建立管制區域來同時監控製程的位置與離散 - 政大學術集成
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(2) 謝辭 在兩年的研究所生涯裡，不管是在知識或生活上皆有許多的成長。最終，要 與大家分享兩年來的成果與喜悅，這一切都要感謝許多人對我的提攜與幫助。 論文能夠順利完成，首先要感謝我的指導老師 楊素芬教授。老師對於我的 論文指導及修改花費了許多心思和時間，同時也在老師辛苦的訓練下，讓我對統 計品管的知識和應用更加精進。在這段學習過程中，也磨練出我對於任何事都要 保持積極且負責任的態度去處理。 再來要感謝我的口試委員 黃榮臣教授、曾勝滄教授與唐正教授，三位老師 在口試時給於我建議和指導，使得本論文能盡善盡美。. 政 治 大. 也要感謝我的同學與朋友，幫我解決了一些事務和課業的問題，也一起渡過. 立. 兩年的碩士生活。. ‧ 國. 學. 最後，要感謝我的家人，有你們在背後支持與鼓勵，讓我能夠專心完成學業。 溫暖的家始終是我的避風港。. ‧. 本研究承蒙科技部補助，計畫編號 NSC 102-2118-M-004 -005 -MY2，謹此. n. al. er. io. sit. y. Nat. 致謝。. Ch. engchi. i n U. v. 胡逢升. 謹致. 中華民國一百零四年七月 ii.
(3) Abstract Control charts are effective tools for monitoring quality of manufacturing processes and service processes. Nowadays, much of the data in service or manufacturing industries comes from processes having non-normal distributions or unknown distributions. The commonly used Shewhart mean and variable control charts, which depend heavily on the normality assumption, are not appropriately used here. In this article, we propose a distribution-free control region of the median and IQR using the kernel density estimation methods– to simultaneously monitor the location and dispersion of an unknown underlying continuous distribution.. 政 治 大. Furthermore, the average run lengths (ARL) of the proposed control region is used to. 立. measure the out-of-control detection performance. The performance of the proposed. ‧ 國. 學. control region and some other non-parametric charts for detecting out-of-control location and scale are compared. The proposed control region of the median and IQR. ‧. shows as well or better detection performance compared to existing non-parametric. y. Nat. io. sit. control charts that can simultaneously monitor the location and scale. Numerical. n. al. er. examples illustrate the application of the proposed control region. Summary and conclusions are offered.. Ch. engchi. iii. i n U. v.
(4) Contents 1. Introduction .............................................................................................................. 1 2. Determination of the True Control Region of the Median and IQR for the Quality Variable with a Specified Distribution ..................................................... 3 2.1 Design of a median control chart .................................................................. 3 2.2 Design of an IQR control chart..................................................................... 4 2.3 Design of the control region of the median and IQR .................................. 5 2.3.1 Determination of the control region of the median and IQR using the joint CDF of the median and IQR ...................................................... 6 2.3.2 Determination of the control region of the median and IQR using joint pdf of the median and IQR................................................................. 7. 政 治 大. 2.4 Performance measurement of the control region of the median and IQR .............................................................................................................................. 11 3. Determination of the Control Region of the Median and IQR for the Quality Variable with Unknown Distribution – the Kernel Density Estimation Method .................................................................................................................................. 15 3.1 Using the kernel density estimation method to estimate the pdf of the distribution-free process data ................................................................... 15. 立. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. 3.2 Determination of the approximated control region using the kernel density estimation method ......................................................................... 17 3.2.1 Determining the control region using the one-dimensional kernel density estimation method ............................................................... 17 3.2.2 Determining the control region using the two-dimensional kernel density estimation method ............................................................... 19. Ch. engchi. i n U. v. 3.3 Performance measurement of the control region of the median and IQR .............................................................................................................................. 19 4. Design the K Control Chart for Monitoring Joint Location and Dispersion ... 29 4.1 Design the K control chart using the one-dimensional kernel density estimation method ...................................................................................... 29 4.2 Design the K control chart using the two-dimensional kernel density estimation method ...................................................................................... 30 5. Performance Comparison ..................................................................................... 30 6. Real Examples ........................................................................................................ 38 6.1 TSMC Company’s stock price data............................................................ 38 6.1.1 Control region of the median and IQR of TSMC company’s stock price using the kernel density estimation method ..................................... 41 6.1.2 The K control chart ............................................................................. 44 iv.
(5) 6.2 Service Times Data....................................................................................... 49 6.2.1 Control region of the median and IQR of service times data using the kernel density estimation method .................................................... 49 6.2.2 The K control chart ............................................................................. 53 6.2.3 Performance comparison of the proposed control charts .................... 55 7. Summary and Concluding Remarks .................................................................... 57 Reference .................................................................................................................... 58. 立. 政 治 大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. v. i n U. v.
(6) List of Tables Table 1. Density function, expectation and variance of a Gamma distribution ........... 12 Table 2.. of the true control region of the median and IQR for Gamma(2, ) with n=5 and. Table 3.. .................................................................................. 14. of the true control region of the median and IQR for Gamma(2, ) distribution with n=15 and. Table 4.. .................................................... 14. of the true control region of the median and IQR for Gamma(2, ) distribution with n=5 and. Table 5.. of the true control region of the median and IQR for Gamma(2, ) with n=15 and. 立. 治 政 ............................................................................. 15 大. of the three control regions of the median and IQR for data from. 學. ‧ 國. Table 6.. ...................................................... 15. Gamma(2, ) distribution with n=5 and Table 7.. ................................ 25. of the three control regions of the median and IQR for data from. ‧. Gamma(2, ) distribution with n=15 and. y. Nat. io. Gamma(2, ) distribution with n=5 and. al. n. Table 9.. sit. of the three control regions of the median and IQR for data from . ................................ 27. er. Table 8.. .............................. 26. i n U. v. of the three control regions of the median and IQR for data from. Ch. engchi. Gamma(2, ) distribution with n=15 and. .............................. 28. Table 10. Performance comparisons between the control region and SL chart for data from Normal. distribution with. ............................................. 33. Table 11. Performance comparisons between the control region and NLE chart for data from Normal. distribution ......................................................... 34. Table 12. Performance comparisons between the control region and SL chart for data from Laplace. distribution with. . ............................................ 35. Table 13. Performance comparisons between the control region and NLE chart for data from. with vi. (location shift) ................ 36.
(7) Table 14. Performance comparisons between the control region and NLE chart for data from. with. (scale shift) ......................... 36. Table 15. Performance comparisons between the control region and NLE chart for data from. distribution. (location shift)................ 37. Table 16. Performance comparisons between the control region and NLE chart for data from. distribution. (scale shift) ...................... 37. Table 17. The fitted time series model. ........................................................................ 39 Table 18. Test result of the Augmented Dickey-Fuller Test......................................... 41. 政 治 大 Table 20. The service times data from Yang and Arnold (2014) ................................. 47 立 Table 19. Contingency table for in-control residuals ................................................... 43. Table 21. The new service times data from Yang and Arnold (2014) .......................... 48. ‧ 國. 學. Table 22. Contingency Table for in-control service times data. .................................. 51. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. vii. i n U. v.
(8) List of Figures Figure 1. The true control region of the median and IQR using their CDF approach ... 7 Figure 2. The true control region of the median and IQR using their pdf approach. .. 11 Figure 3. True control region of the median and IQR using their pdf approach.......... 11 Figure 4. Density curve of Gamma (2, 2) distribution ................................................. 13 Figure 5. The true control region of median and IQR for Gamma (2, 2) distribution with. ............................................................................................. 13. Figure 6. The true control region of the median and IQR for Gamma (2, 2) = 0.0027......................................................................... 14. distribution with. 政 治 大. Figure 7. The control region of the median and IQR for data from Gamma (2, 2) distribution with. 立. ....................................................................... 21. ‧ 國. 學. Figure 8. The control regions of the median and IQR for data from Gamma (2, 2) = 0.0054......................................................................... 21. distribution with. ‧. Figure 9. The control region of the median and IQR for data from Normal (0, 1). y. Nat. . .................................................................... 22. io. sit. distribution with. n. al. er. Figure 10. The control regions of the median and IQR for data from Laplace (0, 1) distribution with. Ch. i n U. v. . ....................................................................... 22. engchi. Figure 11. The control regions of the median and IQR fro data from with. ......................................................................................... 23. Figure 12. The control region of the median and IQRfor data from with. distribution. distribution. ......................................................................................... 23. Figure 13. TSMC company’s stock price from 2009/01/05 to 2013/12/31 ................. 39 Figure 14. Time series analysis for price ..................................................................... 39 Figure 15. Time series analysis for differencing stock price. ...................................... 39 Figure 16. The kernel density (blue curve), and the histogram of in-control residuals .............................................................................................................................. 42 viii.
