Monitoring Multi-Attribute Processes Based on NORTA Inverse Transformed Vectors

Abstract

Although multivariate statistical process control has been receiving a well-deserved attention in the literature, little work has been done to deal with multi-attribute processes. While by the NORTA algorithm one can generate an arbitrary multi-dimensional random vector by transforming a multi-dimensional standard normal vector, in this article, using inverse transformation method, we initially transform a multi-attribute random vector so that the marginal probability distributions associated with the transformed random variables are approximately normal. Then, we estimate the covariance matrix of the transformed vector via simulation. Finally, we apply the well-known T2 control chart to the transformed vector. We use some simulation experiments to illustrate the proposed method and to compare its performance with that of the deleted-Y method. The results show that the proposed method works better than the deleted-Y method in terms of the out-of-control average run length criterion.

Full text

Monitoring Multi-Attribute Processes Based on NORTA Inverse Transformed Vectors
Seyed Taghi Akhavan Niaki, Ph.D., ProfessorF
1
Department of Industrial Engineering, Sharif University of Technology
Babak Abbasi, Ph.D.
Department of Industrial Engineering, Sharif University of Technology
Abstract
Although multivariate statistical process control has been receiving a well-deserved attention in
the literature, little work has been done to deal with multi-attribute processes. While by the
NORTA algorithm one can generate an arbitrary multidimensional random vector by
transforming a multidimensional standard normal vector, in this paper, using inverse
transformation method, we initially transform a multi-attribute random vector so that the
marginal probability distributions associated with the transformed random variables are
approximately normal. Then, we estimate the covariance matrix of the transformed vector via
simulation. Finally, we apply the well-known 2
T
control chart to the transformed vector. We use
some simulation experiments to illustrate the proposed method and to compare its performance
with that of the deleted-Y method. The results show that the proposed method works better than
the deleted-Y method in terms of the out-of-control average run length criterion.
Key words
Quality control; Multivariate Statistics; Multi-attribute control chart; Normal Transformation;
NORTA method
1 Seyed Taghi Akhavan Niaki, Corresponding Author
P.O. Box 11155-9414 Azadi Ave., Tehran, Iran
Phone: (+9821) 66165740, Fax: (+9821) 66022702, Email: HNiaki@Sharif.edu

2
1. Introduction and Literature Review
Statistical control charts, in general, consist of the variable and attribute control charts, for which
researchers have developed various methodologies. When the number of quality characteristics,
in the form of variables or attributes, exceeds unity and there exists a non-zero correlation
between them, then we are dealing with either a multi-variable or multi-attribute quality control
problem. In these problems, if the correlated quality characteristics are monitored separately,
there will be some error associated with the out-of-control detection procedure.
Despite the fact that multi-attribute monitoring has many applications, almost all researchers
have focused on the first category of control charting and only a few methods have been
proposed to monitor multi-attribute processes (see for example Woodall (1997) and Skinner et al.
(2006)). Patel (1973) proposed a Hotelling-type 2

chart to monitor observations from
multivariate distributions in which the marginals are Binomial or Poisson (for time independent
and time dependent samples) and the correlated attributes were monitored. Lu et al. (1998)
proposed a Shewhart-type multivariate np-chart (MNP chart) for dealing with multivariate
Binomial observations, in which the weighted sum of non-conforming counts of each quality
characteristic was defined as X statistic. They showed that this statistic reduces type II errors
better than individual np charts since the correlation of attributes is taken into account. They also
proposed an approach to identify which quality characteristic was the major contributor to an
out-of-control signal. However, in their research there was no discussion on the average run
length (ARL) of the MNP chart and the distribution of the statistic used in this chart. Wu et al.
(2006) proposed an algorithm for the optimization design of the np control chart with
curtailment. Based on several performance studies they showed that the optimal np chart with
curtailment perform better than the conventional np charts. Jolayemi (2000) developed a model
for an optimal design of multi-attribute control charts for processes with multiple assignable

