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| 研究生： |
林郁嘉 Lin, Yu-Chia |
|---|---|
| 論文名稱： |
監控多元品質變數比之EWMA變異數管制圖 EWMA Variability Control Charts for Monitoring Ratios of Multivariate Quality Variables |
| 指導教授： |
楊素芬
Yang, Su-Fen |
| 口試委員: |
葉金田
Yeh, Jin-Tyan 蕭又新 Shiau, Yuo-Hsien |
| 學位類別： |
碩士
Master |
| 系所名稱： |
商學院 - 統計學系 Department of Statistics |
| 論文出版年： | 2024 |
| 畢業學年度： | 112 |
| 語文別： | 中文 |
| 論文頁數： | 195 |
| 中文關鍵詞： | 多元相依變數比 、統計過程管制 、資料深度 、管制圖 、指數加權移動平均 、符號檢定 |
| 外文關鍵詞： | Ratios of multiple correlated variables, Sign method, Quality variable |
| 相關次數： | 點閱：24 下載：0 |
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近年來，維持和改進產品品質的重要議題，也是多元統計過程管制（MSPC）之重要研究領域。但是，在追蹤多變量品質變數比的分散程度的議題文獻中尚少探討。因此，我們提出了無分配假設下的管制圖來追蹤多個相依品質變數比之變異數以了解製程在穩定或失控的狀況。
在本研究中，我們分別提出兩種無分佈假設的方法，以建立兩個不同的EWMA管制圖。第一種我們提出使用資料深度 (data depth) 方法來建立EWMAD管制圖來監控多個相依變數比之變異數，第二種我們考慮符號方法 (sign method)，定義指標變數分配以建立管制圖來監控多個相依變數比之變異數。我們分別再以數值分析評估母體在三元常態和伽瑪分配下，兩個管制圖的管制界線與失控下的偵測效率。最後，我們透過實際多元牛奶數據來展示它們的應用和驗證失控的偵測能力。
In recent years, the variability of ratios of multivariate quality variables has become a crucial study of Multivariate Statistical Process Control (MSPC). However, there has been little exploration for the variability of ratios of dependent quality variables in the MSPC literature. Therefore, we propose control charts considering the distribution free assumption to track the variance stability and instability of the ratios of multiple dependent quality variables.
In this study, we propose two methods of establishing EWMA control charts without the specified distribution of multivariate quality variables. Firstly, we introduce the ZEWMAD control chart using data depth methods monitor the variance of ratios of multiple dependent quality variables. Secondly, we consider the sign method defining indicator variable distribution to establish a control chart for monitoring the variance of ratios of multiple dependent quality variables. We calculate the control limits of the proposed charts and evaluate their out-of-control detection ability considering three-dimensional normal and gamma distributions through numerical analysis. Finally, we demonstrate the out-of-control detection performance of these proposed charts using a real multivariate milk data.
