The Max EWMAMS Control Chart for Joint Monitoring of Process Mean and Variance with Individual Observations Ahmad Ostadsharif Memar, Ph.D. Candidate Department of Industrial Engineering, Sharif University of Technology P.O. Box 11155-9414 Azadi Ave., Tehran 1458889694 Iran Email:
[email protected] Seyed Taghi Akhavan Niaki1, Ph.D., Professor Department of Industrial Engineering, Sharif University of Technology P.O. Box 11155-9414 Azadi Ave., Tehran 1458889694 Iran Tel: (+9821) 66165740, Fax: (+9821) 66022702 Email:
[email protected]
Abstract A traditional approach to monitor both the location and the scale parameters of a quality characteristic is to use two separate control charts. These schemes have some difficulties in concurrent tracking and interpretation. To overcome these difficulties, some researchers have proposed schemes consisting only one chart. However, none of these schemes is designed to work with individual observations. In this research, an exponentially weighted moving average (EWMA)-based control chart that plots only one statistic at a time is proposed to simultaneously monitor the mean and variability with individual observations. The performance of the proposed scheme is compared with the ones of the two other existing combination charts by simulation. The results show that in general the proposed chart has a significantly better performance than the other combination charts. Key Words: Exponentially Weighted Moving Average; Exponentially Weighted Mean Squared Deviation; Simultaneous Monitoring; Individual Observations 1
Corresponding Author
1
1. Introduction Traditional scheme for joint monitoring of the process mean and variance that uses pairs of X and S / R chart, has some possible drawbacks. First, monitoring two separate charts and following their trends causes practical difficulties. Second, and the more important problem, is related to the effects of changes in one of these charts on the other one. As an example, while changes in the X chart have no impact on the S chart, a decrease (increase) in the variance of an unchanged process mean constricts (expands) the control limits of the X chart. In the literature of joint monitoring, a typical approach is to chart the mean and the scale parameter estimators in a same plot such as the works by White and Schroeder1 and Spring and Cheng2. Amin et al.3 employed another approach by plotting two separate exponentially weighted moving average (EWMA)-based statistics on the smallest and the largest observations of each sample in a single chart and named it the Max-Min EWMA control chart. This control scheme has also a diagnosing procedure. In the last two decades, another type of control schemes, namely the single control charts, have been proposed in which only one statistic used to detect both the mean and the variability shifts at the same time. Furthermore, there are few researches in the literature that employ Shewhart-based or CUSUM-based schemes to joint monitoring of the process mean and variance with a single control chart. The semicircle chart of Chao and Cheng4 and the Max chart of Chen and Cheng5 are the two of the Shewhart-based schemes. The VSSI-WLC scheme, which was proposed by Wu et al.6, is a weighted-loss-function-based CUSUM (WLC) scheme using variable sample sizes and sampling intervals (VSSI). Hawkins and Deng7 presented two new methods; one based on the generalized likelihood ratio (GLR) and the other based on the Fisher approach for testing simultaneous null hypothesis. They also proposed two CUSUM charts based on their GLR and Fisher statistics. 2
Many control schemes for joint monitoring of the process mean and dispersion are based on the EWMA procedure. Domangue and Patch8 presented omnibus EWMA, which is an EWMA on absolute value of standardized sample mean of the observations, to detect simultaneous changes in the process. However, this chart does not determine the source and direction of the process changes. Macgregor and Harris9 proposed two EWMA based charts, called exponentially weighted moving mean squared deviation (EWMS) and exponentially weighted moving variance (EWMV) control charts for controlling the process variability. The latter only monitors the shifts in the variance and hence is not a simultaneous monitoring method. However, while the former is designed to detect variability changes, it is sensitive to shifts in both mean and variance. Although the EWMS chart performs well in many cases, it confounds the process mean and variance shifts. The Max EWMA chart that is an extension of the Max chart and the EWMA-SC, which is an extension of the semicircle chart were also proposed by Chen et al.10, 11. The last two control schemes are able to determine the source and direction of the process shifts. For more details on the single control charts, see Cheng and Thaga12 who reviewed and categorized this type of control charting methods. Except the omnibus EWMA and EWMS control charts, none of the aforementioned charts can work with individual observations. The reason is that their control statistics consist of a function of the sample variance that is defined for more than one observation. It means that for these control charts the rational subgroups is always greater than one. The main assumption of the rational subgroups that was first introduced by Shewhart13, is that any shift in the process occurs between the samples rather than during the time a sample is being gathered. Therefore, any shift in the process affects a whole sample not a part of a sample. A recommended size for such a sample is n 4 or 5 (Montgomery14). In other words, if the interval time between two successive observations is equal to d then every d units of time a sample of size n should be collected. However, for at least two reasons this concept may not
3
always work well. First, there are some situations such as 100% inspection and very slow rate production, where the observations should be individually gathered (see Montgomery14 and Reynolds and Stoumbos15 for more details). Second, referring to Reynolds and Stoumbos15, 16, when the mean and standard deviation of a single quality characteristic are simultaneously monitored, even if the observations can be gathered according to rational subgroups concept with n 1 , a better performance can be obtained if one can use a control chart based on individual observations, n 1 . They also studied some combinations of X chart and two EWMA-based charts for mean and variance to monitor individual observations (see Reynolds and Stoumbos17 for more details). In another research, Reynolds and Stoumbos18 concluded that to obtain the best overall performance it is not necessary to use a Shewhart-type chart along with an EWMA/CUSUM type chart, but it is necessary to use an EWMA/CUSUM based on squared deviation from the target. Furthermore, based on Reynolds and Stoumbos19 if an EWMA chart for the mean and an EWMA chart based on squared deviation from the target are used in combination, there is no need to consider any adaptive feature. The objective of this research is to construct a new single variable control chart to detect both the mean and variance shifts of a process while the observations are individual. According to the results obtained by Reynolds and Stoumbos15,
16, 17, 18, 19
, the goal is to
combine two EWMA charts for mean and variance to get the best overall performance. To do this the maximum exponentially weighted moving average and mean squared deviation (Max EWMAMS) chart will be developed. The structure of the rest of the paper is as follows. In section 2 a brief background on the EWMA and EWMS charts that will be used later in the research, is given. Section 3 contains the development of the proposed joint monitoring scheme. In order to compare the performance of the proposed methodology with the ones of other joint monitoring charts, Monte Carlo simulation studies are performed in section 4 and 5 for the detection and
4
diagnosing ability, respectively. The effects of the subgroup size on the performance of the proposed chart are discussed in section 6. A numerical example is also given in section 7 to demonstrate the applicability of the proposed procedure. Finally, the conclusion and recommendations for future research come in section 8.
2. Backgrounds In this section the basis for some of the existing control charts that are used later in the paper, will be briefly described. The notations used for the EWMA-based chart are similar to those of Reynolds and Stoumbos18.
2.1 Notations and assumptions Consider a process of interest with the in control mean and variance denoted by 0 and 02 , respectively. It is assumed that the observations are independent and follow N 0 , 02 distribution. Observations are gathered in rational subgroup of size n and the jth observation at the kth sampling point is denoted by X kj .