3
causes. The development of the control chart of this research is based on the J-approximation
(Jolayemi, 1994), which sums up the number of defectives for all attributes and approximates
this statistic to a Binomial distribution. Then, the traditional np chart was used to monitor the
independent attributes. When the proportions in each quality category are known or estimated,
Marcucci (1985) used a Multinomial distribution to develop a control chart. Since not many
multi-attribute processes follow multinomial distributions, this method may not be widely
applicable. Larpkiattaworn (2003) proposed a back propagation neural network (BPNN) for two-
attribute processes for the case of bivariate Binomial and bivariate Poisson. He detected the out-
of-control condition by an artificial neural network where the output was set to one if the process
was under control and zero otherwise. He also discussed different correlation values between the
two variables and gave some suggestions on using the three-attribute control charts ( 2

, MNP,
and BPNN method). Assuming a multinomial distribution for multi-attribute processes, Gadre
and Rattihalli (2005) used a multi-attribute p (Mp) control chart to monitor a change in the values
of the parameters of the distribution. In their approach, the magnitudes of the intended change in
each parameter must be defined in advance.
Skinner et al. (2006) proposed a new statistic, called deleted-Y, to be computed for each variable.
They applied a k-standard deviations Shewhart-type control chart for each statistic for which the
k-value was obtained by simulation to ascertain a given in-control average run length criterion
for all univariate charts. One of the disadvantages of their method is that only the instances of
process deterioration (not improvement) can be detected. Furthermore, finding the upper control
limits (UCL) of the individual probabilistic control charts is a time-consuming process.
In a more recent research to monitor the proportions of non-conforming items in multi-binomial
distributions, Niaki and Abbasi (2007) proposed a skewness reduction approach and by

4
simulation experiments showed that, in term of the out-of-control criterion, their approach
performs better than the other competing methods.
In this paper, we propose a 2
T
control chart, based on the Patel’s (1973) method, to monitor the
number of defects in multi-attribute processes. However, instead of taking large samples and
using the central limit theorem to show that the limiting marginal distributions of the random
variables in the multivariate Binomial and multivariate Poisson distributions are Normal, we first
propose a data transformation technique and then employ the 2
T
control chart for the
transformed data. The goal is not just to detect process deteriorations but to monitor process
improvements. We note that this research is intended to propose a new method in phase two of
the control charting procedures.
Section 2 proposes a normalizing transformation method, while in section 3 we develop the new
2
Tmulti-attribute control charts based on the transformed data and evaluate its performance
using some numerical examples. In section 4, we provide some simulation experiments to
facilitate for good understanding of the proposed method and to evaluate and compare its
performance with the performance of the method proposed by Skinner et al. (2006). The
conclusion and recommendations for future research come in section 5.
2. The Proposed Normalizing Transformation
Common attribute control charts that are usually used in industries are of Shewhart type. For
these charts, we assume that the quality characteristic of interest follows a normal distribution, if
an appropriate sample size is used. Usually, attribute characteristics have Binomial, Poisson, or
Geometric distribution. Therefore, the assumption of normality causes two problems. The first
and most important problem is that these distributions are usually skewed, as opposed to the

5
symmetric nature of the normal distribution. The second problem arises from the discrete nature
of these distributions.
There are two approaches to reduce skewness in univariate attribute control charts: I) adding
some correction value to the control limits based on the value of the skewness, and II) applying a
normalizing transformation technique. Since using normal data in statistical process control has
many advantages, most researchers prefer to use the second approach.
To solve the skewness problem in the design and application of attribute control charts, some
transformation, such as the square root, inverse, arcsin, and parabolic inverse have been
proposed in the literature (see for example Ryan (1989), Ryan and Schwetman (1997), Box and
Cox (1964) and Johnson and Kotz (1969)). Quesembery (1995) proposed Q-transformation, and
Xie et al. (2000) proposed double square root transformation for Geometric distribution.
Furthermore, Liu et al. (2007) proposed a new exponentially weighted moving average (EWMA)
chart to monitor exponentially-distributed time-between-events (TBE) data. Their chart is based
on transforming the TBE data to approximate normal using the double square root
transformation. Hamasaki and Kim (2007) proposed a Box and Cox power-transformation to
confined and censored non normal responses in regression and discussed an application of the
approach to discrete and ordered categorical responses. Headrick et al. (2008) derived a power-
method-transformation-based methodology to simulate controlled correlation structures between
non normal (a) variates, (b) ranks, and (c) variates with ranks.
Based on the NORmal-To-Anything (NORTA) method, we propose an inverse transformation
technique to transform multi-attribute data to a shape close to a multivariate normal distribution.
The NORTA transformation detailed by Cairo and Nelson (1997) is based on research works by
Marida (1970) and Li and Hammond (1975). More recent references for the NORTA method