(9) Figure 17. The estimate kernel density of the in-control residuals data ...................... 43 Figure 18. The monitoring results of the control region with. using the. one-dimensional kernel density estimation method. .................................... 44 Figure 19. The monitoring results of the control region with. using the. two-dimensional kernel density estimation method. ................................... 44 Figure 20. The K control chart with. using the one-dimensional kernel. density estimation approach ......................................................................... 45 Figure 21. The K control chart with. using the two-dimensional kernel. 政 治 大 Figure 22. Control charts diagnose the out-of-control location and/or dispersion. ..... 46 立 density estimation approach ......................................................................... 46. Figure 23. The kernel density (blue curve), and the histogram of in-control service. ‧ 國. 學. times data. .................................................................................................... 51. ‧. Figure 24. The estimated kernel density of the in-control service times data. ............ 51 using the. sit. y. Nat. Figure 25. The monitoring results of the control region with. io. er. one-dimensional kernel density estimation method . ................................... 52 Figure 26. The monitoring results of the control region with. n. al. Ch. n U engchi. iv. using the. two-dimensional kernel density estimation method. ................................... 52 Figure 27. The K control chart with. using the one-dimension kernel. density estimation method. .......................................................................... 54 Figure 28. The K control chart with. using the two-dimensional kernel. density estimation method. .......................................................................... 54 Figure 29. Control charts diagnose the out-of-control location and/or dispersion. ..... 55 Figure 30. (1) EWMA-AV chart with in-control data (2) EWMA-AM chart with in-control data .............................................................................................. 56 Figure 31. (1) EWMA-AV chart with out-of-control data (2) EWMA-AM chart with out-of-control data ....................................................................................... 56 ix.
(10) 1. Introduction Since the earliest days of statistical process control, the mean and the variance of a process are the two most-often-monitored parameters, and some people use separate. and. charts:. one for the mean and one for the standard. deviation/scale (or dispersion). If either chart exceeds its control limit, then action will be taken to diagnose and resolve any special causes. Due to this problem, many authors have proposed the joint monitoring of the mean and the variance/scale with two charts, but this is sometimes inconvenient and time-consuming.. 政 治 大. Previous studies have set up new methods to employ a single chart or a single. 立. plotting statistic for simultaneously monitoring the mean and variance parameters of a. ‧ 國. 學. normally distributed process. They are useful for situations in which special causes can result in a change in both mean and the variance, and they can avoid the inflated false. ‧. alarm rate that results from simply using two independent control charts. McCracken. Nat. sit. y. and Chakraborti (2011) provided a review of these single charts. Some researchers have. n. al. er. io. developed joint monitoring schemes in which the process data are plotted on a. i n U. v. two-dimensional plane and are considered in-control if they fall within some defined. Ch. engchi. control region; otherwise, the process is declared out-of-control. Gan (1995) presented two combined joint monitoring schemes using EWMA charts with an elliptical control region. Chao and Cheng (2008) built a semi-circular control chart in which “the combined visual effect of points that fall in/out of a certain closed region is more striking than points just crossing the lines.” All the methods listed above are well, but present an important issue. Nowadays, much of the data in service or manufacturing industries come from processes having non-normal distributions or unknown distributions. Being not robust to non-normality is a commonly seen issue for many control charts for mean or variance (see Alwn (2000) 1.
(11) and Braun and Park (2008)). Many authors have recently proposed non-parametric or distribution-free control charts to monitor the location or variability of a process, with research having been done to deal with process location monitoring; see for example, Amin et al. (1995); Chakraborti et al. (2001); Altukife (2003); Chakraborti and Graham (2007); Chakraborti and van der Wiel (2008); Zou and Tsung (2010); Yang et al. (2011); Abbasi et al. (2013). A few research studies have looked at process variability monitoring; see for example, Maravelakis et al. (2005); Das and Bhattacharya (2008); Zou and Tsung. 政 治 大 distribution-free control charts that can simultaneously monitor the location and 立. (2010); Abbasi et al. (2011); Yang and Arnold (2014). However, there are far fewer. (2012); Chowdhury (2014); Zhou et al. (2014).. 學. ‧ 國. scale/variability; see for example, Zou and Tsung (2010); Mukherjee and Chakraborti. ‧. The previous non-parametric approaches are not easy for practitioners to apply,. sit. y. Nat. because they are not statisticians and do not quite understand the proper way to. io. er. implement the schemes. Therefore, in this article we propose a distribution-free control chart - the control region of the median and IQR (Interquartile range) - to. n. al. simultaneously monitor the. v i n C h and dispersionU of an location engchi. unknown underlying. continuous distribution. The monitoring statistics of this approach are quite easy to use and exhibit better performance under non-normally distributions than the SL chart (Mukherjee and Chakraborti (2012)) and NLE chart (Zou and Tsung (2010)). We assume that in-control process data are available, and then we construct the control region of median and IQR using the kernel density estimate method as our in-control region. At each monitoring, we calculate the median and IQR statistics (one for location and one for dispersion) as the plotting statistic. If the plotting statistic falls outside the control region, then an out-of-control signal is produced. Adding in an individual median chart and IQR chart for diagnosing whether the signal is due to a 2.
(12) shift in the location, or in the dispersion, or in both. On the other hand, if the plotting statistic does not fall outside the control region, then the process is in-control. The rest of this article is organized as follows. Section 2 introduces the construction of the true control region of the median and IQRfor a specified distribution. Section 3 sets up the control region of the median and IQR using the kernel density estimate method. Section 4 presents the K control chart. Section 5 offers a performance comparison among some existed joint location and scale control charts. Section 6 illustrates the charting procedure of the proposed control region and control chart with. 政 治 大 2014). Section 7 summarizes the findings. 立. two datasets from TSMC company’s stock price and service times data (Yang et al.. 學. ‧ 國. 2. Determination of the True Control Region of the Median and IQR for the Quality Variable with a Specified Distribution. ‧. 2.1 Design of a median control chart. y. Nat. represent the order statistics of a random sample. sit. Let. al. n. (CDF),. er. io. from a continuous population with cumulated distribution function. i n U. and the probability density function (pdf),. Ch. engchi. and Berger (2002), thepdf of the median of X is:. v. . According to Casella. ,. The CDF of the median of X is:. The control limits (UCL and LCL) for the median chart are obtained by solving:. 3.
(13) 2.2 Design of an IQR control chart Let. represent the order statistics of a random sample from a continuous population with CDF,. the. and pdf,. of X are. According to Casella and Berger (2002), the joint pdf of is. 立. and. ‧ y. sit. n. al. er. io. Therefore,. Nat. Thus,. ‧ 國. So,. 學. Let. 政 治 大 .. Ch. engchi. i n U. v. .. Then, the pdf of IQR,. is derived by .. The CDF of IQR,. is derived by .. The control limits (UCL and LCL) of IQR chart are obtained by letting, .. 4. . Thus,.
(14) When the process data is from a normal distribution, the sample median and IQR from the process are independent. Therefore, we use both the median chart and IQR chart to monitor the process location and dispersion simultaneously. For the median chart and the IQR chart, the respective probability of a type-I error is. when the. process is in-control. The both median chart and IQR chart has a total type-I error probability of. , which is equivalent to having an in-control ARL. of When the process is not from a normal distribution, the median and IQR of the. 政 治 大 monitor the process location and dispersion simultaneously; otherwise, it will increase 立 process are not independent. We cannot use both the median chart and IQR chart to. the occurrence of false alarms. Hence, a solution should be found.. ‧ 國. 學. 2.3 Design of the control region of the median and IQR. ‧. Because the median and IQR of the process are not independent when the process. sit. y. Nat. is not from a normal distribution. In this article we propose a control region of the. io. al. er. median and IQR to simultaneously monitor the process location and dispersion.. n. We hence derive the joint pdf of the sample median and IQR by referring Arnold et. al (1992).. Ch. engchi. i n U. v. The joint pdf of order statistics. is:. where. .. Let, So,. 5.
(15) Thus,. The joint pdf of. ,. and. :. (1). We can obtain the joint pdf of the median and IQR by integrating:. 立. 政 治 大. (2). The joint CDF of the median and IQR is derived as:. ‧ 國. 學. .. 2.3.1 Determination of the control region of the median and IQR using the joint. ‧. CDF of the median and IQR. sit. y. Nat. To implement the control region of the median and IQR, we need to find all pair. io. al. n. which formed the lower control limit (LCL), and. er. values of the median and IQR that respectively fulfill. i n C which formed the upper control limit That is, h e(UCL). ngchi U. v. Therefore, the control region is a region between UCL and LCL. We then assume the in-control process data is from Gamma (2, 2) distribution, and n=15,. , and the control region of the median and IQR is shown in Figure 1.. Intuitively, the control region formed by CDF is normally expected to look like the shape of an ellipse; however, here it looks like a tube. The control region of the median and IQR using the CDF of the median and IQR is considered to be not a good control 6.
(16) region due to the following drawbacks. First, the probability of the control region area is not. Second, the point will not be detected as a signal point when the value of. its median and/or IQR are/is too big. Hence, we try another method for finding a suitable control region of the median and IQR.. 政 治 大. 立. ‧ 國. 學. Figure 1. The true control region of the median and IQR using their CDF approach.. 2.3.2 Determination of the control region of the median and IQR using joint pdf of. ‧. the median and IQR. y. sit. IQR with probability 1-. and minimal area.. io. al. er. Nat. Here, we describe the procedure to determine the control region of the median and. n. Step 1. Set up the control limits of the median chart and IQR chart based on section 2.1-2.2. That is,. Ch. engchi. i n U. v. ； ；. where. .. Step 2. Let median on the x axis, and IQR on the y axis. Step 3. Let upper bound be the upper control limit +1 on x axis/y axis, and lower bound be the lower control limit -1 on the x axis/y axis. If the lower bound of IQR < 0, then let lower bound=0. Consequently, a big rectangular is obtained on the x axis and y axis. 7.