Chapter 1. Introduction 18
1.1 Literature review 18
1.2 Study motivation and proposed research methods 21
Chapter 2. The In-control Distributions of Estimators for Monitoring the 22
Variances of Ratios of Multivariate Quality Variables 22
2.1 The distribution of estimator for monitoring the variances of ratios of 22
multivariate quality variables 22
2.2 The procedure to generate data from a multivariate gamma distribution 22
2.2.1 The method to generate data from a multivariate gamma distribution 22
2.2.2 The pdf and the moments of ratio of multivariate gamma distribution 23
2.2.3 Nine combinations of parameters for the trivariate gamma distribution 25
2.3 The procedure to generate data from a multivariate normal distribution 34
2.3.1 The method to generate data from a multivariate normal distribution 34
2.3.2 Approximated formula for ratios of the multivariate quality variables 36
2.3.3 Different combinations of parameters of the in-control multivariate normal distribution 37
Chapter 3. Data-depth Based EWMA Variability Control Chart for Monitoring the Ratios of Multivariate Quality Variables 39
3.1 Design of the new ZEWMAD chart for monitoring process dispersion of ratios 39
3.2 The procedure to determine the control limits and average run lengths of the ZEWMAD chart 41
Chapter 4. The Sign Based ZEWMA-SN Chart to Monitor Process Dispersion of the Ratios of Multivariate Variables 43
4.1 Design of the sign based ZEWMA-SN chart for monitoring process dispersion of ratios 43
4.2 The procedure to determine the control limits and average run lengths of the ZEWMA-SN chart 44
Chapter 5. The Out-of-control Distributions of the Different Estimators of the Variances of Ratios under Multivariate Quality Variables 47
5.1 The procedure to determine the unknown parameters for an out-of-control multivariate gamma distribution 47
5.1.1 The procedure to find the out-of-control parameters when only one of the parameters (\mathbit{\alpha}\mathbf{1}, \mathbit{\alpha}\mathbf{2}, \mathbit{\alpha}\mathbf{3}) changes 48
5.2 The procedure to determine the unknown parameters for an out-of-control multivariate normal distribution 55
5.2.1 The procedure to determine the unknown parameters in an out-of-control multivariate normal distribution when only change the first variance of ratios 55
5.2.2 The procedure to determine the unknown parameters in an out-of-control multivariate normal distribution when only change the second variance of ratios 80
Chapter 6. Out-of-Control Detection Performance Comparisons of the Proposed EWMA Variances of Ratios Control Charts 97
6.1 Detection performance comparison of the proposed ZEWMAD chart under 2 different multivariate distributions 97
6.1.1 The procedure to determine the detection performance of the ZEWMAD chart 97
6.1.2 The detection performance of the proposed ZEWMAD chart for a multivariate gamma distribution 100
6.1.3 The detection performance of the proposed ZEWMAD chart for a multivariate normal distribution 109
6.2 Detection performance comparison of the proposed ZEWMA-SN chart for the two different multivariate distributions 129
6.2.1 The procedure to determine the detection performance of the ZEWMA-SN chart 129
6.2.2 The detection performance of the proposed ZEWMA-SN chart for a multivariate gamma distribution 129
6.2.3 The detection performance of the proposed ZEWMA-SN chart for a multivariate normal Distribution 138
6.3 The summary of the calculated parameters of the proposed ZEWMA-SN chart for a multivariate gamma distribution 158
6.4 The summary of the calculated parameters of the proposed ZEWMA-SN chart for a multivariate normal distribution 162
6.4.1 The summary of the calculated parameters of the proposed ZEWMA-SN chart when the first variance of ratio changes for a multivariate normal distribution 162
6.4.2 The summary of the calculated parameters of the proposed ZEWMA-SN chart when the second variance of ratio changes for a multivariate normal distribution 168
Chapter 7. Performance Comparison Between the ZEWMAD and ZEWMA-SN Charts 174
7.1 Performance comparison between the ZEWMAD and ZEWMA-SN Charts with the different multivariate distributions 174
7.2 Performance comparison between the two ZEWMAD charts with the different multivariate distributions 179
7.3 Performance comparison between the two ZEWMA-SN charts with the different multivariate distributions 180
Chapter 8. Application of the ZEWMAD and ZEWMA-SN Charts Using a Multivariate Milk Data Set 182
Chapter 9. Summary 192
References 193
List of Tables
Table 2 - 1 The mean vectors, covariance matrices and skewnesses in combinations 1~3 of the trivariate gamma distribution 25