6
include Law and Kelton (2000), Chen (2001), Ghosh and Henderson (2002 & 2003) and Niaki
and Abbasi (2006 & 2008).
The goal of the NORTA algorithm is to generate a k-dimensional random vector X with the
following properties:
 ~ , 1,2,...,
i
iX
X
Fi k, where each i
X
Fis an arbitrary cumulative distribution function
(cdf); and
 Corr[ ] X
X=Σ, where X
Σ is given.
We generate the vector X by a transformation of a k-dimensional standard multivariate normal
(MVN) vector 12
( , ,..., )T
k
Z
ZZZwith correlation matrix Z
Σ. Specifically, the NORTA vector
X is:
1
2
1
1
1
2
1
[()]
[()]
[( )]
k
X
X
Xk
Fz
Fz
Fz













X (1)
where  is the cdf of a univariate standard normal and


uxFxuF XX 
)(:inf)(
1denotes the
inverse cdf.
The transformation 1[(.)]
i
X
F ensures that i
X
has the desired marginal distribution i
X
F.
Therefore, the general problem is to select the correlation matrix Z
Σ that gives the desired
correlation matrix X
Σ.
Non-diagonal elements of Z
Σ,(,), ,
Ziji j


show the correlation between i
Z
and j
Z
. Similarly,
(, ), ,
Xiji j

 denotes the correlation between i
X
and j
X
in X
Σ. That is;


11
(, ) [ , ] [ ( )], [ ( )]
ij
XijXiXj
ijCorrXXCorrFzFz ij


 (2)
Since:

7
()()()
(, ) () ( )
ij i j
ij ij
EXX EX EX
Corr X X Var X Var X

 (3)
and ( )
i
EX ,()
j
E
X,()
i
Var X and ()
j
Var X are fixed by i
X
Fand j
X
F, and since (, )
T
ij
Z
Z has a
standard bivariate normal distribution with ( , ) ( , )
ij Z
Corr Z Z i j


, then we have :

11 11
(, )
( ) [()] [( )] [()] [( )] (, )
ij ijz
ij XiX j XiX jijijij
EXXEFzFz FzFz zzdzdz



 
 
   
 (4)
where (, )
(, )
zij i j
zz


is the standard bivariate normal probability density function (pdf) with
correlation ( , )
Zij

.
In order to generate a k-dimensional random vector by the NORTA algorithm we need to solve
equation (4) for each pair of the variables. Hence, we need to solve (1)/2kk complicated
equations that, for many marginal distributions, are usually unsolvable by analytical methods.
Cario and Nelson (1997) present some theorems that describe the properties of Z
Σ which are
helpful in solving equation (4). Nevertheless, in this research, we propose an inverse
transformation formula for transforming a vector of multi-attribute variables to new variables
with approximate multi-variate normal distribution. The formula is given by

12
11 1
12 1 2
, ,..., ( ( )), ( ( )),..., ( ( ))
k
T
T
kXX Xk
YY Y F x F x F x
 



Y (5)
In other words, based on available historical data from an in-control process, we first estimate
the average number of different defect types along with the correlations between each pair. Then,
assuming marginal Poisson distributions and using equation (5), we transform the original vector
Xto the new one Yand estimate the correlation matrix of the transformed vector Y
Sby the
method of moments.