(17) Step 4. Divide the big rectangular into (g-1)×(g-1) small equal-size rectangles. Therefore, each small rectangle has a length of on the x axis, that is. ,. on the y axis, that is. .. The four points on the corner of the small rectangle are:. 政 治 大. 立. ‧ 國. 學. Step 5. Calculate the corresponding volume (or probability) for each small rectangular. We can use the following formula to calculate the corresponding volume. ‧. of each small rectangular.. n. er. io. sit. y. Nat. al. Ch. i n U. v. However, the calculation is difficult. We hence use an approximation approach to. engchi. calculate the corresponding probability of each small rectangular-that is. (3). where. is the average of four joint density on corner of each small rectangle.. That is,. 8.
(18) (4). Step 6. Find the control region with probability 1- and minimum area. We select the small rectangular by descending its probability until their sum of – . Therefore, the all selected small rectangular with sum of. probabilities is probability. – and minimum area is the constructed control region of the. median and IQR. We can prove the proposed control region with minimum area and probability. –. 政 治 大. using solver algorithm in Excel.. 立. That is, given n, g and ,. st.. ‧. ‧ 國. 學. Min. area of the control region. the probability of the control region=1- .. y. Nat. io. sit. The obtained optimal control region with probability 1- using solver algorithm. n. al. er. in Excel is the same as the control region constructed by above steps. Hence, in. i n U. v. the study, the proposed above procedure is used to determine the control region of the median and IQR.. Ch. engchi. By adding in the individual median chart and IQR control chart, it is also feasible to distinguish the appearance of out-of-control activities. We assume the in-control samples of size n=15, are from Gamma (2, 2) distribution, and the control region. , and of the median. and IQR is shown in Figure 2. Figure 2 shows that any point falling inside area one is an in-control point; a point in area three is an out-of-control point and it also can be detected as a location problem by the individual median control chart; a point in area four is an out-of-control point with a dispersion problem; a point in area five is an 9.
(19) out-of-control point detected as both location and dispersion problems; and a point in area two is an out-of-control point that cannot be detected whether it is a location or dispersion problem by individual charts. With different , the size of control regions will also be different (shown in (blue line with. red line with. Figure 3). It shows that a red control region under blue control region under. . Therefore, as. is smaller than the. decreases, the area of the control. region becomes larger. The in-control ARL is:. 政 治 大 Then, the approximated out-of-control ARL is then 立. ‧ 國. 學. where. (5). (6). and. ‧. T is the number of small rectangles in the control region with probability. Nat. y. , and. n. al. er. io. four corners of a small rectangular.. sit. is the average of four out-of-control joint densities of median and IQR on the. i n U. v. The control region that can simultaneously monitor the process location and. Ch. engchi. dispersion will reduce the required time and effort, but it cannot show the process change over time.. 10.
(20) Figure 2. The true control region of the median and IQR using their pdf approach.. 立. 政 治 大. ‧. ‧ 國. 學 er. io. sit. y. Nat. n. al. Ch. engchi. (blue line with. i n U. v. , red line with. Figure 3. True control region of the median and IQR using their pdf approach.. 2.4 Performance measurement of the control region of the median and IQR To evaluate the performance of the control region of the median and IQR, we assume that the observations of the in-control process is from a skewed Gamma (. ). distribution, and its density function is given in Table 1. Figure 4 also shows the density plot of the Gamma (. ) distribution. Figure 5 and Figure 6 show that there are. four different control regions of the median and IQR under different combinations of n=5, 15and. . 11.
(21) The detection performance index,. , of the control region of the median and. IQR to detect shifts in process location and dispersion are calculated in equation (6). In our case, the process is said to be out-of-control whenever process parameter values shift from an in-control value close to. . Moreover, we expect. for an in-control process.. Table 2 shows that when n=5 and than. to be. , there is quite a few. , and most of them happen when. is between 1.75 to 3 and. 1.75. However, when n increases to 15, overall. greater. is between 1 to. is smaller than. when n=5,. is greater than , shown in Table 3. 政 治 大 Additionally, Table 2 and Table 3 show that decreases when and change 立 and there are less cases in which. results are the same as when. is equal to 370, present that the. 學. ‧ 國. collaboratively. Table 4 and Table 5, in which. . In general, it means that the detection ability. ; furthermore, the detection ability also increases when. being greater and. sit. y. Nat. than. ‧. increases when n increases and reduces the occurrences of. io. n. al. er. the same direction.. v i n Density value C h function Expected engchi U. Table 1. Density function, expectation and variance of a Gamma distribution. Distribution. 12. Variation. change in.
(22) 政 治 大 Figure 4. Density curve of Gamma (2, 2) distribution 立 ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. (1) n=5. i n U. v. (2) n=15. Figure 5. The true control region of median and IQR for Gamma (2, 2) distribution with. 13. ..
(23) (1) n=5. (2) n=15. Figure 6. The true control region of the median for Gamma (2, 2) distribution with 政and IQR治 大 . 立 of the true control region of the median and IQR for Gamma(2, ) with n=5 and. ‧ 國. 學. 1.00 1.50 1.75 2.00 2.25 2.50 3.00. ‧. 1.50. 32.8 114.4 166.3 192.2 162.1 106.2 38.8. 1.75. 85.5 285.4 345.0 262.3 139.7 69.7 22.0. 2.00. 187.5 492.5 417.9 199.9 83.8 39.1 12.6. io. sit. y. 5.1 11.7 16.3 21.5 27.0 32.2 37.3. Nat. 1.00. er. Table 2.. a l312.3 605.5 331.3 118.5 46.5 i22.0 v 2.50 417.3 575.0 207.6 65.8 26.2n 12.9 C 3.00 565.6h 281.6 e n66.0 h i U9.6 5.4 g c 21.5 n. 2.25. Table 3.. 7.6 5.0 2.6. of the true control region of the median and IQR for Gamma(2, ) distribution with n=15 and. 1.00 1.50 1.75 2.00 2.25 2.50 3.00 1.00. 1.0. 1.3. 1.50. 1.4. 5.9 13.8 31.0 57.1 65.2 23.1. 1.75. 2.2 20.9 66.5 148.7 119.9 48.4 9.8. 2.00. 4.3 95.8 310.6 199.8 54.8 18.4 4.5. 2.25. 10.5 425.8 368.3 73.8 19.6 7.6 2.5. 2.50. 29.9 672.5 128.9 25.3. 3.00. 1.7. 237.2 100.7 16.4. 14. 2.3. 5.0. 3.2 4.4 7.7. 8.1 3.7 1.6 2.4 1.6 1.1.
(24) Table 4.. of the true control region of the median and IQR for Gamma(2, ) distribution with n=5 and. 1.00 1.50 1.75 2.00 2.25 2.50 3.00 1.00. 7.4 18.1 25.7 34.5 44.2 53.6 63.9. 1.50 54.2 179.9 261.4 317.0 290.9 199.2 68.6 1.75 125.5 365.9 472.4 436.4 264.2 131.9 37.3 2.00 218.1 530.5 581.2 370.3 163.5 72.3 20.3 2.25 297.2 639.8 542.0 236.7 89.6 39.0 11.7 2.50 358.6 687.0 399.1 133.2 48.6 21.8 7.2 3.00 461.9 515.6 139.9 40.6 16.1. of the true control region of the median and IQR for Gamma(2, ) with n=15 .. 1.00. 治 政 1.50 1.75 2.00 2.25 2.50 3.00 大. 立 1.0. 1.50. 1.6. 8.3 20.7 48.8 93.5 107.6 33.9. 1.75. 2.9. 34.0 115.8 271.1 210.1 76.9 13.4. 2.00. 6.4 279.8 619.0 369.7 89.2 27.1 5.7. 2.25. 17.6 890.6 732.1 125.1 29.4 10.4 2.9. 2.50. 56.3 1427.3 233.5 39.4 11.2. 2.0. 2.8. 3.9. 5.5 10.1. 6.7. 2.9. 4.8 1.8 1.8 1.2. io. sit. 480.4 183.3 25.1. ‧. Nat 3.00. 1.5. 學. ‧ 國. 1.00. y. and. er. Table 5.. 8.1 3.4. 3. Determination of the Control Region of the Median and IQR for the Quality. al. n. v i n C h – the KernelUDensity Estimation Method Variable with Unknown Distribution engchi Most of the data in service or manufacturing industries nowadays come from a. process having non-normal and unknown distributions. Hence, to construct the control region of the median and IQR, we first may find out the approximated in-control process distribution using the kernel density estimation method. 3.1 Using the kernel density estimation method to estimate the pdf of the distribution-free process data Let. be an independent and identically distributed sample drawn. from an unknown in-control density ƒ. Its estimated kernel density is: 15.
(25) (7). where. is the kernel, a non-negative function that integrates to one and has mean. zero, and h > 0 is a smoothing parameter called the bandwidth. Silverman (1986) emphasized the importance of choosing a suitable bandwidth h, because the bias of depends directly on h, but Jones (1990) noted that the specific choice of the kernel is not critical. In this paper, the kernel function is a standard normal distribution. There are many methods to determine h. Azzalini (1981) proposed a simple bandwidth selection:. 立. 政 治 大. Park and Marron (1990) also set up a ‘least squares cross-validation’. Sheather and. ‧ 國. 學. Jones (1991) then improved the ‘least squares cross-validation’ into a ‘plug-in. ‧. approach’. Bowman and Azzalini (1997) verified that the ‘plug-in approach’ has superior performance in these methods. Hence, in this article we select the ‘plug-in. y. Nat. n. al. er. io. its CDF,. sit. approach’ to determine h. We then can integrate the kernel density estimation to obtain. Ch. engchi. For an unknown bivariate density f of vector. i n U. v. (8). , its. kernel density is defined by: (9). where. , and H is the bandwidth matrix. that is symmetric and positive-definite The crucial factor determining the performance of the kernel density estimation is also the bandwidth matrix selection. Here, we still use the ‘plug-in approach’ to determine H, but there are many methods within this approach. Wand and Jones (1994) and Duong and Hazelton (2003) proposed ‘amse’ and 16.