2.2 The EWMA control chart The exponentially weighted moving average (EWMA) statistic at sampling point
k with smoothing parameter is defined as
EkX 1 EkX1 X k
,
E0X 0
(1)
5
X The mean and variance of this statistics for the kth sample are Ε E k 0 and
Var EkX
1 1 n2
2k
2 0
, respectively. Furthermore, the time varying and the
limiting control limits for EkX can be obtained as
CLk 0 hE X
1 1 n 2
2k
0 hE X
n 2
0 k
0
2.3 The EWMS control chart The exponentially weighted moving mean squared deviation (EWMS) control chart was proposed by Macgregor and Harris9 for individual observations, i.e., for n 1 . In the general case where n 1 , the statistic of this chart at the kth sample with smoothing parameter can be defined as follows: X2 k
E
1 E
X2 k 1
n
X
j 1
kj
0 n
2 2
E0X 02
,
(2)
It is easy to show that the mean and variance of this statistic at the kth sample are
2 2 2k 1 1 04 , respectively. Hence, the time Ε EkX 02 and Var EkX n2 2
2
varying and the limiting control limits for EkX can be defined as
CLk 02 h
EX
1 1 n 2 2
2
2k
02 h
2 0 k
EX
2
n 2
2
02
(3)
Using the results of Box20, Macgregor and Harris9 proposed an approximating 2
distribution for EkX when n 1 . Following their approach, it can be shown that for any value of n , E kX
2
02 is approximately distributed as 2 where n 2 (See Appendix 2
for more details). As a result, the limiting control limits for EkX can also be obtained as
6
LCL
UCL
2 2;
02
21 2;
02
Reynolds and Stoumbos17 presented two other EWMA control charts based on squared deviation from target, which are very similar to the EWMS scheme. In these charts, at sampling point k and for smoothing parameter define EkX
2
, max
X 2 ,min k
E
1 min E
Where the EkX while the E kX
2
2
h
2
EX
,max
2 ,min
X 2 ,min k 1
,
2 0
X
kj
j 1
X
2
n
j 1
n
0
kj
0
,
E0X
,
E0X
2
, max
02
(4)
02
(5)
2
n
2
,min
,max
statistic is designed to detect increasing shifts in the process variability
,min
statistic is sensitive to decrease in the process variance. The mean and the
variance of EkX EkX
n
1 max EkX1,max , 02 2
2
and E kX
2
and E kX
2
can be defined as (3) by replacing the coefficient h
,min
,min
2
,max
are the same as those of EkX and hence the control limits for EX
2
with h
EX
2 ,max
and
, respectively.
3. The Proposed Max EWMAMS Control Charts For joint monitoring of the process mean and variance in a single control chart, first, an estimator of each of these parameters should be found and then some transformations on these estimators that map them to a common distribution should be investigated. A possible approach is to set up two EWMA for each of the process mean and variance and then transform these EWMAs to follow standard normal distribution. This idea will be exploited for the new methodology.
7
Consider EkX
in equation (1). It is known that EkX
is distributed as a
2k N 0 , 1 1 02 distribution. Hence, the transformed variable U k n 2
in
equation (9) follows the standard normal distribution.
EkX 0
Uk
n2
1 1
(6)
2k
0
As pointed out in section 2.3, the quantity E kX
2
02 is approximately distributed as
2 where n 2 . Hence, the transformation given in equation (7) results in a variable with approximately standard normal distribution. E X 2 Vk k2 ; 0 1
(7)
Where a; d Pr d2 a
and 1 is the inverse distribution function of a standard
normal variable. Now the maximum of exponentially weighted moving average and exponentially weighted moving mean squared deviation (Max EWMAMS) at sampling point k is defined as
M k max U k , Vk
(8)
The definition in (8) ensures that M k 0 and note that the small values of M k is desirable. Hence, in order to monitor M k , only an upper control limit is needed. However, finding the distribution of M k is not straightforward. The dependency between X k term in the definition of EkX in equation (1) and
X n
j 1
kj
0
2
2
term in the definition of EkX in equation (2) leads
2
to the dependency of EkX and EkX . Therefore, despite the fact that U k and V k share a common distribution, they are dependent of each other and this will complicate finding both 8
the distribution and the moments of M k max U k , Vk . To resolve this complication, the upper control limit for M k such that when M k UCL the chart signals, can be obtained by simulation and is defined in equation (9).
UCL hM
(9)
In a diagnosing procedure, the following algorithm is proposed to determine the source and the direction of the shift: Case 1: U k UCL and Vk UCL . It shows that only the process mean experiences a shift. The shift is increasing if U k 0 and it is decreasing if U k 0 . Case 2: U k UCL and Vk UCL . It indicates that only the process variability experiences a shift. The shift is increasing if Vk 0 and it is a decreasing one if Vk 0 . Case 3: Both U k
and Vk are greater than UCL . This signal occurs due to a
simultaneous change in the process mean and variance. To determine the direction of change in each, the procedures described in the above case 1 and case 2 should be applied.
4. Performance Comparison In this section, the performance of the proposed Max EWMAMS chart is compared to the ones of some other existing schemes. One of the single control charts with very good performance is the Max EWMA chart of Chen et al.10. However, this scheme cannot be implemented for individual observations because it uses the sample variance in its control statistics. Hence, among other combination charts, two schemes proposed by Reynolds and Stoumbos17 were selected for the performance comparison study. The first scheme is the
9
combination of EkX and EkX
2
, max
for detecting changes in the process mean and increase in the
process variance simultaneously and the second one is the combination of EkX , EkX E kX
2
,min
2
, max
and
for detecting simultaneous changes in the process mean and variance. Both of these
combination charts work well with n 1 . The performance of the charts are measured in terms of the average time to signal (ATS) criterion; that is the average length of time to signal an out-of-control condition since the process monitoring was started. The in-control ATS for all of considered comparisons is selected to be ATS0 = 370. Several simulation runs are employed to compare the schemes. The comparisons were performed for 0.05, 0.1, 0.2, 0.3 , n = 1, 2, 5 and some out-of-control scenarios. The shifted process mean and variance are considered out 0 0 and out 0 , respectively and without loss of generality the in-control values of the process parameters were set to be
0 0
and
0 1.
For
0.0, 0.25, 0.5, 0.75, 1.0, 2.0
any and
set
of
and
n
all
combinations
of
0.1, 0.25, 0.5, 0.75, 1.0, 1.1, 1.25, 1.5, 2.0
were used. To find the control chart parameters (control limits), for each set of smoothing parameter and subgroup size, 100,000 simulations are first replicated for each chart. The associated computational error in finding each parameter was then estimated by the standard deviation of the reported ATS. The maximum of these computational errors was 1.3. The results are presented in Table (1).
Insert Table (1) about Here
10
Each out-of-control scenarios was replicated for 10,000 times. The results of comparison are shown in Tables (2)-(5) for the cases of 0.05, 0.1, 0.2, and 0.3 , respectively. Insert Table (2) about Here Insert Table (3) about Here Insert Table (4) about Here Insert Table (5) about Here
4.1 Comparison Results for the Mean Shifts The results in Tables (2)-(5) show that for all values of n , the proposed Max EWMAMS chart outperforms the EkX and EkX outperform the EkX , EkX
2
, max
and E kX
2
,min
2
, max
combination and these two charts
combination chart. For small values of the
smoothing parameter (i.e. 0.05 and 0.1), the performance of the Max EWMAMS is much better than the ones of the two combination charts. However, as the smoothing parameter increases the difference between ATSs of the proposed scheme and the two combination charts becomes smaller. Furthermore, for any value of when the size of the mean shift is small, the performance of all charts under consideration gets better as n increases. Nonetheless, for the large mean shifts, the case n 1 shows the best performance in terms of ATS.
4.2 Comparison Results for the Variability Shifts In limited variability shift scenarios the EkX , EkX
2
, max
combination chart possesses a
slightly better performances than the proposed Max EWMAMS scheme. These scenarios are
11
1. For the case 1.1 , and a. All values of n when 0.05 b. When 0.1 and 0.2 for n 2 and 5 c. For 0.3 with n 5 2. For the case 1.3 , When 0.2 and 0.3 with n 5
In all other cases of increase in variability, the proposed Max EWMAMS outperforms the
EkX , EkX
2
, max
combination scheme. Furthermore, the EkX , EkX
2
, max
and E kX
2
,min
combination
chart never has better performances than the other two charts when the process variance increases. For the cases of decrease in variability, as it is expected, the EkX , EkX
2
, max
combination
chart never detects any shifts. However, for all values of and n , the EkX , EkX E kX
2
,min
2
, max
and
combination scheme outperforms the Max EWMAMS chart. Furthermore, the
difference between their performances becomes smaller as n increases and/or decreases. Moreover, for smaller smoothing parameters and for moderate to large decreases in the variance, the performance of the charts are closer to each other and in some cases they have nearly equal performances within their errors. It should be noted that if any of the three charts detect a variance shift, when the size of the absolute shift from the target ( 0 1 ) is small, the performance of the chart gets better as n increases. Nonetheless, for the moderate to large sizes of absolute shifts from the target, the case n 1 is the best.
