J. Software Engineering & Applications, 2009, 2: 335-343
doi:10.4236/jsea.2009.25044 Published Online December 2009 (http://www.SciRP.org/journal/jsea)
335
An Integrated Use of Advanced T2 Statistics and
Neural Network and Genetic Algorithm in
Monitoring Process Disturbance
Xiuhong WANG
Zhengzhou Institute of Aeronautical Management, Zhengzhou, China.
Received July 31st, 2009; revised September 14th, 2009; accepted September 21st, 2009.
ABSTRACT
Integrated use of statistical process control (SPC) and engineering process control (EPC) has better performance than
that by solely using SPC or EPC. But integrated scheme has resulted in the problem of “Window of Opportunity” and
autocorrelation. In this paper, advanced T2 statistics model and neural networks scheme are combined to solve the
above problems: use T2 statistics technique to solve the problem of autocorrelation; adopt neural networks technique to
solve the problem of “Window of Opportunity” and identification of disturbance causes. At the same time, regarding
the shortcoming of neural network technique that its algorithm has a low speed of convergence and it is usually plunged
into local optimum easily. Genetic algorithm was proposed to train samples in this paper. Results of the simulation ex-
periments show that this method can detect the process disturbance quickly and accurately as well as identify the dis-
turbance type.
Keywords: T2 Statistics, Neural Networks, Statistical Process Control, Engineering Process Control, Genetic Algorithm
1. Introduction
In an intense market competition environment, product
quality plays an important role in facing competition and
gaining competitiveness. Both Statistical Process Control
(SPC) and Engineering Process Control (EPC) are effec-
tive techniques of maintaining and improving the pro-
duce quality. EPC is used to adjust the variables for
compensating the short-term output deviation by uncon-
trollable factors. In regard to long-term process im-
provement, SPC is effective technique which is used to
detect out-of-control conditions and remove the control-
lable factors. So, lots of scholars have proposed the inte-
grated use of SPC/EPC.
However, it is very difficult to monitor the EPC proc-
ess using commonly SPC methods because of the prob-
lem of “Window of Opportunity” and autocorrelation [1].
In the past time, monitored variable of SPC techniques
was only process output. The information of process in-
puts was usually ignored. For the EPC processes, once
output deviation is compensated by feedback-controlled
action, there is only a short window of detecting process
disturbance. Even SPC charts fail to detect out-of-control
when output deviation is small because EPC’s feedback
mechanism can compensate for such small disturbance
quickly and completely. And the optimality of SPC tech-
niques rests on the assumption of time independence.
However, process output of no same time is autocorrela-
tion for each other.
To overcome these shortcomings, a little of papers
have developed some joint-monitoring methods under
the feedback control processes. These methods may be
categorized into two aspects. The first is that various
types of conventional SPC charts are integrated to moni-
tor the process [2–3], such as Huang C.H proposed She-
whart control chart and Cusom control chart simultane-
ously to detect the manufacturing process. This method
can detect out-of-control, also can recognize the distur-
bance type. However, the inherent problems of conven-
tional SPC charts caused by the effects of feedback con-
trol actions have still not been solved. The second is that
the strategy of jointly detecting the controlled outputs
and manipulated inputs using bipartite SPC is suggested
such as multivariate CUSUM chart, multivariate EWMA
chart, T
2
statistics and multivariate profile chart [4,5].
Although, these methods have solved effectively the
autocorrelation problem, but the WO problem has not
been settled completely and effectively because these
methods can not monitor the small process disturbance
quickly within the scope of the WO. Furthermore, these