(26) ‘samse’. In order to simplify the theoretical computations, these authors used univariate pilot bandwidths, which are not optimal for multivariate data. Thus, they sacrificed flexibility for ease of computation. Chacon and Duong (2010) developed unconstrained pilot bandwidth matrices, which are more flexible plug-in selectors. In the two-dimension kernel density estimate method, we select the ‘unconstrained plug-in approach’ to determine H. 3.2 Determination of the approximated control region using the kernel density estimation method In order to construct a control region of the median and IQR based on section 2.3, then. 政 治 大. must be known. If the density is not known, then it is. 立. impossible to construct a control region of the median and IQR. Therefore, we develop. ‧ 國. 學. an approximated control region of the median and IQR using the kernel density. ‧. estimation method.. sit. y. Nat. 3.2.1 Determining the control region using the one-dimensional kernel density estimation method. n. al. er. io. First of all, suppose we have m sets of a sample of observations taken from an unknown in-control distribution. Ch. engchi. small value to a large value and choose calculate. IQR. Therefore,. Second, the in-control pdf. v. have. values from each set, and m. observations. of. can be estimated using the kernel density. estimation approach in equation (7) as (8) to get kernel CDF,. we. i n U. . We then sort each set from a. ; hence, we integrate. in equation. . We use the results to estimate joint pdf. in equation (1):. 17.
(27) by integrating. Moreover, we can get with. . (10). Following the steps in Section 2.3.2 to construct the control region, we need to find the control limits of the Median and IQR control charts using one-dimentional kernel density estimation method. The in-control pdfs of the median and IQR can now be respectively estimated using the one-dimensional kernel density estimation in equation (7) with m. 政 治 大. observations of median and IQR. That is, the estimated kernel densities of the. 立. in-control pdfs of the median and IQR are. n. al. er. sit. to get CDFs. y. ‧. ‧ 國. 學. io. :. Nat. We then integrate. Ch. engchi. i n U. v. The control limits of the approximated Median control chart and approximated IQR control chart are:. where. Employing the same procedure in section 2.3.2, the control region could be obtained. 18.
(28) 3.2.2 Determining the control region using the two-dimensional kernel density estimation method First of all, suppose we have m sets of a sample of observations taken from an unknown in-control distribution. . We then sort each set in. ascending order and choose the values of can be calculated by. from each set, and IQR. Therefore, we have m observations of. Second, we directly use the two-dimensional kernel density estimation method in equation (9) to estimate joint pdf of median and IQR, . That is,. 立. 政 治 大. (11). We also utilize the one-dimensional kernel density estimation method to. ‧ 國. 學. approximate the pdfs of the median and IQR using m observations of Median and IQR.. ‧. We then integrate their pdf to get their CDF. Therefore, the control limits of the approximated median and IQR chart are the same as in section 3.2.1. Next, following. y. Nat. er. io. IQR with probability 1- .. sit. the same approach as in section 3.2.1, we construct the control region of the median and. al. n. v i n 3.3 Performance measurementCofhthe control regionUof the median and IQR engchi. To check whether the performance of the approximated control regions of the. median and IQR is almost the same as the control regions of the median and IQR with the in-control distribution from Gamma (2, 2), Normal (0,1), Laplace (0, 1), t. , and. , we perform the simulation studies. We simulate random samples (m=10,000) of size n=5, 15 from the above distributions. With the data we then construct the approximated control regions of the median and IQR with different. as 0.0054, 0.005,. 0.0027, or 0.002. Figure 7 (a) - Figure 11 (a), which are under n=5, show that the approximated control regions of the median and IQR using the one-dimensional kernel density 19.
(29) estimation approach is close to the control regions with the specified in-control distributions, but using the two-dimensional kernel density estimation approach is not smooth and not close to the control regions with the specified in-control distributions. Moreover, the areas of the control regions using the kernel density estimation methods are smaller than the control regions with the specified in-control distributions. When n increases to 15, the approximated control regions using the one-dimensional kernel density estimation approach is almost the same as the control regions with the specified in-control distributions, and using the two-dimensional. 政 治 大 Figure 11 (b). The area of the control regions using the kernel density estimation 立. kernel density estimation approach is smoother and closer, as shown in Figure 7 (b) -. methods is also almost the same as the control regions with the specified in-control. ‧ 國. 學. distributions. In short, using the kernel density estimation methods under n=15 is more. ‧. robust than under n=5.. n. er. io. sit. y. Nat. al. Ch. engchi. 20. i n U. v.
(30) (1) n=5. (2) n=15. 政 治 大 blue line: the control region using the one-dimentional kernel density estimation method, 立 red line: the control region using two-dimensional kernel density estimation method. black line: the true control region,. ‧ 國. 學. Figure 7. The control region of the median and IQR for data from Gamma (2, 2) distribution with. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. (1) n=5. i n U. v. (2) n=15. black line:the true control region, blue line : the approximated control region using one-dimentional kernel density estimation method, red line : the approximated control region using two-dimensional kernel density estimation method. Figure 8. The control regions of the median and IQR for data from Gamma (2, 2) distribution with .. 21.
(31) (1) n=5 black line: the true control region,. (2) n=15. 政 治 大. blue line: the approximated control region using one-dimentional kernel density estimation method,. 立. red line: the approximated control region using two-dimensional kernel density estimation method.. ‧. ‧ 國. .. 學. Figure 9. The control region of the median and IQR for data from Normal (0, 1) distribution with. n. er. io. sit. y. Nat. al. Ch. engchi. (1) n=5. i n U. v. (2) n=15. black line: the true control region, blue line: the approximated control region using one-dimentional kernel density estimation method, red line: the approximated control region using two-dimensional kernel density estimation method. Figure 10. The control regions of the median and IQR for data from Laplace (0, 1) distribution with .. 22.
(32) (1) n=5. (2) n=15. black line : the true control region,. 政 治 大. blue line: the approximated control region using one-dimentional kernel density estimation method, red line: the approximated control region using two-dimensional kernel density estimation method.. 立. Figure 11. The control regions of the median and IQR for data from. distribution with. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. (1) n=5. v. (2) n=15. black line: the true control region, blue line: the control region using one-dimentional kernel density estimation method, red line: the control region using two-dimensional kernel density estimation method. Figure 12. The control region of the median and IQR for data from. 23. distribution with.
(33) We calculate the performance,. of the control region to detect shifts in the. process location and dispersion using equation (6). In our case, the process is said to be out-of-control whenever process parameter values in-control value. shift from an. , while ARLs for the true control region of the. median and IQR are taken from Table 2 to Table 5. Table 6 presents that a few the approximated control regions of the median and IQR are close to true control region when n=5 and between. of of the. . However, most of the differences. of the true control region and approximated control region are very. is between 1~1.75, and the situation for the 政 治 大 two-dimensional kernel density estimation approach is particularly serious. When n 立 large when. is between 2~3 and. increases to 15, most. of the approximated control region are closer to the true. ‧ 國. 學. control region when n=5, but the two-dimensional kernel density estimation approach is larger and not close, as shown in Table 7. This is. ‧. still has more cases which. are considered. In short, the control region of the. sit. y. Nat. generally true when. io. al. n. increases.. er. median and IQR can be approximated very well by using kernel density when n. Ch. engchi. 24. i n U. v.
(34) Table 6.. of the three control regions of the median and IQR for data from Gamma(2, ) distribution with n=5 and. 1.00. 1.50. 1.75 2.00 2.25 2.50 3.00. 1.0 (1). 5.1. 11.7 16.3 21.5 27.0 32.2 37.3. (2). 4.7. 10.7 14.8 20.0 25.2 30.3 33.4. (3). 19.0. 33.1 49.0 50.0 53.2 61.7 40.9. 1.50(1) (2). 32.8 114.4 166.3 192.2 162.1 106.2 38.8 27.0. 98.4 142.8 171.0 144.7 96.9 37.6. (3) 158.4 331.1 279.0 213.0 153.4 90.4 30.7 1.75(1). 85.5 285.4 345.0 262.3 139.7 69.7 22.0. (2). 68.3 248.8 300.6 262.3 138.4 68.8 23.2. 政 治 大 (2) 152.2 445.0 390.5 199.9 85.8 立 (3) 1056.8 897.3 610.1 199.7 53.6. (3) 363.2 634.6 448.2 187.7 118.0 53.6 17.1 2.00(1) 187.5 492.5 417.9 199.9 83.8 39.1 12.6 42.8 13.2 29.9 9.9. ‧ 國. 學. 2.25(1) 312.3 605.5 331.3 118.5 46.5 22.0 7.6 (2) 267.9 563.6 345.8 124.3 48.6 24.2 8.1. ‧. (3) 2346.7 1106.0 282.3 73.5 31.2 17.3 6.1. 2.50(1) 417.3 575.0 207.6 65.8 26.2 12.9 5.0. y. io. 3.00(1) 565.6 281.6 66.0 21.5. al. 9.6. 9.8 4.2 5.4 2.6. n. v 5.7 i n 4.3 (3) 4008.4 C h140.9 41.3 14.4 U7.5 engchi (2) 514.7 298.7 74.0 23.2 10.4. st. sit. (3) 2453.7 851.2 140.0 47.5 18.6. er. Nat. (2) 337.2 542.0 222.4 73.1 28.5 13.9 5.2. 2.6 2.2. 1 row, (1), is for the true control region;. 2nd row, (2), is for the control region using one-dimensional kernel density estimation method; 3rd row, (3), is for the control region using two-dimensional kernel density estimation method.. 25.