12
4.3 Comparison Results for Simultaneous Shifts For all values of the smoothing parameter when the mean change is greater than 0.5 (i.e. for moderate and large mean shifts), the Max EWMAMS chart detects the simultaneous shifts faster than the other two combination charts. In these cases the EkX , EkX combination scheme outperforms the EkX , EkX
2
, max
and E kX
2
,min
2
, max
combination chart. However,
when 0.25, 0.5 and 1 there are some cases that the EkX , EkX
2
, max
and E kX
2
,min
combination chart has better performance than the proposed chart. These cases are 1. When 0.05, 0.1 , the rational differences are negligible (except one case in which 0.1 , 0.25 and 0.75 ). 2. When 0.2, 0.3 , number of cases with significant difference in ATS becomes
greater and the EkX , EkX
2
, max
and E kX
2
,min
combination outperforms the Max EWMAMS
chart. For large smoothing parameter when the shift in the mean is low and the variance decreases, there are some cases that the Max EWMAMS cannot detect the simultaneous shifts while for the EkX , EkX
2
, max
and E kX
2
,min
combination scheme there is only one such
case. In general, while the Max EWMAMS chart has a better performance than the other two combination charts, the following notes on the performance of the proposed scheme need special attention: 1. The rational performance of the proposed chart improves as the smoothing parameter deceases. This fact is due to the approximation that is used in the transformation given in equation (7). As was mentioned before, it is assumed that the quantity E kX
2
02 approximately follows a 2 distribution with
13
n 2 degrees of freedom. The defined degree of freedom is a monotonic decreasing function of the smoothing parameter and as the smoothing parameter increases, the limits of 2 becomes more accurate. As a result, the control limits for lower values of is more accurate and this leads to a better rational performance of the Max EWMAMS chart for these values of smoothing parameter. 2. The degree of freedom is also a monotonic increasing function of the subgroup number. In other words, the performance of the chart gets better as n increases. However, this is only true for low and moderate shifts. For the large shifts, the chart usually signals very fast and detects a shift in some early samples. Hence, if the chart works with a given sample size of n , it cannot alarm until it uses a multiple of n observations. This can increase the ATS of the chart. For example, for 0.05 , 2.0 and 2.0 , the proposed chart signals in average at time 2.6, 3.3 and 5.5 for n = 1, 2, 5, respectively. It means that it uses in average 2.6 1 2.6 , 3.3 2 1.65 and 5.5 5 1.1 samples for n = 1, 2, 5, respectively. As a result, although the cases n = 2 and 5 use less samples than the case n = 1, they need more time to make a signal. More details on interpreting the impacts of n on the performance of the proposed chart will be discussed in section 6. 3. As it has been mentioned in Box20 and Macgregor and Harris9, the approximated distribution of 2 has less precision in lower bound than its upper bound. Due to this fact, the rational performance of the Max EWMAMS for decrease in the process variance is not as good as the one for increasing process variance.
14
5. Diagnosing Comparison
For all scenarios of the shifts in Tables (2)-(5), the diagnosing procedures of the schemes were also implemented. Table (6) shows the results for the case 0.05 and
n 1 . In this table for each scheme there are five columns named m, v+, v-, mv+, and mvthat represent the mean shift, the variance increase, the variance decrease, the combination of the mean shift-variance increase, and the combination of the mean shift-variance decrease, respectively. After each replication of an out-of-control scenario, the source of the shift was diagnosed and categorized in one of the m, v+, v-, mv+, and mv- columns. While all of the replications (10,000 times) were performed, the sum of each shift type was calculated and the corresponding correct diagnosing percentage was recorded in Table (6).
Insert Table (6) about Here
5.1 The Mean-Shifts Diagnosing Performances
The results in Table (6) show that for very low and very high mean shifts the correct diagnosis percentages of the proposed chart are slightly less than the ones for the other shifts. Furthermore, it can be seen that the two combination charts cannot effectively diagnose the large shifts. This may be due to the property of the EkX
2
, max
and E kX
2
,min
statistics
that react to the large mean shifts very fast (See Reynolds and Stoumbos18 for more details). Moreover, in all the cases of the mean shifts the Max EWMAMS chart diagnoses the shifts better than the other two combination charts.
5.2 The Variability-Shifts Diagnosing Performances
When the variance decreases, the EkX and EkX
2
, max
combination chart does not detect
and diagnose the shift at all. However, both the Max EWMAMS and the EkX , EkX 15
2
, max
and
E kX
2
,min
combination schemes diagnose decreasing variability in all of the times.
Furthermore, for smaller increases in the variance, both of the combination charts detect the direction of the shift better than the Max EWMAMS chart and as the size of the variance shift becomes greater the precision of the detection procedures of the combination charts increases. While this is not true for the proposed Max EWMAMS scheme, it has considerable false signals on the mean shifts of this case.
5.3 The Simultaneous-Shifts Diagnosing Performances
When the process experiences a simultaneous shift, none of the charts under consideration can effectively detect the shift and the performance of the simultaneous shift detection procedure gets slightly better as the size of the shift increases. In general, the charts do not signal a simultaneous shift but they signal either a mean or a variance shift. For the cases of simultaneous mean changes and variance decreases, the following results can be concluded for the charts under consideration: 1. For the Max EWMAMS chart:
For 0.25 or 0.5 with large decrease in variability, the chart detects only a decrease in the process variance.
For other cases, the chart detects only a change in the process mean. For the EkX and EkX
2
, max
combination scheme:
If this chart signals an alarm, it is diagnosed as a mean shift for 2.0 . If 0.25 there is a probability that the chart has a false signal containing either an increase in the variance or simultaneous mean and variance increase. The chance of this false alarm becomes larger as gets close to 1. 2. For the EkX , EkX
2
, max
and E kX
2
,min
chart:
16
For 0.25 or 0.5 with large decreases in variability, the chart detects only a decrease in the process variance.
When the mean shift is large, it signals only a mean-shift. However, there is a probability of false alarm involved.
For other cases, the chart detects only a change in the process mean. For the cases of simultaneous mean changes and variance increases, all of the three charts signal a mean shift, a variance increase, or simultaneous mean shift and variance increase. While the probability that the Max EWMAMS chart signals only a mean shift is higher than the ones of the other two combination charts, the probability of signaling a variance increase alarm for the combination charts is greater than the one for the proposed chart. As was mentioned in section 5.1, this can be due to the high sensitivity of the EkX
2
, max
and E kX
2
,min
statistics for the large mean shifts.
6. The Effect of the Sample Size n on the Performance of the Max EWMAMS Chart
Since the proposed MAX EWMAMS chart can be applied with any rational subgroup n, the question arises on possible advantages of choosing a special value of n. In this
section, we elaborate on this question. Based on the results in Tables (2)-(5), none of the n-values ends with the best performance in all shift cases. In general, as the size of shift becomes larger, the smaller value of n is preferred. To show this, consider the MAX EWMAMS scheme as a function of smoothing parameter and rational subgroup n, for convenience denoted by F , n from this point forward. At any sampling point k in a F , n scheme, n observations are collected
17
and according to equations (1) and (2) the statistics E kX and E kX
2
are generated. These
equations can be rewritten as follows.