8
3. 2
TMulti-Attribute Control Charts Based on Transformed Data
A 2
T
control chart, due to its excellent performance in multivariable quality control
environments, may perform well in multi-attribute processes in which the attributes are
transformed to multi-normal variables. However, to avoid the problems in Patel (1973), we first
transform the vector of the original quality attributes to a new attribute vector with approximate
multivariate normal distribution using equation (5). The covariance matrix of the transformed
variables is then estimated using the method of moments. Finally, we determine the control
limits for our multi-variate control chart. If the plotted points


21
T
T
 
Y
YYS YY (where
12
( , ,..., )T
k
YY YY) fall within the control limits, the process is in control. Otherwise, it is out of
control. To detect the attribute(s) causing out-of-control condition, we may use some appropriate
techniques like Murphy (1987), Mason et al. (1997), and Niaki & Abbasi (2005).
The procedure for detecting mean shifts of number of defects in multi-attribute processes is
performed in two phases as follows:
Phase I
1. From historical data at hand (or generated data from estimated distributions), estimate the
mean of the number of each defect and the correlation between each pair and form the
estimated mean vector and the correlation matrix of the original variables.
2. Assuming a Poisson distribution for each defect type and using either equation (4) or the
transformed historical vector using equation (5), estimate the correlation matrix of the
transformed normal variables


Y
S.
Phase II
3. For each sample taken, count the number of different defect types and arrange the results
in a sample vector

X.

9
4. Apply equation (5) to transform the sample vector X to the transformed vector Y.
5. Employ the Hotelling T2control chart to monitor Ywhere the correlation matrix is Y
S.
Figure (1) shows the algorithm.
Insert Figure (1) about here
In the next section, numerical examples are presented to demonstrate and evaluate the
performance of the proposed normalizing transformation method.
3.1. Numerical Examples
Three different numerical examples are employed in this section. The differences result from the
magnitude of the means of quality characteristics. Examples 1, 2, and 3 contain Poisson
parameters with medium, large, and small sizes, respectively.
For the first example, suppose

12
,T
XXX, where )5(~),4(~ 21 PoissonXPoissonX and
covariance of 1
X
and 2
X
is equal to 2. In the second example let

12
,T
XXX, where
12
~(8),~(8)
X
Poisson X Poisson , and 12
(, )4Cov X X

. Finally, for the third example suppose

12
,T
XXX, where 12
~(3), ~(4),X Poisson X Poisson and 12
(, )1Cov X X . In order to
estimate the mean vector and the covariance matrix of the transformed process, using the
NORTA algorithm proposed by Johnson et al. (1997), we generate 5000 observations for Xin
each example. Using equation (5), we also generate a new vector 12
(, )
T
YYYwith an
approximate bi-variate normal distribution. We then apply the method of moments to estimate
the covariance of the variables using the new data. The Jarque-Bera test (1987), which is an
asymptotic test, evaluates the hypothesis that a random variable has a normal distribution with

10
unspecified mean and variance, against the alternative that the variable does not have a normal
distribution. The test is based on the sample skewness and kurtosis. For a true normal
distribution, the sample skewness should be near zero and the sample kurtosis should be near
three. This test determines whether the sample skewness and kurtosis are significantly different
from their expected values, as measured by a chi-square statistic.
The general statistics of the original and the transformed vector data sets are summarized in
Table (1).
Insert Table (1) about here
The results in Table (1) show that while the normality of the original variables is rejected in two
of the three examples, the transformed variables follow approximate normal distributions. It
means that the proposed normalizing transformation method works well for Poisson distributions
with medium and large-size parameters. However, in the third example while the transformed
variables are closer to normality than the original variables, due to the fact that the parameters of
the Poisson distributions are chosen to be too small, the transformation technique has not been
very effective.
4. Simulation Experiments
In this section, we perform two simulation experiments to evaluate and compare the performance
of the proposed multi-attribute chart with the one from the method proposed by Skinner et al.
(2006). In these experiments, we need to generate data from a joint Poisson distribution. There
are several joint distributions of Poisson type. However, if we do not use the NORTA method,
generating data from them is a difficult task. Since the proposed method is based on the NORTA

11
algorithm, we intentionally do not want to use the NORTA method to generate data. Skinner et
al. (2006) used Johnson et al.’s (1997) method to generate data from a joint Poisson distribution
in which the joint probability distribution is given by equation (6).
12
min( )
12 0
12
012
0
12 12
( ) exp( ( )) !
!!
ii
x
xx
i
xx
fi
ii
xx