Table 2 - 2 The mean vectors, covariance matrices and skewnesses in combinations 4~6 of the trivariate gamma distribution 26
Table 2 - 3 The mean vectors, covariance matrices and skewnesses in combinations 7~9 of the trivariate gamma distribution 27
Table 2 - 4 Two combinations of parameters of mean vector and covariance matrices of trivariate distribution and ratios 35
Table 3 - 1 The values of LCL, ARL0, MRL0, SDRL0 for the ZEWMAD chart. 42
Table 4- 1 The values of UCL, LCL, ARL0, MRL0 and SDRL0 under different \lambda and nominal ARL0 for ZEWMA-SN chart 46
Table 5 - 1 Out-of-control variance of R2 based on different \delta2 in the first combination of parameters 49
Table 5 - 2 Out-of-control covariance of ratios based on different \delta1,\delta2 in the first combination of parameters 49
Table 5 -3 \alpha1\ast values based on different \delta2\ in the first combination of parameters 50
Table 5 -4 \alpha2\ast values based on different \delta2\ in the first combination of parameters 51
Table 5 - 5 \alpha3\ast values based on different \delta2\ in the first combination of parameters 52
Table 5 - 6 The out-of-control variance of R2 based on different \delta2 in the second combination of parameters 52
Table 5 -7 Out-of-control covariance of ratios based on different \delta1,\delta2 in the second combination of parameters 53
Table 5 - 8 \alpha1\ast values based on different \delta2\ in the second combination of parameters 54
Table 5 - 9 \alpha2\ast values based on different \delta2\ in the second combination of parameters 54
Table 5 - 10 \alpha3\ast values based on different \delta2\ in the second combination of parameters 55
Table 5 - 11 Variance of R1 under different scale parameters in the first combination of parameters under a trivariate normal distribution 58
Table 5 - 12 \sigma X1\ast values based on different \delta1\ in the first combination of parameters under a trivariate normal distribution 60
Table 5 - 13 \sigma X1\ast2, \sigma X1,X2\ast and \sigma X1,X3\ast values based on different \delta1\ in the first combination of parameters under a trivariate normal distribution 61
Table 5 - 14 Variance of R1 under different scale parameter in the second combination of parameters under a trivariate normal distribution 62
Table 5 - 15 \sigma X1\ast values based on different \delta1\ in the second combination of parameters under a trivariate normal distribution 63
Table 5 - 16 \sigma X1\ast2, \sigma X1,X2\ast and \sigma X1,X3\ast values based on different \delta1\ in the second combination of parameters under a trivariate normal distribution 64
Table 5 -17 \sigma X3\ast values based on different \delta1\ in the first combination of parameters under a trivariate normal distribution 67
Table 5 - 18 \sigma X3\ast, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta1\ in the first combination of parameters under a trivariate normal distribution 68
Table 5 - 19 \sigma X3\ast values based on different \delta1\ in the second combination of parameters under a trivariate normal distribution 69
Table 5 - 20 \sigma X3\ast, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta1\ in the second combination of parameters under a trivariate normal distribution 70
Table 5 - 21 \sigma X1\ast and \sigma X3\ast values based on different \delta1\ in the first combination of parameters under a trivariate normal distribution 75
Table 5 - 22 \sigma X1\ast2, \sigma X3\ast2,\sigma X1,X2\ast, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta1\ in the first combination of parameters under a trivariate normal distribution 76
Table 5 - 23 \sigma X1\ast and \sigma X3\ast values based on different \delta1\ in the second combination of parameters under a trivariate normal distribution 78
Table 5 - 24 \sigma X1\ast2, \sigma X3\ast2,\sigma X1,X2\ast, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta1\ in the second combination of parameters under a trivariate normal distribution 79
Table 5 - 25 Variance of R2 under different scale parameter in the first combination of parameters under a trivariate normal distribution 81
Table 5 - 26 \sigma X2\ast values based on different \delta2\ in the first combination of parameters under a trivariate normal distribution 81
Table 5 - 27 \sigma X1\ast2, \sigma X1,X2\ast and \sigma X2,X3\ast values based on different \delta2\ in the first combination of parameters under a trivariate normal distribution 82
Table 5 -28 Variance of R2 under different scale parameter in the second combination of parameters under a trivariate normal distribution 84
Table 5 - 29 \sigma X2\ast values based on different \delta2\ in the second combination of parameters under a trivariate normal distribution 84