E kX 1 E kX1
E kX 1 E kX1
2
2
n
X n j 1
X n
kj
n
j 1
kj
, 0
E 0X 0 2
,
(10)
2
E 0X 02
(11)
In these equations, the weight n is assigned to the sum of the same function of each of the newly n added observations. For example in a F ,1 scheme, the weight is assigned to a function of the last observation and in the F , 5 scheme, the weight 5 is assigned to a function of the last 5 observations. As a result, in a F , 5 scheme the last 5 observations get a weight equal to 5 5 5 5 5 in total and in a F ,1 scheme
the
total
weight
of
the
last
5
observations
is
1 1 1 1 1 1 5 . If is set to be 0.1 then the 2
4
4
last 5 observations of the F , 5 and F ,1 schemes receive the total weight of 0.1 and 0.40951, respectively. This assignment of different weights in F , 5 and F ,1 schemes affects their ATS. To overcome the above problem that is raised in an EWMA based schemes that can be represented by similar equations like (10) or (11), Reynolds and Stoumbos15, 16 presented an adjustment in smoothing parameter depending on the value of n. They used the n notation for smoothing parameter of an EWMA scheme with subgroup size of n and proposed that if a comparison between two EWMA schemes with subgroup sizes 1 and n is interested, the smoothing parameter should be selected in a way that the last collected observations of each
18
scheme receive the same weight in total. Thus, if a F 1 ,1 scheme is compared with a
F n , n scheme then the relation n 1 1 1 should be used. n
According to the above discussion, 1 0.05 , 2 1 1 0.05 0.09750 and 2
5 1 1 0.05 0.22622 are selected to compare F 0.05,1 scheme with F 0.9750, 2 5
and F 0.22622, 5 schemes. Using 100,000 replications, the control chart parameters of
F 0.9750, 2 and F 0.22622, 5 are obtained as 2.66903 and 2.52223, respectively. The results of the comparison are given in Table (7). Although the ATS values in the third column of this table should be identical to the ones in the third column of Table (2), the differences are due to computational errors of recalculating theses values.
Insert Table (7) about Here
Table (7) shows that the F 0.05,1 scheme has better overall performance than the
F 0.9750, 2 and F 0.22622, 5 schemes and the F 0.9750, 2 scheme has better overall performance than the F 0.22622, 5 scheme. Hence, it can be concluded that the smaller value of n results in better performance of the Max EWMA chart and the chart with individual observations is the best. This result is similar to the results reported by the Reynolds and Stoumbos15, 16 for their proposed EWMA based charts. The only criterion that should be considered in using small values of n is the corresponding costs. If a reduction of subgroup size leads to an increase in the operational and managerial costs, a trade-off between n and the cost should be considered and the smallest value of n with acceptable associated costs should be selected (see Reynolds and Stoumbos15, 16 for more details).
19
7. An Illustrative Example
To illustrate how to use the proposed Max EWMAMS chart, in this section a numerical example is given. Consider a quality characteristic that follows a normal distribution with in-control mean 0 0 , in-control standard deviation 0 1 , and that individual observations are gathered from an out-of-control process with out 0 0 and
out 0 . For the 300 generated observations the values of and are set as follows:
For the first 50 observations, 0 and 1 (in-control observations)
For observations 51 to 100, 1.0 and 1 (mean shift)
For observations 101 to 150, 0 and 1 (in-control observations)
For observations 151 to 200, 0.0 and 1.5 (variance increase)
For observations 201 to 250, 0 and 1 (in-control observations)
Observations 251 to 300, 0.5 and 1.5 (mean shift and variance increase) The data stream is used in the Max EWMAMS chart and both of the combination
charts as well. The in-control average time to signal, ATS0, the smoothing parameter and the subgroup size were set to 370, 0.1, and 1, respectively. Figure (1) depicts the reaction of the charts to these observations.
Insert Figure (1) about Here
Figure (1) shows that all of the three charts signal at observation 62. Applying the diagnosing
procedure
of
the
Max
EWMAMS
chart
reveals
that
U 62 3.4340 2.9161 UCL and V62 2.3004 2.9161 UCL . It means that a mean
20
increase
in
the EkX and EkX
the 2
, max
process
has
occurred.
Moreover,
the
statistic
EkX
of
combination scheme alarms at observation 62 as well and concludes that 2
there is a mean shift. Note that the EkX This is due to the property of the EkX
2
similar conclusion for the EkX and EkX
, max
, max
2
, max
also has an out-of-control signal at observation 71.
, which is sensitive to large mean shifts. There is a statistics of the EkX , EkX
2
, max
and E kX
2
,min
combination
chart. At the 163rd observation, the Max EWMAMS statistic exceeds its upper control limit. More investigation shows that at this point
U163 -0.7105 2.9161 UCL and
V163 2.9821>2.9161 UCL . Hence, a variance increase has occurred. At this time, EkX of the EkX and EkX
2
, max
2
, max
combination chart also signals and hence this scheme shows an
increase in the process variance. However, the EkX , EkX
2
, max
and E kX
2
,min
combination
scheme alarms later at observation 171. Since this alarm has been resulted from the EkX
2
, max
statistic, it indicates an increase in the process variability.
Finally, all of the three charts signal at observation 263. Following the diagnosing procedure
of
the
Max
U 263 0.90862.9161 UCL . It means that there is only a variance increase. At this time, both of the combination charts show an increase in the process variability with their EkX
2
, max
statistic. The EkX statistic of these schemes also alarms later (at observation 271). Note that the EkX
2
, max
statistic of the EkX , EkX
2
, max
and E kX
2
,min
combination chart never falls below its
LCL due to lack of decrease in the process variance.
21
8. Conclusion and Further Researches
In this paper, a new single control chart called the Max EWMAMS with the ability of working with individual observations was proposed. The control statistics of the chart was developed by taking the maximum between the absolute values of two standard normal variables that are estimators of the process mean and variance. The mean estimator is an EWMA statistic with normal distribution and can easily be standardized by a transformation equation (6). However, the variance estimator is a more complicated statistic. This statistic 2
is set up first by transforming the statistic EkX into a continuous uniform distribution by taking the approximated distribution function. This uniform variable is then mapped to the standard normal variable by implementing the inverse distribution function of the standard normal. The performance of the proposed chart was compared to the ones of two existing combination charts. The results of the comparison study showed that the Max EWMAMS chart generally detects the mean shifts, the variance increase, and the simultaneous changes of the process mean and variability faster than the other two charts. However, it is not able to detect a decrease in the variance as quick as the EkX , EkX
2
, max
and E kX
2
,min
combination
scheme, specially for large smoothing parameters. The effect of choosing different values of subgroup size on the performance of the proposed chart was then studied. It was concluded that regardless of managerial and operational costs, the best results are obtained with n = 1 (individual observations). If the costs associated with individual observations are not satisfactory, the smallest value of n with acceptable costs should be chosen. 2
The distribution of EkX statistic was obtained as a chi-square distribution with
n 2 degrees of freedom. The dependency of to causes reduction in the
22
chart performance for large values of the smoothing parameter. The approximation is also more inaccurate in the lower tail than in the upper tail. This is the reason that the chart signals longer than EkX , EkX
2
, max
and E kX
2
,min
combination scheme in some cases in which
decreases in process variance occur. These drawbacks can be improved by finding a more 2
accurate approximation for the distribution of EkX statistic in future research.
9. Acknowledgment
The authors are thankful for constructive comments of the referees that improved the presentation of the paper.
Appendix 2
2
Consider the E kX statistic in equation (2). It is straightforward to rewrite the E kX in terms of all gathered observations up to the sample k as follows
E
X2 k
1 E k
X2 0
k
1
k i
i 1
n
X
ij
0
2
(A-1)
n
j 1
For sufficiently large k, the term 1 E0X tends to be zero. Using this fact and dividing k
2
both sides of (A-1) by 02 we have
EkX
2
02
k
i 1
1 n
k i
n
j 1
X
ij
0
2
(A-2)
02
Since X kj is a N 0 , 02 random variable, X ij 0 02 is distributed as a chi-square 2
random variable with 1 degree of freedom and
X n
j 1
23
0 02 follows a n2 distribution 2
ij
(because the observations are independent). Hence, equation (A-2) can be considered as some of independent weighted chi-square random variables as follows EkX
2
02
1
k
k i
n
i 1
k
Bi ci Bi
(A-3)
i 1
where c i 1
k i
n ; i 1, 2,..., k are the coefficients and Bi s are independent
weighted chi-square random variables with n degree of freedom. Now, from the results of Box20 the quantity E kX
2
02 is approximately distributed as
g 2 where k
g
c i 1 k
c i 1
k i ci i k1 i ci2
2 i i
,
i i
k
nc n i
1
k i
k i
i 1
n
k
k
2 1
i 1
i 1
i 1
nci2 n
(A-4)
i 1
Knowing c i 1 k
2
n and i n , the following can then be obtained
1 1 1 1 k 1 1 1 1 i 1
2 k i
n2
k
k
k i
k
2 1
2 k i
n
i 1
2 1 1 n 1 2 1 2k
1 1 2 k n2 n2 Using these results, the equations in (A-4) reduce to
n2 g 1 n 2
As a result, g 1
,
1
2
n2
n2 and E kX
2
(A-5)
02 is approximately distributed as 2 with
n 2 .