 

 

Xx (6)
where 12
and


are the parameters of the marginal distribution and 0

is the covariance. We
note that the mean of the first and the second marginal distribution are 10 20
and


,
respectively. In order to generate simulation data of the multivariate Poisson distribution given in
equation (6), in this paper we use the method of Johnson et al. (1997). For a tri-variate case, we
may refer to Johnson et al. (1997).
4.1. Simulation Experiment 1
Consider a process with two attributes. Based on a set of available historical data we estimate the
parameters of the marginal distributions for each defect type as 12
3 and 2



with covariance
being two. To monitor both attributes simultaneously, we first use simulation to generate 5000
data sets for a distribution with the above estimated parameter values. We treat these data to
come from an in-control process of phase one monitoring. Then we transform the vector to an
almost multivariate normal vector using equation (5). Based on the proposed transformation
method we have
^0.9525 0.4083
[0.2238,0.2523] , and ( ) 0.4083 0.9286
TCov





YY.
The p-values of the Jarque-Bera normality test on the marginal distributions of the transformed
vector are 0.0738 and 0.1610, respectively and the upper control limit of 2
T
chart

12
is 59.10
2
2,995.0 

. Figures (2) and (3) show the joint probability distribution of the original and
the transformed vectors, respectively.
Insert Figure (2) about here
Insert Figure (3) about here
Moreover, we use the ARL criterion and perform a further evaluation of the proposed method
along with comparisons with the Patel and deleted-Y methods. To do this a replication of 10000
data sets resulted in in-control average run length ( 0
ARL ) values of 238.3556, 119.0999, and
264.53 for the proposed, Patel, and deleted-Y methods, respectively. We see that when we use
the original data in Patel’s method, the 0
ARL value is very low, whereas for the transformed data,
the 0
ARL will have an appropriate value. Knowing that:
 Different methods can be compared to each other based on out-of-control average run
length ( 1
ARL ) criterion if they share an almost equal 0
ARL , and
 The 0
ARL value of the Patel’s method is not compatible with the other two methods
we estimate 1
ARL values and the standard deviations of run lengths of the proposed method
along with the deleted-Y procedure for different scenarios of mean shifts determined in multiples
of their standard deviations and summarize the results in Table (2).
Insert Table (2) about here

13
The results of Table (2) show that in majority of the mean-shift cases the proposed method
performs better than the deleted-Y method in terms of 1
ARL criterion. In situations in which there
is no shift in the mean of the second variable, the deleted-Y method performs better than the
proposed method. In cases where there are simultaneous positive shifts in the means of the
variables, the deleted-Y method does not perform well at all.
4.2. Simulation Experiment 2
We consider another example in which there are three correlated attributes in a given production
process. Assuming the number of defects of three types follow a multivariate Poisson
distribution, we estimate the parameters on an in-control process as 123
1, 1, 3



with the
estimated variance-covariance matrix of 










511
142
124
^
.
To monitor all three attributes simultaneously, first we generate 5000 observations for a vector
from the above distribution. Then we perform the NORTA inverse transformation to arrive at the
following estimates
^0.9244 0.4694 0.2130
[0.2768,0.2555,0.2307] and ( ) 0.4694 0.9139 0.2085 .
0.2130 0.2085 0.9609
TCov







YY
The p-values of the Jarque-Bera normality test for the marginal distributions of the transformed
vector are 0.8307, 0.3702, and 0.3534, respectively and the upper control limit of 2
Tchart
is 83.12
2
3,995.0 

.
The 0
ARL value of the proposed control chart based on 10000 generated data sets is 215.6202,
while this figure is 88.7483 and 217.1561 for the Patel’s and deleted-Y methods, respectively.