Table 5 -30 \sigma X2\ast2, \sigma X1,X2\ast, \sigma X2,X3\ast values based on different \delta2\ in the second combination of parameters under a trivariate normal distribution 85
Table 5 - 31 \sigma X3\ast values based on different \delta2\ in the first combination of parameters under a trivariate normal distribution 87
Table 5 - 32 \sigma X3\ast2, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta2\ in the first combination of parameters under a trivariate normal distribution 88
Table 5 - 33 \sigma X3\ast values based on different \delta2\ in the second combination of parameters under a trivariate normal distribution 89
Table 5 - 34 \sigma X3\ast2, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta2\ in the second combination of parameters under a trivariate normal distribution 90
Table 5 - 35 \sigma X2\ast, \sigma X3\ast values based on different \delta2\ in the first combination of parameters under a trivariate normal distribution 91
Table 5 - 36 \sigma X2\ast2, \sigma X3\ast2,\ \ \sigma X1,X2\ast, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta2\ in the first combination of parameters under a trivariate normal distribution 93
Table 5 - 37 \sigma X2\ast, \sigma X3\ast values based on different \delta2\ in the second combination of parameters under a trivariate normal distribution 94
Table 5 - 38 \sigma X2\ast2, \sigma X3\ast2,\ \ \sigma X1,X2\ast, \sigma X1,X3\ast and \sigma X2,X3\ast values based on different \delta2\ in the second combination of parameters under a trivariate normal distribution 96
Tables 6 - 1 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 200 under the first combination of parameters for a trivariate gamma distribution 101
Tables 6 - 2 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 200 under the second combination of parameters for a trivariate gamma distribution 103
Tables 6 - 3 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 370.4 under the first combination of parameters for a trivariate gamma distribution 105
Tables 6 - 4 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 370.4 under the second combination of parameters for a trivariate gamma distribution 107
Tables 6 - 5 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 200 when the first variance of ratio changes under the first combination of parameters for a trivariate normal distribution 110
Tables 6 - 6 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 370.4 when the first variance of ratio changes under the first combination of parameters for a trivariate normal distribution 113
Tables 6 - 7 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 200 when the first variance of ratio changes under the second combination of parameters for a trivariate normal distribution 116
Tables 6 - 8 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 370.4 when the first variance of ratio changes under the second combination of parameters for a trivariate normal distribution 118
Tables 6 - 9 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 200 when the second variance of ratio changes under the first combination of parameters for a trivariate normal distribution 121
Tables 6 - 10 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 370.4 when the second variance of ratio changes under the first combination of parameters for a trivariate normal distribution 123
Tables 6 - 11 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 200 when the second variance of ratio changes under the second combination of parameters for a trivariate normal distribution 125
Tables 6 - 12 ARL1s, MRL1s, SDRL1s of the proposed ZEWMAD chart with nominal ARL0 = 370.4 when the second variance of ratio changes under the second combination of parameters for a trivariate normal distribution 127
Tables 6 - 13 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 200 under the first combination of parameters for a trivariate gamma distribution 130
Tables 6 - 14 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 370.4 under the first combination of parameters for a trivariate gamma distribution 132
Tables 6 - 15 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 200 under the second combination of parameters for a trivariate gamma distribution 134
Tables 6 - 16 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 370.4 under the second combination of parameters for a trivariate gamma distribution 136
Tables 6 - 17 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN charts with nominal ARL0 = 200 when the first variance of ratio changes under the first combination of parameters for a trivariate normal distribution 139