24
Note that Box20 and Macgregor and Harris9 mentioned that the error of this type of approximation in its lower bound is more than the error in its upper bound and the results of simulations in Tables (2)-(5) are in favor of this finding.
References
1.
White EM, Schroeder R. A simultaneous control chart. Journal of Quality Technology 1987; 19: 1-10.
2.
Spiring FA, Cheng SW. An alternate variables control chart: The univariate and multivariate case. Statistica Sinica 1998; 8: 273–287.
3.
Amin RW, Wolf H, Besenfelder W. EWMA control charts for the smallest and largest observations. Journal of Quality Technology 1999; 31: 189-206.
4.
Chao MT, Cheng SW. Semicircle control chart for variables data. Quality Engineering 1996; 8: 441-446.
5.
Chen G, Cheng SW. Max-chart: Combining X-bar chart and S chart. Statistica Sinica 1998; 8: 263-271.
6.
Wu Z, Zhang S, Wang P. A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance. Quality & Reliability Engineering 2007, 23: 157-170.
7.
Hawkins DM, Deng Q. Combined charts for mean and variance information. Journal of Quality Technology 2004; 41: 415-425.
8.
Domangue R, Patch SC. Some omnibus exponentially moving average statistical process monitoring scheme. Technometrics 1991; 33: 299-313.
9.
MacGregor JF, Hariss TJ. The exponentially weighted moving variance. Journal of Quality Technology 1993; 25: 106-118.
25
10. Chen G, Cheng SW, Xie H. Monitoring process mean and variance with one EWMA chart. Journal of Quality Technology 2001; 33: 223-233. 11. Chen G, Cheng SW, Xie H. A new EWMA control chart for monitoring both location and dispersion. International Journal of Quality and Quantitative Management 2004; 1: 217-231. 12. Chen G, Thaga K. A new single variable control charts: an overview. Quality and Reliability Engineering International 2006; 22: 811-820. 13. Shewhart WA. Economic control of quality of manufacturing product. Van Nostrand: New York, 1931. 14. Montgomery DC. Introduction to statistical quality control (5th ed.). Wiley: New York, 2005. 15. Reynolds MR, Stoumbos ZG. Should observations be grouped for effective process monitoring? Journal of Quality Technology 2004; 36: 343-366. 16. Reynolds MR, Stoumbos ZG. Control Charts and the Efficient Allocation of Sampling Resources. Technometrics 2004; 46: 200-214. 17. Reynolds MR, Stoumbos ZG. Monitoring the process mean and variance using individual observations and variable sampling intervals. Journal of Quality Technology 2001; 33: 181-205. 18. Reynolds MR, Stoumbos ZG. Should exponentially weighted moving average and cumulative sum charts be used with Shewhart limits? Technometrics 2005; 47: 409424. 19. Reynolds MR, Stoumbos ZG. Comparison of Some Exponentially Weighted Moving Average Control Charts for Monitoring the Process Mean and Variance. Technometrics 2006; 48: 550-567.
26
20. Box GEP. Some theorems on quadratic forms applied in study of analysis of variance problems: Effect of inequality of variance in one-way classification. Annals of Mathematical Statistics 1954; 25: 290-302.
27
Table (1): The parameters of the compared control charts for ATS0 = 370.
n
1
2
5
2
, max
X 2 ,min
λ
hM
hE X
0.05
2.74166
2.739358
3.287557
2.902278
3.564138
2.158465
0.10
2.91628
2.92138
3.88097
3.07317
4.19948
1.98884
0.20
3.05227
3.05141
4.61311
3.19607
5.00804
1.69643
0.30
3.10635
3.09822
5.10301
3.24172
5.56051
1.47771
0.40
3.12335
3.11489
5.45187
3.25856
5.96817
1.30630
0.05
2.48514
2.456362
2.715727
2.635822
2.974887
2.161551
0.10
2.67521
2.67197
3.17245
2.83555
3.44940
2.11469
0.20
2.82202
2.82774
3.69204
2.97915
4.00613
1.93365
0.30
2.88074
2.89053
4.03064
3.03574
4.37631
1.76529
0.40
2.90855
2.91899
4.27406
3.06268
4.65210
1.61777
0.05
2.11737
2.026830
2.067924
2.228374
2.316079
1.968758
0.10
2.32443
2.28734
2.43378
2.47227
2.68985
2.07084
0.20
2.49751
2.48805
2.80686
2.65637
3.07485
2.04877
0.30
2.57323
2.57447
3.02829
2.73183
3.30728
1.97189
0.40
2.61031
2.61886
3.17924
2.77311
3.47363
1.88592
h
EX
2 ,max
28
, max
EkX & EkX
Comb.
EkX & EkX
2
Max EWMAMS
hE X
h
& Ek
EX
2 ,max
Comb.
h
EX
2 ,min
Table (2): The comparison results for ATS0 = 370 and 0.05 Max EWMAMS
EkX & EkX
2
, max
EkX & EkX
2
, max
X 2 ,min
& Ek
Comb.
n=1
n=2
n=5
n=1
n=2
n=5
0.00
1.00
371.0
373.5
375.9
370.6
369.1
375.9
372.6
374.2
371.8
0.25 0.50 0.75 1.00 1.50 2.00
1.00 1.00 1.00 1.00 1.00 1.00
82.9 25.9 12.7 7.8 4.0 2.5
69.9 23.4 11.9 7.6 4.2 2.8
60.7 21.5 11.7 8.1 5.6 5.0
89.4 31.5 18.1 12.8 8.0 5.7
79.1 32.0 19.4 13.9 8.6 5.9
80.0 37.2 24.1 18.0 11.7 8.1
104.2 35.3 19.8 13.7 8.5 6.0
90.6 35.4 21.2 15.1 9.3 6.3
91.3 41.2 26.6 19.7 12.8 8.8
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
15.0 15.9 22.7 66.3 130.8 41.1 15.6 6.3
18.0 19.6 26.0 57.5 128.6 41.6 17.1 7.5
25.0 25.1 33.3 62.1 124.9 44.8 20.5 10.3
128.0 44.9 18.6 8.5
122.6 44.0 19.4 8.8
121.2 48.3 23.1 11.4
15.0 15.9 22.0 56.9 165.6 53.5 20.6 9.1
18.0 18.9 25.3 54.6 154.5 51.1 21.3 9.4
25.0 25.0 32.8 61.7 149.6 55.3 25.5 12.3
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
16.1 17.6 26.3 64.7 59.1 30.9 14.4 6.2
20.0 21.2 28.9 52.1 54.6 31.5 15.5 7.2
25.0 28.6 35.0 49.0 51.6 33.4 18.7 10.0
165.0 63.4 35.1 17.5 8.5
219.7 130.7 101.4 60.9 35.7 18.2 8.6
87.5 88.2 88.3 86.8 65.3 40.8 22.0 11.2
16.1 17.5 25.2 63.3 76.3 40.5 19.3 9.0
19.5 20.7 28.4 56.8 71.6 40.7 20.1 9.2
25.0 27.8 36.1 63.0 76.0 46.2 24.4 12.1
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
23.4 25.5 28.1 29.0 23.1 18.2 11.4 5.8
25.3 24.5 24.3 24.6 22.0 18.3 12.4 6.8
20.4 20.7 20.8 21.2 21.0 19.3 14.8 9.4
42.5 41.3 38.2 35.1 28.8 22.9 14.8 8.1
32.7 32.9 33.2 33.1 29.9 24.1 15.6 8.2
35.6 36.1 36.6 37.3 35.3 29.1 19.2 10.8
22.6 25.5 32.0 37.2 32.2 25.5 16.2 8.5
25.6 27.9 32.1 35.5 33.2 26.9 17.1 8.8
33.8 34.8 38.2 40.8 39.3 32.6 21.2 11.6
0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
14.5 14.5 14.3 13.6 12.3 11.1 8.6 5.2
12.1 12.2 12.3 12.2 11.8 11.2 9.3 6.1
10.0 10.6 10.9 11.4 11.9 12.1 11.1 8.5
18.7 18.8 18.9 18.8 17.5 15.7 12.1 7.5
19.1 19.2 19.4 19.7 18.8 16.9 12.8 7.6
24.6 23.7 23.8 24.1 23.5 21.3 16.2 10.0
20.4 20.5 20.5 20.4 19.0 17.1 13.1 7.9
20.8 20.9 21.1 21.4 20.5 18.6 13.9 8.1
25.1 25.8 26.1 26.5 26.0 23.5 17.8 10.8
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
8.1 8.2 8.2 8.1 7.7 7.3 6.4 4.6
7.3 7.3 7.4 7.5 7.6 7.5 7.0 5.3
5.6 6.6 7.1 7.6 8.3 8.6 8.6 7.7
12.8 12.8 12.9 13.0 12.5 11.7 9.8 6.8
13.9 13.7 13.9 14.0 13.7 12.7 10.4 6.9
18.8 18.0 17.9 18.1 17.6 16.3 13.5 9.3
13.7 13.7 13.9 13.9 13.5 12.6 10.5 7.2
14.7 14.9 15.0 15.2 14.8 13.7 11.3 7.4
20.0 19.9 19.7 19.8 19.3 17.9 14.7 9.9
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
2.2 2.4 2.4 2.5 2.6 2.6 2.6 2.6
2.0 2.2 2.5 2.7 2.9 3.0 3.2 3.3
5.0 5.0 5.0 5.0 5.1 5.2 5.3 5.5
6.1 6.3 6.2 6.0 5.6 5.4 5.2 4.6
6.2 6.6 6.5 6.2 5.7 5.5 5.1 4.5
10.0 10.0 9.2 8.6 7.9 7.7 7.3 6.6
6.9 6.7 6.5 6.3 5.9 5.8 5.4 4.8
8.0 7.5 7.0 6.6 6.1 5.9 5.5 4.7
10.0 10.0 9.7 9.3 8.6 8.3 7.8 6.9
29
Comb.