14
Again we see that when we use original data the 0
ARL value is very low and hence the method is
not applicable, however when we transform the data, the 0
ARL will be of appropriate value.
Similar to simulation experiment 1, while the Patel’s method cannot be compared to the
proposed method in terms of 1
ARL criterion, we compare the 1
ARL of the proposed method with
the ones from the deleted-Y method and summarize the result in Table (3). We note that as soon
as we apply a shift in the means of the Multivariate Poisson distribution, we inherently shift the
co-variances as well.
The results of Table (3) show that the proposed method performs well even in situations where
there are both positive and negative shifts around the Poisson parameters. In addition, in majority
of mean-shift cases, the proposed method outperforms the deleted-Y method.
Insert Table (3) about here
4.3. Sensitivity Analysis on the Parameter Values
In section 4.1 we observed that the proposed normalizing transformation did not perform well in
situations in which the parameter values were small. In this section, we perform some simulation
experiments to see the effect of parameter values on the performance of the proposed method.
Furthermore, some simulation experiments are performed to compare the performance of the
proposed method to the ones from the deleted-Y procedure. In these examples, small and large
values of the parameters are chosen.
For a two-dimensional case, based on an intended in-control average run length of 200, Table (4)
shows the observed in-control average run lengths along with the standard deviations of the run
lengths of the proposed method for different small values of the parameters. The results of Table
(4) show that as the parameter values increase, the proposed method performs better. As stated in

15
section 4.1, this is due to the fact that the proposed normalizing transformation does not work
well for small parameters. In cases where the values of the parameters are too small, we need to
apply control charting methods proposed for high-quality production systems (see for example
Niaki and Abbasi, 2007).
Insert Table (4) about here
In order to compare the performances of the proposed method with the ones from the deleted-Y
procedure for small and large parameter values, first consider a process with two attributes in
which 12 0
0.3, 1, and 1.5


  (the 11th column of Table 4). The 0
ARL of the proposed
method and the deleted-Y method are estimated as 207.84 and 172.06, respectively. The
NORTA inverse transformation method leads us to










0.85720.5838
0.58380.8130
Cov(y)
0.3347] 0.4053[
ˆy











2.45001.4831
1.48311.8076
Cov(x)
2.5074] 1.8156[
ˆx

And Table (5) summarizes the results of the simulation experiment.
Insert Table (5) about here
The results of Table (5) show that while the proposed method performs well in situation in which
the parameters posses small values, in majority of mean-shift cases it outperforms the deleted-Y
procedure.
For the case in which the parameters are relatively large, we perform a similar simulation
experiment. In this case, 12 0
10.0, 15.0, and 4.0


  and the 0
ARL of the proposed method

16
and the deleted-Y method are estimated as 200.94 and 200.34, respectively. The results are
summarized in Table (6).
Insert Table (6) about here
Same conclusion as in the previous cases can be made here.
5. Conclusion and Recommendations for Future Research
Monitoring multi attribute processes, where there exists some correlations between attributes, is
an important issue in statistical quality control. There are some well-established methods to carry
out such monitoring. One of these methods is to approximate the distribution of the correlated
attributes with a multivariate normal distribution and then use a multivariate control chart, such
as 2
T
. However, one of the drawbacks of this method is the skewness of the distributions of the
attributes. In this paper, firstly we proposed a new transformation technique to approximate the
skewed distribution to a joint probability distribution in which the marginals are normal, and
then applied a multivariate control charting technique on the transformed data. In addition, using
simulation experiments we showed that when we use original data the 0
ARL value is very low
and hence the method is not applicable. However, when we transform the data, the 0
ARL will
have a more appropriate value. Moreover, when we apply the proposed methods on the processes
with intentional shifts in the mean vector we obtained satisfying 1
ARL values for most of the
values of the parameters of the marginal distributions. Furthermore, by simulation we showed
that the proposed method performs better than the deleted-Y method in most of the mean-shift
scenarios.