(35) Table 7.. of the three control regions of the median and IQR for data from Gamma(2, ) distribution with n=15 and. 1.00 1.50 1.75 2.00 2.25 2.50 3.00 1.00(1) (1)1.0. 1.3. 1.7. 2.3. 3.2 4.4. 7.7. (2). 1.0. 1.4. 1.8. 2.4. 3.3 4.5. 7.9. (3). 1.1. 1.6. 2.1. 3.1. 4.2 5.8. 9.0. 1.50(1). 1.4. 5.9 13.8 31.0 57.1 65.2 23.1. (2). 1.4. 5.9 13.7 29.6 53.8 63.3 23.1. (3). 1.7. 8.4 20.6 44.6 61.3 67.2 24.3. 1.75(1). 2.2 20.9 66.5 148.7 119.9 48.4. (2). 2.2 20.1 62.2 137.7 127.3 49.6 10.1. (3). 3.1 33.7 97.6 171.6 106.3 46.9. 9.6. 2.00(1). 4.3 95.8 310.6 199.8 54.8 18.4. 4.5. (2). 治 4.4政 90.5 284.3 199.3. 58.0 19.1. 大 52.5 19.8. 4.6. 19.6 7.6. 2.5. 7.1 143.1 467.9 199.9 立 2.25(1) 10.5 425.8 368.3 73.8 (3). 4.8 2.6. (3) 21.0 637.8 255.3 83.4 20.4 7.8. 2.5. 8,1 3.7. (2) 28.7 644.5 135.7 27.0. 8.4 3.9. (3) 61.8 666.2 117.7 24.4. 8.2 3.9. 1.7. 1.6. y. 1.7. sit. Nat. 2.50(1) 29.9 672.5 128.9 25.3. ‧. ‧ 國. 學. (2) 10.4 386.6 374.8 77.9 20.6 8.0. 2.4 1.6. 1.1. (2) 206.8 107.7 17.7. 5.2. 2.5 1.6. 1.1. al. 4.9. 2.5 1.6. v ni. 1.1. n. Ch. engchi U. 1st row, (1), is for the true control region;. er. 5.0. io. 3.00(1) 237.2 100.7 16.4 (3) 665.1 72.5 13.7. nd. 9.8. 2 row, (2), is for the control region using one-dimensional kernel density estimation method; 3rd row, (3), is for the control region using two-dimensional kernel density estimation method.. 26.
(36) Table 8.. of the three control regions of the median and IQR for data from Gamma(2, ) distribution with n=5 and. .. 1.00. 1.50 1.75 2.00 2.25 2.50 3.00. 1.00(1). 7.4. 18.1 25.7 34.5 44.2 53.6 63.9. (2). 8.7. 21.2 30.6 40.4 51.1 59.2 64.2. (3). 23.4. 38.7 47.5 59.8 63.1 58.6 48.3. 1.50(1). 54.2 179.9 261.4 317.0 290.9 199.2 68.6. (2). 61.6 196.8 267.7 316.0 264.1 179.1 64.3. (3). 179.3 307.7 288.4 311.2 228.0 96.8 34.9. 1.75(1). 125.5 365.9 472.4 436.4 264.2 131.9 37.3. (2). 132.3 373.9 458.2 405.3 245.3 130.4 38.0. (3). 425.6 1188.2 481.7 307.5 146.7 62.3 20.2. 政 治 大 (3) 1159.4 1811.0 802.1 369.8 80.5 立 2.25(1) 297.2 639.8 542.0 236.7 89.6. 218.1 530.5 581.2 370.3 163.5 72.3 20.3. (2). 223.6 528.4 562.3 369.8 175.7 73.4 20.5. (2). 37.9 10.8 39.0 11.7. 學. ‧ 國. 2.00(1). 301.3 631.3 527.1 248.4 91.5 44.2 13.2. (3) 1603.7 1167.7 368.7 113.2 47.4 20.6 7.2 358.6 687.0 399.1 133.2 48.6 21.8 7.2. (2). 365.5 702.8 433.5 145.3 51.3 23.3 7.9. ‧. 2.50(1). (2). 467.7 574.5 157.7 46.0 17.6. n. al. (3) 36505.6 249.8 56.6 20.1. Ch. 1st row, (1), is for the true control region; nd. engchi U. 8.1 3.4 8.9 3.6. er. 461.9 515.6 139.9 40.6 16.1. io. 3.00(1). sit. y. Nat. (3) 7768.6 848.2 260.6 66.1 24.7 12.4 4.8. v 5.0 i n. 8.2. 2.5. 2 row, (2), is for the control region using one-dimensional kernel density estimation method; 3rd row, (3), is for the control region using two-dimensional kernel density estimation method.. 27.
(37) Table 9.. of the three control regions of the median and IQR for data from Gamma(2, ) distribution. with n=15 and. 1.00. 1.50 1.75 2.00 2.25 2.50 3.00. 1.00(1). 1.0. 1.5. 2.0. 2.8. 3.9. 5.5 10.1. (2). 1.0. 1.5. 2.1. 2.9. 4.2. 6.1 10.7. (3). 1.1. 1.8. 2.5. 3.4. 4.9. 6.4 10.9. 1.50(1). 1.6. 8.3 20.7 48.8 93.5 107.6 33.9. (2). 1.6. 8.3 20.7 48.4 90.6 103.6 32.9. (3). 2.3. 11.8 24.8 51.1 83.2 73.1 28.9. 1.75(1). 2.9. 34.0 115.8 271.1 210.1 76.9 13.4. (2). 2.8. 31.6 111.2 246.1 207.9 76.9 13.4. (3). 4.7. 40.1 104.9 220.1 153.1 66.5 12.8. 政 治 大 (3) 10.8 165.1 383.0 369.8 81.7 23.5 立 2.25(1) 17.6 890.6 732.1 125.1 29.4 10.4 2.00(1). 6.4 279.8 619.0 369.7 89.2 27.1 5.7. (2). 6.4 165.4 593.7 369.4 93.7 28.0 5.7 2.9. ‧ 國. 學. (2) 16.0 828.4 774.1 131.8 29.8 10.7 3.0. 2.50(1) 56.3 1427.3 233.5 39.4 11.2. 4.8 1.8. (2) 49.4 1472.1 245.6 41.1 11.8. 4.9 1.9. (3) 88.5 879.7 150.9 33.2 10.5. 4.7 1.9. sit. y. ‧. 9.7 3.0. Nat. (3) 29.1 610.9 426.6 106.3 25.6. 2.9. 1.8 1.2. (2) 512.0 194.6 25.8. 6.9. 3.0. 1.8 1.2. 6.5. 2.9. n. al. Ch. 1st row, (1), is for the true control region;. engchi U. er. 6.7. io. 3.00(1) 480.4 183.3 25.1 (3) 967.0 141.2 22.0. nd. 5.7. v1.8 i n. 1.2. 2 row, (2), is for the control region using one-dimensional kernel density estimation method; 3rd row, (3), is for the control region using two-dimensional kernel density estimation method.. 28.
(38) 4. Design the K Control Chart for Monitoring Joint Location and Dispersion Due to the drawback of the control region, which is that it cannot show the process changes over time, we determine to use the value of. as the. monitoring statistic for location and dispersion. We call it the K control chart. However, if a process is detected as out-of-control, then the K control chart does not distinguish whether it is a location or dispersion problem. Therefore, we add in an individual Median control chart and IQR control chart to detect the appearance of out-of-control activities.. 政 治 大. 4.1 Design the K control chart using the one-dimensional kernel density. 立. estimation method. ‧ 國. 學. First of all, suppose we have m sets of a sample of observations. taken. from an unknown in-control distribution. We then sort each set from a small value to a. ‧. large value and choose. Therefore, we have m observations of Median and IQR.. sit. y. Nat. IQR by. values from each set. We also calculate. io. al. er. Second, we can get m observations of the kernel density value (k-value) in. n. equation (10) with m pair observations of. Ch. engchi U. v ni. That is,. Moreover, we use these k-values to construct the kernel control chart. In this chart, if the k-values are less than lower control limit (LCL), then they are considered to be out-of-control. The values computed from using equation (10) with the pair-observation of Median and IQR on the control region line are almost the same and form the lower control limit (LCL). The in-control k-values, which are calculated with points in the control region, are larger than LCL, however, the out-of-control values with points outside the control region are below LCL. In the end, we may m observations of Median and IQR to construct the Median 29.
(39) control chart and IQR control chart for identifying whether the out-of-control point is a location and/or dispersion problem. 4.2 Design the K control chart using the two-dimensional kernel density estimation method First of all, suppose we have m sets of a sample of observations taken from an unknown in-control distribution. We then sort each set from a small value to large value and choose by. values from each set. We calculate IQR. Therefore, we have m observations of Median and IQR. Second, we get m observations of the kernel density value (. 政 治 大. ) using the. two-dimension kernel density estimation method in equation (11) with m pair. 立. observations of Median and IQR. That is,. ‧. ‧ 國. 學. Moreover, we use these. to construct the kernel control chart. We. sit. y. Nat. follow a similar approach as in section 4.1 to find the control limit (LCL) for the K. al. n. diagnosis.. er. io. control chart. We also construct the median control chart and IQR control chart for. 5. Performance Comparison. Ch. engchi. i n U. v. We consider comparing the control region of the median and IQR with some other nonparametric methods. The NLE chart is proposed by Zou and Tsung (2010) , while the he SL chart is proposed by Mukherjee and Chakraborti (2012). In order to investigate the out-of-control performance of these charts, we consider five scenarios: (I) the thin-tailed symmetric normal distribution with in-control distribution N( =0,. =1) versus out-of-control distribution N( 0 , 1 )); (II) the heavy-tailed symmetric Laplace distribution with the in-control distribution Laplace ( =0, =1) versus the out-of-control distribution Laplace ( 0 , 1 ); (III) the heavy-tailed 30.