n=1
n=2
n=5
Table (3): The comparison results for ATS0 = 370 and 0.1 Max EWMAMS
EkX & EkX
2
, max
EkX & EkX
2
, max
X 2 ,min
& Ek
Comb.
n=1
n=2
n=5
n=1
n=2
n=5
0.00
1.00
364.2
371.3
372.1
363.4
371.0
369.8
365.4
370.1
373.3
0.25 0.50 0.75 1.00 1.50 2.00
1.00 1.00 1.00 1.00 1.00 1.00
106.9 31.3 14.5 8.7 4.4 2.8
84.3 26.5 13.4 8.4 4.5 3.1
69.1 24.2 13.2 8.9 5.8 5.1
110.9 34.6 17.6 11.7 7.1 5.1
88.9 31.0 17.4 12.1 7.3 5.1
79.2 33.5 21.0 15.5 10.1 7.2
134.5 39.9 19.4 12.6 7.5 5.4
103.7 34.6 18.9 13.0 7.8 5.4
90.4 36.9 22.8 16.8 10.8 7.6
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
12.0 13.2 22.5 122.2 134.0 43.9 15.9 6.3
14.0 16.0 22.5 65.0 133.2 42.7 16.8 7.2
20.0 20.1 27.7 57.2 130.5 44.8 19.7 9.7
135.2 47.6 18.5 8.1
128.2 44.2 18.1 8.0
121.8 45.6 21.0 10.4
11.5 12.6 19.3 69.1 175.8 58.6 20.8 8.6
14.0 14.7 20.9 55.8 166.6 52.6 20.1 8.6
20.0 20.0 27.1 55.6 153.4 52.3 23.1 11.0
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
13.5 15.3 28.6 135.7 66.2 32.9 14.4 6.1
16.0 17.4 26.1 64.4 60.5 32.4 15.2 7.1
20.0 23.2 30.2 51.0 55.9 33.8 18.0 9.5
69.6 36.2 16.9 8.0
162.8 63.2 34.7 16.8 7.9
133.5 122.6 106.0 93.7 62.2 37.3 19.9 10.2
12.8 14.2 23.3 89.0 86.3 43.2 19.0 8.4
15.1 16.4 24.1 62.5 76.0 40.1 18.6 8.5
20.0 22.2 30.1 57.5 72.6 42.4 21.8 10.9
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
26.2 33.1 48.6 43.2 26.5 19.4 11.6 5.8
22.9 25.0 27.7 29.2 24.3 19.2 12.3 6.6
24.3 23.8 23.7 24.3 23.4 20.3 14.6 9.0
86.4 47.8 30.0 22.6 14.2 7.6
39.6 39.0 36.5 34.0 28.3 22.5 14.2 7.5
32.6 32.8 33.3 33.9 31.6 26.0 17.3 9.8
21.1 25.6 44.2 52.0 34.1 25.6 15.6 8.1
21.3 24.1 30.7 36.0 31.7 25.2 15.6 8.0
26.9 29.1 33.1 36.3 35.1 29.0 18.9 10.5
0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
21.3 20.6 18.6 16.6 13.7 11.8 8.8 5.2
14.3 14.3 14.3 14.0 13.1 12.0 9.5 5.9
11.6 12.2 12.6 12.9 13.3 13.0 11.4 8.2
22.9 22.5 20.9 19.3 16.7 14.7 11.2 7.0
17.4 17.5 17.8 17.8 16.8 15.1 11.5 6.9
20.0 20.4 20.7 21.0 20.5 18.7 14.4 9.2
28.2 26.7 23.8 21.4 18.5 16.2 12.2 7.4
19.1 19.2 19.4 19.4 18.4 16.5 12.4 7.4
21.5 22.1 22.5 22.8 22.4 20.4 15.6 9.7
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
9.7 9.7 9.6 9.2 8.4 7.8 6.6 4.6
8.2 8.3 8.4 8.5 8.3 8.0 7.2 5.3
9.1 8.2 8.1 8.4 9.1 9.1 8.9 7.5
12.1 12.2 12.2 12.1 11.3 10.6 9.0 6.4
11.9 11.9 12.0 12.2 11.7 11.0 9.2 6.2
15.0 15.1 15.4 15.6 15.1 14.2 12.0 8.5
13.1 13.2 13.3 13.1 12.3 11.5 9.7 6.7
12.5 12.8 12.9 13.1 12.7 11.9 10.0 6.6
15.0 15.8 16.5 16.8 16.4 15.4 13.0 9.0
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
2.8 2.7 2.7 2.7 2.8 2.8 2.8 2.7
2.1 2.5 2.8 2.9 3.1 3.2 3.3 3.4
5.0 5.0 5.0 5.0 5.1 5.2 5.4 5.5
5.2 5.3 5.3 5.2 5.0 4.9 4.7 4.3
6.0 5.9 5.5 5.3 5.0 4.8 4.6 4.1
10.0 9.3 8.2 7.6 7.1 6.9 6.7 6.2
5.8 5.7 5.6 5.5 5.3 5.2 4.9 4.5
6.0 6.0 5.8 5.6 5.3 5.1 4.9 4.3
10.0 9.8 8.9 8.1 7.5 7.3 7.0 6.4
30
Comb.
n=1
n=2
n=5
Table (4): The comparison results for ATS0 = 370 and 0.2 Max EWMAMS
EkX & EkX
2
, max
EkX & EkX
2
, max
X 2 ,min
& Ek
Comb.