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 336
methods are not easy to identify the disturbance types
which are crucial links of confirm and remove the con-
trollable factors.
In this research, we put forward a new program of in-
tegrated use of T
2
statistics technique, artificial neural
networks and genetic algorithm: use T
2
statistics tech-
nique to solve the problem of autocorrelation and infor-
mation missing; adopt neural networks technique and
genetic algorithm to solve the problem of “Window of
Opportunity” and identification of disturbance causes.
2. Feedback-Controlled Process
For better understanding, we consider the following
process under the feedback mechanism shown in Figure 1.
We consider an integrated-moving-average noise mo-
del (ARMA (1, 1)).
1
1
tt
B
da
B
θ
φ
−
=− (1)
Where θ, Φ are constants. аt represents white noise which
complies with a standard normal distribution with mean
=0 and σ2=1. Also, let B be the usual backward shift op-
erator, i.e., Bа
t
=а
t-1. mt
represents random form of the
process disturbance such as step change and process
drift.
yt is the measured output value. Without loss of gener-
ality, the target value is assumed to be zero. Then, y
t
represents the output deviation from the target value.
1
ttttttt
ydmxdmu
−
=++=++ (2)
ut is the feedback control action decided by the feed-
back process mechanism. In the industrial practice, sev-
eral feedback controllers are used such as PI controllers,
I controllers, PID controller and EWMA controllers in
which PID controllers are the most extensively adopted.
Its feedback control rules can be expressed as [6].
1
0
()
tPtItjDtt
j
ukykykyy
∞
−−
=
=−−−−
∑ (3)
where k
p
,kD ,kI are constants. Function 3 can also be
expressed as
112
()(2)
ttPIDtPDtDt
uukkkykkyky
−−−
−=−+++++
(4)
Process
Feedback
Mechanism
y
t
d
t
x
t
u
t
y
t
m
t
Figure 1. Feedback-controlled process
In light of the function 4, output at the different times
is autocorrelation, and input at the different times is auto-
correlation. Moreover, output and input are autocorrela-
tion for each other. So, traditional SPC control charts,
such as Shewhart chart, EWMA chart and Cusum chart
are invalid to monitor the above process.
3. Design of the New Methodology
3.1 Design of Standard T2 Statistics Technique
Standard T
2
statistics method is used to deal with the
multiple-input process. In this paper, the devised ap-
proach is similar to the standard T2 statistics but the data
vectors are made up of the process input and output at the
different times. It can measure the overall distance of
observation from reference values including process
output, input and covariance of output and input, hence,
it will come to the most commonly used schemes. Ac-
cording to the function 4, complete monitoring informa-
tion should include control action at time t and t-1, the
process output at time t, t-1 and t-2. However, to detect
the closed-loop process, the five sets are co-linear. In
other words, arbitrary set is equal to a linear combination
of other sets. So, we can only select two sets, three sets
or four sets from the above five sets to make the moni-
toring scheme. We design the monitoring model of T
2
statistics as follows
1T
ttt
DZZ
−
=
∑
(5)
Where Σ is the covariance matrix of Zt . In light of the
above analysis, the options N of Zt are equal to
432
555
NCCC
=++ (6)
There is not commonly admitted approach for con-
firming the best Z
t
selection in the T2 statistics model.
Selection of the model parameter is based on the problem
which will be solved. Therefore, the design of model is
scientific as well as art. Hotelling, Montgomery and Alt
discussed the possibilities and advantages of the T2 sta-
tistics method used to monitor the EPC process [7–9].
They designed the simplest and the most basic form of Zt,
i.e. Zt = [yt,ut]. On the basis of the above study, FUGEE
TSUNG elaborated on the problem and proposed that
one could define Z
t
=[yt,ut,ut-1,ut-2]T or Z
t
=[yt, u
t
, y
t-1,
yt-2]T [1]. However, in these methods, ∑ is not estimated
from the historical data directly, but obtained from a very
complex function based on the parameter of Φ, θ,
kp ,kD ,kI. So the T2 control chart is not available for these
methods.
According to function 2 and 3, since outputs and in-
puts are correlated, all inputs can be expressed as the
combination of the process outputs at different times. In
other words, all information concluding the process in-
puts and the process outputs can be monitored as long as
we detect the outputs at different times.

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 337
pt-1It-1-jDt-1t-2
j0
-ky -ky- k(y-y)
ttt
ydm
∞
=
=+ ∑ (7)
We proposed to define Zt = [yt, yt-1, yt-2,…, yt-s]T. Se-
lection of s value is a very difficult and challenging task.
Now there is no universally recognized method for con-
firming the value. In this research, simulation experim-
ents are implemented to determine the value of s. Aiming
at each choice, experiments simulate the feedback-con-
trolled process with the step-change step=5, 2 and 0.8.
Value of the parameter Φ, θ, KP, KI and KD is randomly
set to 0.8, 0.5, 0.5, 0.5 and -0.3. Simulation results are
shown in Figures 2–14.
Figure 2. T2 chart detects the disturbance with the parameter step=5 and s=1
Figure 3. T2 chart detects the disturbance with the parameter step=5 and s=2
Figure 4. T2 chart detects the disturbance with the parameter step=5 and s=3
Figure 5. T2 chart detects the disturbance with the parameter step=5 and s=4

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 338
Figure 6. T2 chart detects the disturbance with the parameter step=5 and s=5
Figure 7. T2 chart detects the disturbance with the parameter step=2 and s=1
Figure 8. T2 chart detects the disturbance with the parameter step=2 and s=2
Figure 9. T2 chart detects the disturbance with the parameter step=2 and s=3
From Figure 2 to Figure 6 is T2
chart with the same
step 5 and no same s values. From Figure 7 to Figure 10
is T2 chart with the same step 2 and no same s values.
From Figure 11 to Figure 14 is T2 chart with the same
step 0.8 and no same s values.
In light of these figures, when step values are the same,
the larger is s, larger is the value of T2 and the quicker is
to detect the disturbance. However, the larger is s, the
greater is false alarm such as Figures 2 to 6. To the proc-
ess with step=5, when s is equal to 4, there are two out-of

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 339
Figure 10. T2 chart detects the disturbance with the parameter step=2 and s=4
Figure 11. T2 chart detects the disturbance with the parameter step=0.8 and s=1
Figure 12. T2 chart detects the disturbance with the parameter step=0.8 and s=2
Figure 13. T2 chart detects the disturbance with the parameter step=0.8 and s=3
control points. However, when s is equal to 5, there are
four out-of control points in which two points fall into
false alarm. In the same way, to the process with step 2
and step 0.8, when s increases from s=2 to s=4, the dots
of false alarm grow from 0 to 2.
To the different step changes, the smaller is step value,
the larger is to need the value of s to detect the process.
For example, it only need s=1 to monitor the process
with step=5, but need s=3 to detect the process with
step=2 and step =0.8.

