國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
30
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 values from each set. We calculate IQR by Therefore, we have m observations of Median and IQR.
Second, we get m observations of the kernel density value ( ) using the two-dimension kernel density estimation method in equation (11) with m pair observations of Median and IQR. That is,
Moreover, we use these to construct the kernel control chart. We follow a similar approach as in section 4.1 to find the control limit (LCL) for the K control chart. We also construct the median control chart and IQR control chart for diagnosis.
5. Performance Comparison
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‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
31
symmetric t distribution with the in-control distribution versus the 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 versus the out-of-control distribution ); 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 distributions. With the data we construct the proposed control region with different as 0.0054, 0.0027or 0.002. Thus for scenario (I), the results are presented in Table 10 and Table 11. We note from Table 10 that the proposed control region clearly outperforms 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 method is especially more effective than the other three charts. As location parameter 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 region is only larger than the NLE chart when n is 5 and (
0.25,
1.1) except 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.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
32
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 control regions of the kernel density estimation methods perform better than the NLE chart. In short, using the two-dimensional kernel density estimation method is more effective than the NLE chart in detecting the process scale when the process distribution is a heavy-tailed symmetric t distribution.
Table 15 presents the results of scenarios (V). When n is 15 and location shift is larger than 0.25, the control regions have a better performance than the NLE chart.
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 more effective than the control regions.
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 distribution, the control regions do not perform better than the NLE chart.
‧
Table 10. Performance comparisons between the control region and SL chart for data from Normal distribution with
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
34
Table 11. Performance comparisons between the control region and NLE chart for data from Normal distribution
Control region of the median and IQR NLE
n=5 n=15 λ
True CR
One-dimensional kernel approach
Two-dimensional kernel approach
True CR
One-dimensional kernel approach
Two-dimensional 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 2.0 1.1 1.2 1.1 4.9
2.00 1.80 1.4 1.5 1.4 1.0 1.0 1.0 3.1
3.00 2.00 1.1 1.1 1.1 1.0 1.0 1.0 1.6
4.00 3.00 1.0 1.0 1.0 1.0 1.0 1.0 0.9
‧
Table 12. Performance comparisons between the control region and SL chart for data from Laplace distribution with .
‧
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 n=15 λ
Table 14. Performance comparisons between the control region and NLE chart for data from with (scale shift)
Control region of the median and IQR NLE
n=5 n=15 λ
‧
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 NLE
n=5 n=15 λ
Table 16. Performance comparisons between the control region and NLE chart for data from distribution. (scale shift)
Control region of the median and IQR NLE
n=5 n=15 λ
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
38