(40) symmetric t distribution with the in-control distribution out-of-control distribution:. ; (IV) the in-control distribution. versus the out-of-control distribution:. ; (V) the right-skew chi-square. distribution with the in-control distribution distribution. versus the. versus the out-of-control. ); and (VI) the in-control distribution. versus the out-of-control distribution. ). Here, we just compare the. best results of the NLE chart and SL chart with our proposed control region. We then simulate random samples (m=10,000) of size n=5, 15 from above as 政 治 大 0.0054, 0.0027or 0.002. Thus for scenario (I), the results are presented in Table 10 and 立 distributions. With the data we construct the proposed control region with different. Table 11. We note from Table 10 that the proposed control region clearly outperforms. ‧ 國. 學. the SL chart if there is a shift in the scale parameter along with a small shift in location.. ‧. The proposed control region using the two-dimensional kernel density estimation. sit. y. Nat. method is especially more effective than the other three charts. As location parameter. io. er. increases, these charts become more or less equally effective - that is, their ARL values become close to each other. We observe form Table 11 that the ARL value of the control. al. n. v i n C hchart when n is 5Uand ( 0.25 , 1.1 ) except region is only larger than the NLE engchi when using the two-dimensional kernel density estimation method. When n increases to 15, the proposed control regions are more effective. Table 12 presents the results of scenarios (II). It shows that the proposed control region using the two-dimensional kernel density estimation method is also more effective than the SL chart and other control regions. If there is a small shift in location, then using the one-dimensional kernel density estimation method always has a better performance than the SL chart. However, the out-of-control ARL of the true control region is smaller than the SL chart when there is a moderate to large shift in the scale parameter. 31.
(41) Table 13 presents the results of scenarios (III). When n is 5 and location shift. is. larger than 0.75, the control regions using the kernel density estimation methods have a better performance than the NLE chart. As n increases, the control regions become more effective than n=5 and the NLE chart. Table 14 presents the results of scenarios (IV). We see that all control regions clearly outperforms the NLE chart when n is 15, and using the two-dimensional kernel density estimation method is especially the best. However, the performance of the true control region is better than NLE chart only for large shift scale when n is 5, but the. 政 治 大 chart. In short, using the two-dimensional kernel density estimation method is more 立 control regions of the kernel density estimation methods perform better than the NLE. effective than the NLE chart in detecting the process scale when the process. ‧ 國. 學. distribution is a heavy-tailed symmetric t distribution.. ‧. Table 15 presents the results of scenarios (V). When n is 15 and location shift. is. sit. y. Nat. larger than 0.25, the control regions have a better performance than the NLE chart.. io. er. Among these charts, using the control region constructed by two-dimensional kernel density estimation method is the best. However, as n decreases, the NLE chart becomes. n. al. Ch. more effective than the control regions.. engchi. i n U. v. Table 16 presents the results of the last scenarios (VI). We see that the NLE chart clearly outperforms the control regions when n is 15 if scale shift is not very large. In short, when the process distribution is a right-skew do not perform better than the NLE chart.. 32. distribution, the control regions.
(42) Table 10. Performance comparisons between the control region and SL chart for data from Normal distribution with SL chart. Control region of the median and IQR True. One-dimensional. Two-dimensional. CR. kernel approach. Kernel approach. 0. 1. 499.6. 499.5. 499.6. 499.7. 0.25. 1. 292.7. 318.5. 405.3. 173.9. 0.50. 1. 94.7. 118.2. 194.4. 79.2. 1.00. 1. 9.3. 15.3. 25.4. 11.8. 1.50. 1. 2.3. 3.6. 5.0. 3.3. 2.00. 1. 1.3. 1.6. 1.9. 1.5. 0. 1.25. 106.2. 63.9. 109.1. 46.2. 0.25. 1.25. 73.6. 51.0. 0.50. 1.25. 35.4. 1.00. 1.25. 7.4 立. 1.50. 1.25. 86.6 治 政 29.7 49.6 大. 39.5 22.4. 12.5. 7.0. 2.6. 3.1. 4.2. 2.9. 1.25. 1.4. 1.6. 2.0. 1.50. 36.82. 18.22. 28.8. 1.50. 30.64. 16.26. 24.7. 1.50. 19.0. 12.1. 17.4. 1.00. 1.50. 6.5. 5.5. 7.3. y. 5.0. 1.50. 1.50. 2.8. 2.7. 3.5. sit. 2.5. 2.00. 1.50. 1.6. 1.7. 1.9. 0. 1.75. 0.25. 1.75. 16.7. 0.50. 1.75. 12.1. 1.00. 1.75. 1.50. 0. io. n. al. 18.5. 8.3. C h7.8 engchi. ‧. 0.25. i n U 10.9 11.8. 1.6 14.8. er. 2.00. Nat. 0.50. ‧ 國. 學. 8.5. v. 13.4 10.0. 1.6 6.9 6.5. 6.6. 9.3. 5.7. 5.7. 4.0. 5.2. 3.6. 1.75. 2.9. 2.4. 3.0. 2.2. 2.00. 1.75. 1.8. 1.6. 1.9. 1.6. 0. 2.00. 11.3. 5.0. 6.6. 4.5. 0.25. 2.00. 10.3. 4.8. 6.1. 4.3. 0.50. 2.00. 8.5. 4.3. 5.6. 3.8. 1.00. 2.00. 4.9. 3.1. 3.8. 2.8. 1.50. 2.00. 2.9. 2.1. 2.5. 2.0. 2.00. 2.00. 1.9. 1.6. 1.8. 1.5. 33.
(43) Table 11. Performance comparisons between the control region and NLE chart for data from Normal distribution Control region of the median and IQR n=5. NLE λ. n=15. True One-dimensional Two-dimensional True One-dimensional Two-dimensional CR kernel approach. kernel approach. CR. kernel approach. kernel approach. 0.05. 0.00 1.00 184.9. 184.8. 184.7 184.7. 184.8. 184.5 184.7. 0.25 1.10 58.2. 63.5. 48.1 32.2. 31.2. 32.8 54.5. 0.50 1.20 19.0. 20.8. 16.0. 7.2. 7.4. 8.3 22.1. 0.75 1.30. 8.1. 8.6. 7.1. 2.9. 2.9. 3.2 12.6. 1.00 1.40. 4.4. 4.6. 4.1. 1.7. 1.7. 1.8. 8.7. 1.50 1.60. 2.1. 2.2. 1.2. 1.1. 4.9. 2.00 1.80. 1.4. 1.5. 1.0. 1.0. 3.1. 3.00 2.00. 1.1. 1.1. 1.0. 1.0. 1.6. 4.00 3.00. 1.0. 1.0. 1.0. 1.0. 0.9. 立. 2.0 1.1 政 1.4 治 1.0 大 1.1 1.0 1.0. 1.0. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 34. i n U. v.
(44) Table 12. Performance comparisons between the control region and SL chart for data from Laplace distribution with. .. SL chart. Control region of the median and IQR True. One-dimensional. Two-dimensional. CR. kernel approach. kernel approach. 0. 1. 493.2. 500.0. 496.4. 495.5. 0.25. 1. 403.1. 854.5. 363.1. 136.6. 0.50. 1. 235.2. 382.2. 195.5. 92.0. 1.00. 1. 36.1. 151.0. 59.3. 21.4. 1.50. 1. 5.93. 36.8. 13.1. 6.9. 2.00. 1. 2.0. 8.3. 3.7. 2.6. 0. 1.25. 156.8. 45.6. 0.25. 1.25. 128.8. 0.50. 1.25. 79.8. 169.7 81.4 治 政 173.9 74.8 大 125.8 58.7. 1.00. 1.25. 19.9. 57.2. 24.6. 1.25. 5.2. 20.0. 9.0. 1.25. 2.2. 6.7. 3.4. 1.50. 65.9. 57.7. 31.0. 1.50. 59.1. 56.0. 28.3. 29.8 12.5. 1.50. 42.1. 47.0. 23.4. y. 7.8. 1.00. 1.50. 14.2. 26.8. 13.1. sit. 學. 0.50. 6.2. 1.50. 1.50. 12.6. 6.6. 2.00. 1.50. 0. 1.75. 35.6. 0.25. 1.75. 33.0. 0.50. 1.75. 24.5. 1.00. 1.75. 1.50. 2.00. io. 4.70. n. al. 2.30. 5.5. C h26.0 engchi 25.2. ‧. 0. i n U14.7 3.1. 5.5 2.5. er. 1.50. Nat. 0.25. ‧ 國. 立. 35.3. v. 10.0 8.9. 4.3 2.3 10.5. 14.3. 9.8. 22.4. 13.1. 8.8. 10.4. 15.1. 8.4. 5.6. 1.75. 4.50. 8.6. 5.0. 3.5. 2.00. 1.75. 2.40. 4.6. 2.8. 2.2. 0. 2.00. 22.1. 14.4. 8.9. 6.9. 0.25. 2.00. 20.7. 14.1. 8.8. 6.0. 0.50. 2.00. 17.0. 13.0. 8.0. 5.7. 1.00. 2.00. 8.60. 9.7. 6.0. 4.2. 1.50. 2.00. 4.30. 6.4. 4.0. 2.9. 2.00. 2.00. 2.50. 3.9. 2.6. 2.1. 35.