17
As an extension, we may consider processes with multivariate binomial distributions.
Furthermore, after the transformation phase, instead of 2
Tcontrol chart we may want to examine
other multivariate control charting techniques such as MEWMA and MCUSUM as well.
6. Acknowledgement
The authors would like to thank the referees for their valuable comments and suggestions that
improved the presentation of this paper. Moreover, this research was supported by the Iranian
National Science Foundation under grant number 85034/11, which is greatly appreciated.
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21
Figure (1): The Application Procedure of the Proposed Method
From historical data, estimate the
average number of each defect and the
correlations between each pair of them
Assume a Poisson distribution for each
defect
Estimate the correlation matrix of the
transformed normal variables


Y
S
We can do it in two ways:
1-Using equation (4).
2-Using simulation.
For each sample, count the number of
each defect and arran
g
e them in X
Transform Xto Yusing equation (6)
Employ the Hotelling T2control chart
to monitor Y where the correlation
matrix is Y
S
Start
End
Phase I
Phase II

22
5
10
15
0
5
10
15
0
50
100
150
200
250
X1
x2
Figure (2): The Joint Probability Distribution of the Original Vector in Simulation 1
4
2
0
-2
-4
-4
-2
0
2
4
0
350
700
x1
x2
(0,0,766)
Figure (3): The Joint Probability Distribution of the Transformed Vector in Simulation 1

23
Table (1): General Statistics of the Original and the Transformed Vectors
Data Mean and Covariance
Skewness Kurtosis P-Value (JB test)
1st
variable
2nd
variable
1st
variable
2nd
variable
1st
variable
2nd
variable
#1
Original data
^
[3.940, 4.954]
4.022 1.902
() 1.902 4.969
T
Cov






X
X
0.531 0.423 3.327 3.278 <0.0
٠
001 <0.0
٠
001
Transformed
data
^
[0.225, 0.207]
0.939 0.401
() 0.401 0.946
T
Cov






Y
Y
0.079 0.023 2.952 2.897 0.056 0.259
#2
Original data
^
[8.067,8.067]
7.815 4.104
() 4.104 8.177
T
Cov






X
X
0.356 0.363 3.129 3.214 <0.00
٠
01 <0.0
٠
001
Transformed
data
^
[0.203,0.201]
0.939 0.490
() 0.490 0.982
T
Cov






Y
Y
0.019 0.030 3.016 2.960 0.835 0.578
#3
Original data
^
[2.992,3.967]
2.985 0.975
() 0.975 3.963
T
Cov






X
X
0.6102 0.4789 3.3760 3.1574 <0.00
٠
01 <0.0001
Transformed
data
^
[0.294,0.240]
0.889 0.258
() 0.258 0.925
T
Cov






Y
Y
0.148 0.055 2.861 2.854 <0.0001 0.0297

24
Table (2): 1
ARL Values for Different Shifts in Simulation Experiment 1
Shift→ 1
ARL )( Std
Mean
)0,
1
(

),0( 2

)
2
,
1
(

)0,
1
2(

)2,0( 2

)
2
2,
1
2(

Proposed
method
24.128 23.828 20.325 5.206 5.222 4.439
23.828 24.027 20.829 4.726 4.815 4.128
deleted-Y
method
19.322 63.429 67.278 4.248 9.957 29.645
18.38 65.273 64.57 3.646 9.2753 29.483
Shift→ )0,
1
3(

)3,0( 2

)
2
3,
1
3(

)5.0,
1
5.0( 2

)5.0,
1
5.0( 2

 ),
1
(2


Proposed
method
2.263 2.303 1.903 48.110 49.595 8.685
1.719 1.782 1.370 48.932 50.272 8.324
deleted-Y
method
1.9675 3.339 17.204 30.007 93.953 12.169
1.3268 2.7625 16.904 29.723 90.772 12.267

25
Table (3): 1
ARL Values for Different Shifts in Simulation Experiment 2
Shift→
1
ARL )( Std
Mean
(,0,0)
1

2
(0, , 0)

3
(0, 0, )

12
(, ,0)

3
(,0, )
1


23
(0, , )


23
(, , )
1


Proposed
method
27.402 25.283 38.784 23.030 17.030 15.592 16.608
26.304 25.055 39.007 23.366 16.325 15.189 16.311
deleted-Y
method
28.011 27.658 35.096 36.902 35.016 35.182 58.608
26.027 26.105 34.931 36.001 34.582 34.271 57.010
Shift→ 2( , 0, 0)
1

2
2(0, , 0)

3
2(0, 0, )

12
2( , , 0)

