1.	Combined EWMA M-chart and EWMA V-chart	6
3.2.	Max-EWMA chart	8
3.3.	Combined   chart and  chart	11
3.4.	ELR control chart	13
4.	Development of RMax-MEWMA Charts	16
5.	Comparison of the detecting performance for RMax-MEWMA and Other 
Multivariate Control Charts	20
5.1.	Simulation procedure	20
5.2.	In-control performance	23
5.3.	Determining the minimum number of samples	31
5.4.	Out-of-control performance	36
6.	Conclusions and Future works	43
References	45

List of Tables
Table 1. A summary of the characteristics for various multivariate EWMA control charts	15
Table 2. The upper control limit value that produces an overall in-control ARL of 200 for various control charts under different p = 2, n=5 and λ=0.1, 0.2 when number of samples approaches infinity.	21
Table 3. The upper control limit value that produces an overall in-control ARL of 200 for various control charts under different p (p = 3, 4), n=5 and λ=0.1, 0.2  when number of samples approaches infinity.	22
Table 4. The simulation results of in-control ARL and SDRL for the various multivariate control charts with p=2, n=5,  λ=0.1.	25
Table 5. The simulation results of in-control ARL and SDRL for the various multivariate control charts with p=3, n=5, λ=0.1.	26
Table 6. The simulation results of in-control ARL and SDRL for the various multivariate control charts with p=4, n=5,  λ=0.1.	27
Table 7. The simulation results of in-control ARL and SDRL for the various multivariate control charts with p=2, n=5,  λ=0.2.	28
Table 8. The simulation results of in-control ARL and SDRL for the various multivariate control charts with p=3, n=5,  λ=0.2.	29
Table 9. The simulation results of in-control ARL and SDRL for the various multivariate control charts with p=4, n=5,  λ=0.2.	30
Table 10. Minimum numbers of samples required for various multivariate control charts under p=2	35
Table 11. Minimum numbers of samples required for various multivariate control charts under p=3, 4	35
Table 12. The simulation results of out-of-control ARL for the various multivariate control charts under m=300, p=2, n=5,λ=0.1 given the in-control ARL=200 when number of samples approaches infinity	37
Table 13. The simulation results of out-of-control ARL for the various multivariate control charts under m=300, p=2, n=5,λ=0.2 given the in-control ARL=200 when number of samples approaches infinity	38
Table 14. The simulation results of out-of-control ARL for the various multivariate control charts under m=300, p=3, n=5,λ=0.1 given the in-control ARL=200 when number of samples approaches infinity	40
Table 15. The simulation results of out-of-control ARL for the various multivariate control charts under m=300, p=3, n=5,λ=0.2 given the in-control ARL=200 when number of samples approaches infinity	41


 
List of Figures
Figure 1. The relationship between in-control ARLs and different numbers of samples (m) for various multivariate control charts under p=2, n=5, .	31
Figure 2. The relationship between in-control ARLs and different numbers of samples (m) for various multivariate control charts under p=3, n=5, .	32
Figure 3. The relationship between in-control ARLs and different numbers of samples (m) for various multivariate control charts based on p=4, n=5, .	32
Figure 4. The relationship between in-control ARLs and different numbers of samples (m) for various multivariate control charts under p=2, n=5, .	33
Figure 5. The relationship between in-control ARLs and different numbers of samples (m) for various multivariate control charts under p=3, n=5, .	33
Figure 6. The relationship between in-control ARLs and different numbers of samples (m) for various multivariate control charts under p=4, n=5, .	34



                                    

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