(45) Table 13. Performance comparisons between the control region and NLE chart for data from with. (location shift) Control region of the median and IQR. NLE. n=5 True. One-dimensional Two-dimensional True. CR. kernel approach. kernel approach CR. 368.6. 367.7 369.4. 0.25 335.6. 234.7. 102.9. 0.50 223.2. 71.3. 0.75. 88.7. 20.0. 1.00. 24.8. 6.3. 1.50. 2.9. 1.5. 2.00. 1.2. 1.1. 3.00. 1.0. 1.0. 1.0. 4.00. 1.0. 1.0. 1.0. 立. One-dimensional Two-dimensional kernel approach kernel approach. 0.05. 368.3. 368.0. 372. 60.5. 69.2. 53.1. 72.1. 38.4. 6.4. 7.4. 6.8. 28.4. 13.4. 1.8. 2.0. 1.9. 17.9. 4.9 1.1 政1.5 治 1.0 大 1.0 1.0. 1.2. 1.2. 13.3. 1.0. 1.0. 8.4. 1.0. 1.0. 6.0. 1.0. 1.0. 1.0. 3.7. 1.0. 學. 0.00 369.9. 1.0. 2.7. 1.0. ‧. ‧ 國. λ. n=15. Table 14. Performance comparisons between the control region and NLE chart for data from. CR. al. y. v i n Capproach kernel U approach h e n gCRc h ikernel. One-dimensional Two-dimensional True kernel approach. NLE λ. n=15. n. True. Control region of the median and IQR. sit. io. n=5. (scale shift). er. Nat with. One-dimensional Two-dimensional kernel approach. 0.05. 1.00 369.9. 369.1. 368.4. 369.6. 369.0. 368.7 370.0. 1.10 297.9. 113.1. 117.6. 150.0. 158.7. 90.5 218.0. 1.20 214.5. 65.1. 62.6. 68.4. 69.7. 41.2 127.0. 1.30 147.2. 41.8. 40.0. 35.2. 35.4. 24.7 80.7. 1.40 100.6. 27.4. 30.1. 20.2. 21.0. 13.8 56.7. 1.60. 50.2. 14.8. 15.5. 8.5. 8.5. 6.6 34.3. 1.80. 28.1. 9.2. 9.7. 4.7. 4.6. 3.8 24.2. 2.00. 17.4. 6.5. 7.2. 3.0. 3.0. 2.7 18.8. 3.00. 4.3. 2.4. 2.5. 1.3. 1.3. 1.2. 36. 9.2.
(46) Table 15. Performance comparisons between the control region and NLE chart for data from the distribution. (location shift) Control region of the median and IQR n=5 True CR. NLE λ. n=15. One-dimensional Two-dimensional True kernel approach. kernel approach CR. One-dimensional Two-dimensional kernel approach. 0.1. kernel approach. 369.7. 369.8 369.2. 368.3. 367.2 370.0. 0.25 303.2. 240.5. 135.2 116.2. 126.7. 78.6 88.7. 0.50 206.8. 112.2. 71.5. 23.1. 24.8. 17.0 26.8. 0.75 118.7. 48.9. 29.1. 5.6. 6.2. 4.6 18.4. 1.00. 60.1. 23.7. 2.2. 1.9 14.4. 1.50. 13.6. 5.6. 1.0. 1.0. 9.8. 2.00. 3.6. 1.8. 1.0. 1.0. 7.1. 3.00. 1.0. 1.0. 1.0. 1.0. 1.0. 1.0. 4.2. 4.00. 1.0. 1.0. 1.0. 1.0. 1.0. 2.8. 立. 15.1 2.1 政4.1 治 1.0 大 1.6 1.0. 學 1.0. ‧. ‧ 國. 0.00 369.9. Table 16. Performance comparisons between the control region and NLE chart for data from. Nat CR 1.00 369.90. 368.33. 1.10 268.76. λ. er. al. v i n Capproach kernel h e n gCRc h kernel i Uapproach. One-dimensional Two-dimensional True kernel approach. NLE. n=15. n. True. sit. Control region of the median and IQR. io. n=5. y. distribution. (scale shift). One-dimensional Two-dimensional. 0.1. kernel approach. 369.4 369.16. 369.45. 368.20 372.00. 124.92. 70.81 122.43. 119.6. 102.56 23.40. 1.20 103.33. 47.34. 33.90 45.04. 43.4. 38.65 10.90. 1.30 42.28. 22.23. 20.46 20.42. 20.2. 20.70. 7.36. 1.40 22.05. 13.69. 12.23 11.10. 11.1. 10.72. 5.67. 1.60. 9.61. 6.74. 6.89. 4.79. 4.8. 4.63. 4.06. 1.80. 5.83. 4.35. 4.29. 2.83. 2.8. 2.80. 3.30. 2.00. 4.17. 3.22. 3.12. 2.01. 2.0. 2.02. 2.92. 3.00. 1.93. 1.62. 1.56. 1.10. 1.1. 1.10. 2.03. 37.
(47) 6. Real Examples We use two real data set of TSMC company’s stock price and the service time of a Bank branch to illustrate how to construct the proposed control region of Median and IQR using the kernel density estimation approach and the kernel control chart, and the process monitoring. 6.1 TSMC Company’s stock price data We obtain the data from TSMC company’s stock closing prices in Taiwan from January 5, 2009 to December 31, 2013, see Figure 13. It was not until January 2, 2012. 政 治 大. that its share price broke through NT$80; therefore, we take this time as a breakpoint.. 立. Thus, we take the share price from January 5, 2009 to December 30, 2011 as the. ‧ 國. 學. in-control data, which have 749 observations, and from December 30, 2011 to December 31, 2013 as the out-of-control data, which have 495 observations.. ‧. Because the TSMC’s stock price data is dependent on time, we fit it with a time. y. Nat. sit. series model in order to extract meaningful statistics, residuals. First of all, we take time. n. al. er. io. series analysis for in-control data, shown as in Figure 14. We can observe that the. i n U. v. in-control data is not stationary and it displays a trend and seasonality. However, the. Ch. engchi. common assumptions in time series model are that the data should be stationary and no seasonality. Moreover, we difference the in-control data with a year period to eliminate the seasonality and difference again to eliminate the trend. The in-control data after differencing is now stationary and does not display trend and seasonality, see Figure 15. We next fit it with ARMA (0, 2) model, see Table 17. That is, ARMA (0, 2) model is. 38.
(48) Through the model residuals assumptions checking and using the Augmented Dickey-Fuller Test to test a unit root in the residuals of ARMA (0, 2) model, the residuals showed stationary in Table 18. Hence, the ARMA (0, 2) model is adequate. Here, we use the residuals of the ARMA (0, 2) model to implement the control region. We let each sample of size 15 residuals; hence, we have 35 samples of in-control residuals. The out-of-control residuals are calculated by. ; they are 14. samples of size 15 for out-of-control residuals.. 立. 政 治 大. ‧. ‧ 國. 學 er. io. sit. y. Nat. al. v. n. Figure 13. TSMC company’s stock price from 2009/01/05 to 2013/12/31. Ch. engchi. 39. i n U.
(49) 立. 政 治 大. Figure 14. Time series analysis for price. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 15. Time series analysis for differencing stock price. Table 17. The fitted time series model.. MA1. MA2. coefficient. 0.0123. -0.1432. s.e.. 0.0436. 0.0442. AIC=1700.89 BIC=1713.67 40.
(50) Table 18. Test result of the Augmented Dickey-Fuller Test.. Augmented Dickey-Fuller Test Dickey-Fuller = -7.36 Lag order = 8 p-value = 0.01. 6.1.1 Control region of the median and IQR of TSMC company’s stock price using the kernel density estimation method We sort the price for each of the 35 in-control samples and calculate statistics Q1,. Median, Q3 and IQR. Therefore, we have 35 paired Median and IQR. To implement the approximated control region of the median and IQR using the one-dimensional kernel density estimation method, we first estimate kernel density. 政 治 大 to get its CDF We follow the Sturges rule (see 立. for the in-control residuals using equation (7), as shown in Figure 16.. We then integrate. ‧ 國. 學. Sturges (1926)) to separate the in-control residuals into 10 groups, and the interval length of residuals in each group is 0.78. We also examine how well the kernel density testing result is in Table 19, it shows. ‧. fits the residuals using goodness of fit test. The. sit. y. Nat. the kernel density fits the in-control residuals very well.. io. er. We now can get estimated joint pdf of median and IQR,. al. using order statistic method based on the approach described in Section 3.2.1. We then. n. v i n C h of the median andUIQR respectively, see equation estimate the in-control kernel density engchi. (7), using the thirty-five paired Median and IQR, as illustrated in Figure 17. We next integrate. to get CDFs. and. . It is feasible to find the UCL and LCL of the median and IQR charts with respectively, and total. .. Moreover, the control limits of the approximated Median and IQR control charts are:. 41.