3
2( , 0, )
1


),,0(2 32

),,
1
(2 32

Proposed
method
5.825 5.573 8.613 5.117 3.377 3.207 3.479
5.1716 5.129 8.3424 4.5361 2.8165 2.6558 2.955
deleted-Y
method
5.438 5.479 6.987 9.623 9.518 9.392 26.107
5.025 5.193 6.466 9.190 8.913 8.741 25.312
Shift→ )0,0,
1
(3

)0,,0(3 2

),0,0(3 3

)0,,(3 21

),0,
1
(3 3

23
3(0, , )


),,
1
(3 32

Proposed
method
2.473 2.427 3.310 2.105 1.530 1.483 1.532
1.8999 1.7949 2.7395 1.5083 0.88844 0.88758 0.92707
deleted-Y
method
2.220 2.233 2.720 4.151 4.122 4.202 15.127
1.939 2.109 2.170 3.826 3.692 3.857 14.442
Shift→ )0,,
1
(4.0 2


)0,,
1
(4.0 2


),0,
1
(4.0 3


),,0(4.0 32


),,0(4.0 32


),0,
1
(4.0 3


23
(0.4 , 0.4 , )
1



Proposed
method
71.671 81.687 108.904 99.063 118.405 104.902 44.685
72.618 82.532 109.06 95.994 116.86 102.31 43.324
deleted-Y
method
80.400 81.104 101.190 83.191 99.223 82.696 39.244
78.983 80.923 100.023 82.390 97.891 81.474 38.836

26
Table (4): 0
ARL of the Proposed Method for Different Small Values of the Parameters

1

0.1 0.1 0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.3 0.3 0.3 0.7 0.7

2

0.2 0.3 0.4 0.5 1.0 1.2 0.5 0.7 0.9 1.0 1.2 1.5 0.7 1.0
0

 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
0
ARL
)( Std
Mean
60.745 65.706 97.162 135.52 151.74 223.38 141.48 153.62 150.6 207.84 241.31 233.23 203.88 211.14
61.941 64.401 99.482 137.72 154.63 217.11 139.91 155.21 157.5 219.75 231.53 240.47 212.79 211.10
Table (5): 1
ARL Values for Different Shifts in Small-Valued Parameters
Shift→ 1
ARL )( Std
Mean
)0,
1
(

),0( 2

)
2
,
1
(

)0,
1
2(

)2,0( 2

)
2
2,
1
2(

Proposed
method
9.061 9.520 9.007 2.682 2.764 2.494
8.437 8.887 9.193 2.330 2.502 2.302
deleted-Y
method
12.088 13.355 43.397 2.749 2.941 18.867
11.604 12.614 43.027 2.197 2.415 18.202
Shift→ )0,
1
3(

)3,0( 2

)
2
3,
1
3(

)5.0,
1
5.0( 2

)5.0,
1
1.0( 2

 ),
1
1.0( 2


Proposed
method
1.678 1.788 1.587 18.503 29.054 8.861
1.064 0.958 1.009 17.942 28.680 8.313
deleted-Y
method
1.518 1.495 11.281 13.307 18.597 3.475
0.895 0.862 10.777 12.801 18.045 2.891

27
Table (6): 1
ARL Values for Different Shifts in Large-Valued Parameters
Shift→ 1
ARL )( Std
Mean
)0,
1
(

),0( 2

)
2
,
1
(

)0,
1
2(

)2,0( 2

)
2
2,
1
2(

Proposed
method
34.613 33.837 23.442 6.694 6.573 4.015
33.824 34.011 23.162 6.200 6.080 3.480
deleted-Y
method
51.856 23.955 77.288 10.007 5.204 39.193
51.461 23.790 76.422 9.473 4.665 38.842
Shift→ )0,
1
3(

)3,0( 2

)
2
3,
1
3(

)5.0,
1
5.0( 2

)5.0,
1
5.0( 2

 ),
1
(2


Proposed
method
2.498 2.395 1.581 61.022 56.611 11.628
1.918 1.832 0.948 60.688 57.252 11.155
deleted-Y
method
3.571 2.136 23.552 60.924 33.634 6.949
3.011 1.526
22.715 60.089 33.009 6.536