Tables 6 - 18 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 370.4 when the first variance of ratio changes under the first combination of parameters for a trivariate normal distribution 142
Tables 6 - 19 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 200 when the first variance of ratio changes under the second combination of parameters for a trivariate normal distribution 145
Tables 6 - 20 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 370.4 when the first variance of ratio changes under the second combination of parameters for a trivariate normal distribution 147
Tables 6 - 21 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 200 when the second variance of ratio changes under the first combination of parameters for a trivariate normal distribution 150
Tables 6 - 22 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 370.4 when the second variance of ratio changes under the first combination of parameters for a trivariate normal distribution 152
Tables 6 - 23 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 200 when the second variance of ratio changes under the second combination of parameters for a trivariate normal distribution 154
Tables 6 - 24 ARL1s, MRL1s, SDRL1s of the proposed ZEWMA-SN chart with nominal ARL0 = 370.4 when the second variance of ratio changes under the second combination of parameters for a trivariate normal distribution 156
Table 6 - 25 \delta1,\ \ \sigma R1\ast2,\ \ p1 and \alpha i\ast, i = 1, 2, 3 of the proposed ZEWMA-SN chart under the first combination of parameters for a trivariate gamma distribution 159
Table 6 - 26 \delta1,\ \ \sigma R1\ast2,\ \ p1 and \alpha i\ast, i = 1, 2, 3 of the proposed ZEWMA-SN chart under the second combination of parameters for a trivariate gamma distribution 160
Table 6 - 27 The relationship of \delta2 with calculated parameters in the multivariate gamma distribution 161
Tables 6 - 28 \delta2,\ \ \sigma R2\ast2,\ \ p1 and covariance matrices of the proposed ZEWMA-SN chart when the first variance of ratio changes under the first combination of parameters for a trivariate normal distribution 163
Tables 6 - 29 \delta2,\ \ \sigma R2\ast2,\ \ p1 and covariance matrixes of the proposed ZEWMA-SN chart when the first variance of ratio changes under the second combination of parameters for a trivariate normal distribution 165
Table 6 - 30 The relationship of \delta1 with calculated parameters in the multivariate normal distribution 167
Table 6 - 31 \delta2,\ \ \sigma R2\ast2,\ \ p1 and covariance matrices of the proposed ZEWMA-SN chart when the second variance of ratio changes under the first combination of parameters for a trivariate normal distribution 169
Table 6 - 32 \delta1,\ \ \sigma R1\ast2,\ \ p1 and covariance matrices of the proposed ZEWMA-SN chart when the second variance of ratio changes under the second combination of parameters for a trivariate normal distribution 171
Table 6 - 33 The relationship of \delta2 with calculated parameters in the multivariate normal distribution 173
Table7- 1 ARL1 comparison for the proposed charts for a trivariate gamma distribution 175
Table 7- 2 ARL1 comparison for the proposed charts when \delta1 changes for a trivariate normal distribution 177
Table7- 3 ARL1 comparison for the proposed charts when \delta2 changes for a trivariate normal distribution 178
Table7- 4 ARL1 comparison for ZEWMAD chart with various \delta2 under a trivariate distribution 179
Table 7- 5 ARL1 comparison for ZEWMA-SN chart with various \delta2 under a trivariate distribution 180
Table 8- 1 The control group of the multivariate milk data 183
Table 8- 2 The treatment group of the multivariate milk data 184
Table 8- 3 The three normality tests for the in-control data of milk components 185
Table 8- 4 The three normality tests for the out-of-control data of milk components 185
List of Figures
FIGURE 1 SCATTER PLOT OF DIFFERENT PARAMETERS COMBINATION FOR 3-DIMENSIONAL GAMMA DISTRIBUTION AND 2-DIMENSIONAL RATIOS 28
FIGURE 2 SCATTER PLOT OF DIFFERENT PARAMETERS COMBINATION FOR 3-DIMENSIONAL GAMMA DISTRIBUTION AND 2-DIMENSIONAL RATIOS (CONTINUE THE FIGURE 1) 29
FIGURE 3 SCATTER PLOT OF DIFFERENT PARAMETERS COMBINATION FOR 3-DIMENSIONAL GAMMA DISTRIBUTION AND 2-DIMENSIONAL RATIOS (CONTINUE THE FIGURE 2) 30
FIGURE 4 THE IC MILK CHARTING STATISTICS ON THE ZEWMAD CHART 188
FIGURE 5 THE OC MILK CHARTING STATISTICS ON THE ZEWMAD CHART 189
FIGURE 6 THE IC MILK CHARTING STATISTICS ON THE ZEWMA-SN CHART 190
FIGURE 7 THE OC MILK CHARTING STATISTICS FOR THE ZEWMA-SN CHART 191
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全文公開日期 2029/08/27