n=1
n=2
n=5
n=1
n=2
n=5
0.00
1.00
367.5
365.1
373.2
364.4
366.8
371.1
367.3
371.0
374.1
0.25 0.50 0.75 1.00 1.50 2.00
1.00 1.00 1.00 1.00 1.00 1.00
149.4 43.8 18.5 10.2 4.8 3.0
110.9 32.2 15.1 9.3 4.9 3.2
82.0 27.1 14.4 9.6 6.0 5.1
150.7 45.9 20.4 12.0 6.5 4.6
113.3 34.3 17.0 11.0 6.3 4.4
86.2 31.7 18.6 13.4 8.8 6.5
185.1 56.1 23.7 13.3 7.0 4.9
136.9 40.0 18.9 11.9 6.8 4.7
101.3 35.2 20.1 14.3 9.3 6.9
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
10.0 12.1 39.1 137.6 47.0 17.0 6.3
12.0 12.9 22.1 120.2 137.3 45.2 16.9 7.0
15.0 16.1 23.3 60.3 136.3 45.2 18.8 9.1
142.1 51.4 19.4 7.8
135.7 46.8 18.0 7.5
126.7 44.7 19.4 9.5
9.0 10.1 19.8 103.1 184.5 65.2 22.5 8.5
10.0 11.6 18.2 69.0 178.8 57.9 20.4 8.0
15.0 15.1 22.1 54.0 162.4 53.0 21.4 10.1
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
12.9 17.1 67.9 80.0 36.1 15.2 6.2
13.3 15.1 28.2 134.8 69.0 34.4 15.5 6.8
16.4 19.5 26.4 58.0 61.5 34.4 17.3 9.0
82.7 39.4 17.5 7.8
70.8 35.9 16.6 7.4
282.9 135.6 63.4 35.8 18.3 9.4
10.4 12.3 26.5 146.3 106.5 48.8 20.2 8.4
12.0 13.1 22.2 89.9 88.2 42.9 18.6 7.9
15.0 17.9 25.0 58.5 75.8 41.1 20.1 9.9
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
113.8 33.3 21.8 12.2 5.8
25.3 32.6 48.9 43.7 27.6 20.6 12.4 6.5
24.0 24.2 26.2 28.1 25.4 21.1 14.2 8.6
116.9 35.4 24.2 14.1 7.3
86.2 47.2 29.5 22.4 13.5 7.0
33.0 33.5 33.7 33.2 29.5 23.8 15.6 9.0
96.6 48.2 102.1 130.7 42.5 28.3 15.9 7.9
19.6 24.2 43.4 52.5 34.2 25.4 15.0 7.5
21.8 24.6 30.2 35.2 33.0 26.5 16.9 9.5
0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
54.4 26.9 16.2 13.1 9.2 5.2
20.3 20.0 18.4 16.9 14.3 12.6 9.5 5.8
14.6 13.9 14.2 14.4 14.4 13.6 11.3 8.0
55.7 28.7 18.1 15.1 11.0 6.7
20.9 20.9 19.7 18.5 16.1 14.2 10.6 6.4
18.2 17.9 18.3 18.6 18.2 16.6 12.9 8.5
85.7 35.0 20.7 17.1 12.1 7.2
25.9 25.0 22.6 20.7 17.9 15.6 11.5 6.8
20.0 19.6 19.8 20.2 19.7 18.0 13.9 8.9
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
26.3 18.5 14.3 11.9 9.6 8.6 6.9 4.7
9.6 9.7 9.7 9.5 9.0 8.5 7.3 5.2
10.0 9.3 9.0 9.3 9.8 9.7 9.0 7.3
27.3 19.6 15.7 13.5 11.4 10.4 8.6 6.1
10.8 11.0 11.2 11.2 10.7 10.0 8.4 5.7
14.2 13.2 13.1 13.3 13.2 12.5 10.7 7.8
107.2 27.8 18.7 15.4 12.6 11.4 9.3 6.4
11.9 12.0 12.2 12.2 11.7 10.9 9.0 6.1
15.0 14.5 14.0 14.3 14.1 13.4 11.5 8.2
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
3.0 2.9 2.9 3.0 3.0 3.0 2.9 2.8
2.9 3.0 3.0 3.1 3.3 3.4 3.4 3.4
5.0 5.0 5.0 5.0 5.2 5.3 5.4 5.5
4.9 4.7 4.7 4.7 4.6 4.5 4.4 4.1
4.0 4.2 4.5 4.5 4.3 4.3 4.2 3.8
5.0 5.5 6.3 6.5 6.4 6.4 6.2 5.9
5.0 5.0 5.0 4.9 4.8 4.8 4.6 4.3
4.3 4.7 4.8 4.8 4.6 4.5 4.4 4.0
6.6 7.1 7.3 7.1 6.8 6.7 6.5 6.1
31
Comb.
n=1
n=2
n=5
Table (5): The comparison results for ATS0 = 370 and 0.3 Max EWMAMS
EkX & EkX
2
, max
EkX & EkX
2
, max
X 2 ,min
& Ek
Comb.
n=1
n=2
n=5
n=1
n=2
n=5
0.00
1.00
370.4
360.8
371.3
368.9
357.6
364.6
365.1
364.5
370.1
0.25 0.50 0.75 1.00 1.50 2.00
1.00 1.00 1.00 1.00 1.00 1.00
181.3 58.1 23.9 12.2 5.2 3.1
137.0 38.8 17.0 9.9 5.0 3.3
95.2 29.2 15.2 10.0 6.1 5.1
180.6 59.0 25.2 13.6 6.6 4.5
138.7 40.3 18.4 11.0 5.9 4.1
97.8 32.0 17.8 12.5 8.0 5.9
219.1 76.2 30.5 15.7 7.2 4.7
168.8 48.9 21.1 12.2 6.3 4.3
116.2 36.1 19.4 13.3 8.5 6.2
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
9.2 14.4 147.2 140.0 49.1 17.8 6.5
10.0 11.9 26.7 266.6 139.2 47.2 17.5 6.9
15.0 15.0 21.6 72.8 141.7 47.2 18.7 8.9
145.0 52.8 19.9 7.9
140.0 49.0 18.5 7.4
130.0 46.2 19.1 9.1
7.8 9.2 23.2 139.6 188.8 68.7 23.6 8.6
8.0 10.0 18.0 89.1 182.1 62.0 21.3 7.9
15.0 15.0 19.6 58.6 170.8 55.4 21.1 9.6
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
22.4 34.3 336.6 88.9 39.4 16.3 6.4
12.2 14.9 39.9 298.9 77.2 36.2 16.0 6.8
15.0 16.0 25.5 74.3 66.1 35.4 17.2 8.7
91.5 42.3 18.3 7.8
78.5 37.9 16.9 7.2
207.9 66.4 36.2 17.8 9.0
9.8 12.5 34.3 192.4 119.4 53.5 21.4 8.5
10.1 11.7 23.5 119.7 100.4 46.3 19.3 7.8
15.0 15.2 22.9 66.6 81.5 42.0 19.7 9.5
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
266.9 39.8 24.2 12.7 5.9
233.8 87.8 154.6 73.9 30.9 21.7 12.6 6.4
21.4 24.4 29.2 32.3 26.9 21.6 14.1 8.3
261.7 41.4 26.0 14.5 7.3
386.7 79.0 32.4 23.0 13.5 6.9
50.4 46.3 40.2 36.2 29.3 23.1 15.0 8.6
136.3 161.1 248.1 51.5 31.3 16.6 8.0
26.6 32.1 75.2 89.2 39.0 26.8 15.2 7.4
20.1 22.8 31.0 38.6 33.2 26.0 16.4 9.1
0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
339.6 49.5 19.3 14.6 9.7 5.4
67.2 28.6 21.2 15.6 13.3 9.7 5.9
15.0 15.2 15.5 15.4 14.8 13.9 11.3 7.8
325.6 50.0 20.9 16.2 11.3 6.7
70.8 29.7 22.3 16.8 14.4 10.5 6.3
16.5 17.2 17.8 18.0 17.3 15.7 12.3 8.2
68.7 24.7 18.8 12.6 7.2
177.9 39.0 26.4 19.1 16.1 11.5 6.7
19.4 19.0 19.4 19.6 18.8 17.1 13.3 8.5
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
32.4 17.2 11.0 9.4 7.3 4.7
11.6 11.8 11.3 10.6 9.6 8.8 7.3 5.2
10.0 9.6 9.5 9.7 10.0 9.8 9.0 7.1
32.8 18.3 12.5 10.9 8.7 6.0
12.1 12.4 12.0 11.7 10.6 9.7 8.1 5.6
10.3 11.3 11.9 12.4 12.3 11.6 10.0 7.5
49.6 22.6 14.2 12.1 9.6 6.4
14.2 14.4 13.7 13.0 11.6 10.7 8.8 5.9
12.9 12.7 12.8 13.2 13.1 12.5 10.8 7.8
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
3.0 3.1 3.1 3.1 3.1 3.1 3.0 2.8
3.4 3.2 3.1 3.2 3.4 3.4 3.4 3.4
5.0 5.0 5.0 5.0 5.2 5.3 5.4 5.5
4.4 4.5 4.6 4.5 4.5 4.4 4.3 4.0
4.0 4.0 4.1 4.1 4.0 4.0 3.9 3.6
5.0 5.0 5.2 5.6 5.9 5.9 5.9 5.7
4.9 4.7 4.8 4.8 4.7 4.6 4.5 4.2
4.0 4.1 4.3 4.3 4.3 4.2 4.1 3.8
5.0 5.0 5.5 5.9 6.2 6.2 6.1 5.9
32
Comb.