(51) We follow the same approach as in Section 3.2.1 to get the approximated control region of the median and IQR with probability. . We next plot the 35. in-control paired Median and IQR as black points and 14 out-of-control paired Median and IQR as red points into the control region in Figure 18. The figure presents that there is no false alarm for the in-control samples but there are three out-of-control samples that are detected as out-of-control signals, which are a dispersion problem. Here, we directly use the two-dimensional kernel density estimation method, in. 政 治 大 that is,. equation (9), to estimate the joint pdf,. 立. .Next, following the same approach in Section 3.2.2 to get the approximated control. ‧ 國. 學. region of the median and IQR with probability. , shown as Figure 19.. ‧. The figure illustrates that there are 4 points detected as out-of-control signals. Here, the control region of the median and IQR using the two-dimensional kernel density. y. Nat. n. al. Ch. engchi. er. io. one-dimensional kernel density estimation method.. sit. estimation method has better detection ability than the control region using the. i n U. v. Figure 16. The kernel density (blue curve), and the histogram of in-control residuals. 42.
(52) Table 19. Contingency table for in-control residuals < -3.41. -3.41~. -2.62~. -1.83~. -1.04~. -0.25~. 0.54~. 1.33~. 2.12~. -2.62. -1.83. -1.04. -0.25. 0.54. 1.33. 2.12. 2.91. >2.91. Expected probability. 0.009. 0.017. 0.050. 0.124. 0.222. 0.255. 0.189. 0.089. 0.033. 0.012. Expected frequency. 4.69. 8.77. 26.25. 65.29. 116.44. 133.64. 99.17. 46.88. 17.53. 6.34. Observed frequency. 5. 7. 27. 59. 120. 141. 103. 41. 17. 5. goodness of fit test df=9 p-value=0.9749. 立. 政 治 大. ‧. ‧ 國. 學 er. io. sit. y. Nat. al. n. v i n Figure 17. The estimateC kernel density of the in-control residuals data hengchi U (1) median kernel density. (2) IQR kernel density. 43.
(53) 政 治 大 Figure 18. The monitoring results of the control region with 立 one-dimensional kernel density estimation approach.. (Black points: in-control points, red points: out-of-control points) using the. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. (Black points: in-control points, red points: out-of-control points) Figure 19. The monitoring results of the control region with. using the. two-dimensional kernel density estimation approach.. 6.1.2 The K control chart Using the one-dimensional kernel density in equation (10) we may find the lower control limit (LCL) of the K control chart with. = 0.005. The LCL, 0.0043, was. calculated by plug in all pairs of the Median and IQR on the blue control region line in 44.
(54) equation (10), shown in Figure 18. Second, the corresponding k-values for in-control and out-of-control paired median and IQR were also calculated using equation (10). Finally, we plot 35 in-control k-values as black points and 14 out-of-control k-values as red points on the K chart, see Figure 20. In Figure 20, the second, fifth and seventh points of the 14 out-of-control points are detected as signals. Same procedure to construct the above K chart, we use the two-dimensional kernel density estimation method to construct the K control chart, shown as Figure 21 that presents 4 red points detected as out-of-control signals. Here, we found the K control. 政 治 大. chart using the two-dimensional kernel density estimation method has better detection ability.. 立. We next use the median control chart and IQR control chart to distinguish the. ‧ 國. 學. signals for whether it’s a location and/or dispersion problem shown as Figure 22. Figure. ‧. 22 illustrates that they have a dispersion problem on the second, fifth and seventh points. n. al. er. io. sit. y. Nat. but the eleventh point cannot be diagnosed as location/dispersion problem.. Ch. engchi. i n U. v. (Black points: in-control points; red points: out-of-control pints) Figure 20. The K control chart with. using the one-dimensional kernel density. estimation approach. 45.
(55) 政 治 大. 立. Figure 21. The K control chart with. 學. ‧ 國. (Black points: in-control points; red points: out-of-control pints) using the two-dimensional kernel density. estimation approach.. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. (1) Median control chart. i n U. v. (2) IQR control chart. Figure 22. Control charts diagnose the out-of-control location and/or dispersion.. 46.
(56) Table 20. The service times data from Yang and Arnold (2014). t 1. 0.88. 5.06. 11.59. 1.2. 0.89. 2 3 4 5 6 7 8 9 10 11 12 13 14 15. 3.82 13.4 5.16 3.2 32.27 3.68 3.14 1.4 3.89 10.88 30.85 0.54 8.4 5.1 16.8 8.77 8.36 3.55 7.76 1.81 1.11 0.24 9.57 0.66 1.15 2.34 0.57 8.94 4.21 8.73 11.44 2.89 19.49 1.2 8.01 15.08 7.43 4.31 6.14 10.37 2.33 1.97 13.89 0.3 3.21 11.32 9.9 4.39 10.5 0.03 12.76 2.41 7.41 1.67 3.7 4.31 12.89 17.96 2.78 3.21 1.12 12.61 4.23 7.71 1.05 1.11 0.22 3.53 0.81 0.41 5.81 6.29 3.46 2.66 4.02 10.95 1.59 2.89 1.61 1.3 2.58 18.65 10.77 18.23 1.36 1.92 0.12 11.08 8.85 3.99 4.32 21.52 0.63 8.54 3.37 6.94 3.44 3.37. 1.58 2.63 5.91 5.54 6.19 1.08 1.7 2.45 6.18 3.73 5.58 3.13 1.71 6.37. 2.72 7.71 9.17 3.94 8.26 7.19 11.69 6.58 7.48 0.07 4.27 14.08 10.74 1.46 3.57 3.33 2.33 6.92 0.08 2.55 0.55 4.1 3.38 6.34 1.77 1.94 1.28 12.83. 立. 5.45. 2.93. 6.11. 政 治 大. 3.82 6.29 10.88 30.85 9.9 3.99 1.59 0.24 12.76 11.44 3.2 3.53 0.57 18.23 3.82 7.43 0.12 3.37 1.12 12.61 1.59 13.89 3.89 5.16 11.32 4.02 0.57 8.01 5.81 12.76 2.41 1.15 3.53 0.81 11.59 12.89 8.73 10.88 2.89 18.65 10.95 0.41 2.89 0.63 0.12 0.22 4.02 10.95 8.01 16.8 1.05 1.3 3.2 2.34 0.81 4.32 4.21 17.96 5.06 0.22 4.02 3.99 8.01. 10 11 12 13 14 15. 12.89 0.88 7.71 7.71 2.89 1.36. v. 8.26 2.72 0.89 1.77 4.27 4.27 10.74 0.08 0.55. 11.44 7.41 1.12 1.81 4.32 5.06 3.55 8.85 10.95 18.23 0.12 2.58 1.12 2.33 4.23 0.12 30.85 7.76 1.81 3.14 2.41 11.32 32.27 8.4 1.97 3.46 11.32 0.54 10.95 4.23. 5.58 5.54 2.63 1.71 2.45 2.45. 0.89 14.08 2.33 6.58 4.27 3.33 2.72 14.08 11.69 12.83 2.33 6.34. sit. n. al. er. io. 8.77 8.77 7.43 9.57 1.05 0.63. 3.21. 1.71 2.45 1.08 6.19 5.91 3.13 1.08 3.13 5.91. Nat. 1 2 3 4 5 6 7 8 9. ‧. ‧ 國. 學. t. y. 0.78. Ch. engchi. 47. i n U. 4.1 6.92 0.07 6.58 3.33 7.71 4.1 1.46 3.33.
(57) Table 21. The new service times data from Yang and Arnold (2014). t 1. 3.54. 0.01. 1.33. 7.27. 5.52. 0.09. 1.84. 1.04. 2.91. 0.63. 2 3 4 5 6 7 8 9 10. 0.86 1.45 1.37 3 1.59 5.01 4.96 1.08 4.56. 1.61 0.19 0.14 2.46 3.88 1.85 0.55 0.65 0.44. 1.15 4.18 1.54 0.06 0.39 3.1 1.43 0.91 5.61. 0.96 0.18 1.58 1.8 0.54 1 4.12 0.88 2.79. 0.54 0.02 0.45 3.25 1.58 0.09 4.06 2.02 1.73. 3.05 0.7 6.01 2.13 1.7 1.16 1.42 2.88 2.46. 4.11 0.8 4.59 2.22 0.68 2.69 1.43 1.76 0.53. 0.63 0.97 1.74 1.37 1.25 2.79 0.86 2.87 1.73. 2.37 3.6 3.92 2.13 6.83 1.84 0.67 1.97 7.02. 0.05 2.94 4.82 0.25 0.31 2.62 0.13 0.62 2.13. 0.63 1.25 1.37 1.37 0.97. 6.83 7.02 3.6 1.97 7.02. 0.25 0.62 2.62 0.31 0.63. 0.63 0.63 2.87 2.79 1.74. 2.13 2.13 7.02 2.91 6.83. 0.13 4.82 4.82 0.62 4.82. al. 1.42 1.16 6.01 2.88. 4.11 4.11 4.59 4.11. 4.12 7.27 1 1.58 0.88. 4.06 0.45 0.45 0.54 1.73. 6.01 1.42 2.46 1.7 3.05. 2.69 0.53 2.22 1.84 1.43. Ch. engchi. 48. y. sit. ‧ 國. 0.91 1.54 1.33 0.91 0.91. 1.73 5.52 4.06 0.09. ‧. 0.14 2.46 0.44 0.01 0.44. 0.54 1 1 7.27. 學. 4.96 4.96 1.37 1.59 5.01. 4.18 0.39 1.43 1.43 1.54. n. 6 7 8 9 10. 立. 3.88 0.19 0.65 0.65 0.01. io. 5.01 1.59 1.59 3.54 5.01. Nat. 1 2 3 4 5. 政 治 大 0.88 4.06 3.05 4.59. er. t. i n U. v.
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