n=1
n=2
n=5
Table (6): The diagnosing percentage of the charts for ATS0 = 370, 0.05 and n = 1
Max EWMAMS
0.25 0.50 0.75 1.00 1.50 2.00
1.00 1.00 1.00 1.00 1.00 1.00
89 95 96 96 93 88
7 2 1 1 0 0
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
0 0 0 1 35 25 26 31
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
m
+
-
v
v
+
EkX & EkX -
m
+
mv
mv
2 0 0 0 0 0
2 2 3 4 7 12
0 0 0 0 0 0
84 90 88 80 51 18
12 6 6 8 21 45
0 0 0 0 61 70 65 48
100 100 100 99 0 0 0 0
0 0 0 0 3 5 9 21
0 0 0 0 0 0 0 0
26 12 5 1
0 0 1 36 69 41 31 31
0 0 0 0 26 51 58 48
100 100 99 64 0 0 0 0
0 0 0 0 5 8 11 22
0 0 0 0 0 0 0 0
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
0 20 67 95 85 63 41 33
0 0 0 0 8 25 42 42
100 79 33 5 0 0 0 0
0 0 0 0 6 13 17 25
0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
100 100 100 100 90 74 52 37
0 0 0 0 4 11 26 34
0 0 0 0 0 0 0 0
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
100 100 100 100 91 80 60 39
0 0 0 0 2 6 16 27
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
0.10 0.25 0.50 0.75 1.10 1.25 1.50 2.00
100 100 100 96 83 77 64 46
0 0 0 0 0 1 2 7
v
v
, max
+
EkX & EkX
-
m
+
mv
mv
0 0 0 0 0 0
3 4 7 12 28 37
0 0 0 0 0 0
82 92 90 83 54 19
10 5 4 7 19 44
71 84 91 94
0 0 0 0
3 4 5 5
0 0 0 0
0 0 0 0 24 11 4 1
100 59 26 8 1
0 35 66 85 93
0 0 0 0 0
0 6 8 7 6
0 0 0 0 0
0 0 0 0 0 0 0 0
100 100 100 100 74 43 13 2
0 0 0 0 17 43 74 90
0 0 0 0 0 0 0 0
0 0 0 0 9 14 13 8
0 0 0 0 7 14 22 30
0 0 0 0 0 0 0 0
100 100 100 100 74 48 18 2
0 0 0 0 13 33 63 86
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 8 15 24 34
0 0 0 0 0 0 0 0
100 100 100 98 67 45 18 3
0 0 0 0 15 31 57 83
0 0 0 0 0 0 0 0
0 0 0 4 17 23 34 48
0 0 0 0 0 0 0 0
91 63 42 28 15 11 6 2
0 3 18 32 50 56 65 77
33
-
2
2
, max
-
v
v
X 2 ,min
& Ek +
-
mv
mv
6 0 0 0 0 0
2 3 5 10 27 37
0 0 0 0 0 0
0 0 0 0 71 86 92 95
100 100 100 100 3 0 0 0
0 0 0 0 3 4 4 4
0 0 0 0 0 0 0 0
0 0 0 21 61 26 7 1
0 0 0 0 32 66 86 94
100 100 100 79 1 0 0 0
0 0 0 0 6 8 7 5
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 5 49 90 77 45 13 2
0 0 0 0 15 41 75 91
100 95 51 10 0 0 0 0
0 0 0 0 8 14 12 7
0 0 0 0 0 0 0 0
0 0 0 0 12 19 19 11
0 0 0 0 0 0 0 0
100 100 100 100 77 51 19 2
0 0 0 0 12 31 63 87
0 0 0 0 0 0 0 0
0 0 0 0 11 18 18 11
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1 18 25 25 14
0 0 0 0 0 0 0 0
100 100 100 99 71 48 19 3
0 0 0 0 13 29 57 84
0 0 0 0 0 0 0 0
0 0 0 1 17 24 24 13
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
9 34 39 40 35 33 29 21
0 0 0 0 0 0 0 0
71 67 46 31 16 12 6 1
0 1 14 29 49 56 65 79
0 0 0 0 0 0 0 0
29 33 40 39 35 33 28 20
0 0 0 0 0 0 0 0
Table (7): The comparison results of F 0.05,1 scheme with F 0.9750, 2 and F 0.22622, 5
schemes for ATS0 = 370 F 0.05,1
F 0.9750, 2
F 0.22622, 5
0.0
1.0
0.05 370.0
0.2 0.5 0.7 1.0 1.5 2.0
1.0 1.0 1.0 1.0 1.0 1.0
82.4 25.7 12.7 7.8 4.0 2.5
83.8 26.3 13.4 8.3 4.5 3.0
84.9 27.6 14.7 9.7 6.0 5.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.2 0.5 0.7 1.1 1.2 1.5 2.0
15.0 15.8 22.6 66.5 129.3 41.1 15.7 6.3
14.0 16.1 22.5 65.0 132.1 42.5 16.7 7.2
15.0 15.2 22.7 63.0 137.3 45.1 18.7 9.1
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.1 0.2 0.5 0.7 1.1 1.2 1.5 2.0
16.1 17.6 26.3 64.7 58.4 31.2 14.4 6.2
16.0 17.6 26.1 63.0 59.6 32.3 15.4 7.0
15.0 18.1 25.9 60.8 62.5 34.8 17.3 8.9
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.1 0.2 0.5 0.7 1.1 1.2 1.5 2.0
23.4 25.5 28.1 29.0 23.3 18.3 11.4 5.8
23.0 24.9 27.6 29.0 24.2 19.1 12.3 6.7
23.0 24.1 26.7 29.2 25.8 21.3 14.2 8.5
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.1 0.2 0.5 0.7 1.1 1.2 1.5 2.0
14.4 14.5 14.2 13.6 12.2 11.1 8.6 5.2
14.2 14.3 14.2 13.9 12.9 11.9 9.4 6.0
14.8 14.2 14.5 14.6 14.5 13.6 11.3 7.9
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.1 0.2 0.5 0.7 1.1 1.2 1.5 2.0
8.1 8.2 8.2 8.1 7.6 7.3 6.4 4.5
8.1 8.3 8.4 8.4 8.3 8.0 7.2 5.4
10.0 9.4 9.1 9.4 9.8 9.7 9.0 7.3
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
0.1 0.2 0.5 0.7 1.1 1.2 1.5 2.0
2.2 2.4 2.4 2.5 2.6 2.6 2.6 2.6
2.1 2.5 2.8 2.9 3.1 3.2 3.3 3.3
5.0 5.0 5.0 5.0 5.2 5.3 5.4 5.5
0.0975 369.0
34
0.22622 365.4
2.9163
(a)
0
62
163
263
0.6702
(b ) −0.6702 1
62
(b ) 2
271
2.2592 0
71
163
263
0.705 (c ) −0.705 1
62
271
2.3625
(c ) 2
(c ) 3
0
71
171
263
0.3547 50
100
150
200
250
Observation No
Figure (1): Plots of data stream for ATS0 = 370, 0.05 and n = 1 for 2 (a) for the Max EWMAMS, (b1) and (b2) for the EkX , EkX ,max combination, and
(c1), (c2) and (c3) for the EkX , EkX
35
2
, max
, E kX
2
,min
combination
300