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 340
Figure 14. T2 chart detects the disturbance with the parameter step=0.8 and s=4
Figure 15. Feedback-controlled process
According to Figure 2–14 and the above analysis, Z
t
can be expressed as Zt = [yt, yt-1, yt-2, yt-3]T. In light of the
function 7, yt is a linear combination of the yt-i, (i=0, 1, 2,
3). So Zt accords with a multivariate normal distribution
and Zt has a chi-squared distribution with p degrees-of-
freedom. The control limit UCL for Zt should be χ2α,p.
Dt contains the information of output, input and corre-
lation for each other. So the advanced T2 statistics can
solve effectively the problem of autocorrelation and re-
duce the problem of “Window of Opportunity”. More-
over, it is difficult to interpret the results and search for
the root cause of process disturbance once system moni-
tored out-of-control such as Figure 15.
Figure 15 shows the process with the drift disturbance
slope=1. But it has not essential distinction between Fig-
ure 15 and Figure 2–14 to identify the disturbance type
such as significant upward or downward trend. So, ad-
vanced T2 statistics technique can not be used solely.
3.2 Artificial Neural Networks
Artificial neural networks are modeled following the
neural activity in human brain and rapidly developed
since the last century 80’s [10]. The main characteristics
of neural networks are the overall use of network, Large-
scale parallel distributed processing, Ability to study
association, high degree of fault tolerance and robustness.
However, neural networks are easy to fall into local op-
timum, slow convergence and cause oscillation effect.
Genetic algorithm [11] has strong macro-search capabili-
ties and greater probability of finding the global optimal
solution. So, genetic algorithm can overcome the short-
comings of neural networks if it is used to finish the
pre-search. In this paper, a novel algorithm combining
neural networks algorithm and genetic algorithm was
proposed. The framework of neural network is shown in
Figure 16.
Network is composed of an input layer, a hidden layer
and an output layer. Input layer has three neurons which
are expressed as D
t
, D
t-1 and D t-2 representing the pa-
rameter value of T2 statistics at time t, t-1 and t-2 based
on the function 4. Output layer has two neurons which
are expressed as O1 and O2, where O1 represents distur-
bance classification values and O2 represents disturbance
causes. Hidden layer has s neurons which is expressed as
HR∈
S
(H=（H1, H2,…, Hs）
T. WIH represents the relation
weight between input layer and hidden layer. W
OH
Input layer
H
1
O
2
W
IH
W
OH
H
s
O
1
D
t
Hidden layer
D
t-1
D
t-2
Output layer
Figure 16. A three-layer neural network

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 341
represents the connection weight between output layer
and hidden layer.
Input layer of the network is a key decision which has
a great impact on the effectiveness of the network. Now
there is no commonly accepted method for selecting the
input layer. In this paper, an all-possible-regression
analysis [12,13] is used to define the input layer accord-
ing to the R2
P
, AIC and CP criterion. It is assumed that
input layer is a possible combination from Dt , Dt-1 , Dt-2 ,
Dt-Dt-1 , Dt-1-Dt-2 , Dt -Dt-2. Purpose of this method is to
select a good combination so that a detailed examination
can be made of the regression models, leading to the se-
lection of the final input vectors to be utilized [12]. The
result is shown in Table 1. In light of the R2P, AIC and CP
criterion, we select the (Dt, D
t-1, D t-2) combination be-
cause it has the largest R2P, the smallest AIC and CP val-
ues in the Table 1.
3.3 Neural Network Training Based on Genetic
Algorithm
1) Determination of fitness function
Purpose of which genetic algorithm is used to optimize
the network weights and threshold of neural network is to
obtain the optimum combination of weights value and
threshold. Output error measures the effect of combina-
tion. Hence, fitness function of individual chromosome
should be the function of output error of BP network.
Ideal output value is expressed as Dj and actual output
value is expressed as Aj. The fitness function
()
fE
can
be written as
2
1
()(())
(1) jj
fEEDA
E
==−
+∑ (8)
2) Genetic manipulation
Assumed that Group size is M and Fitness of individ-
ual i is Fi. Individual probability of being selected can be
expressed as follows:
1
i
si M
i
i
F
P
F
=
=
∑
(9)
Arithmetic crossover operator is adopted which is spe-
cially used to solve floating-point cross, and uniform
mutation operator is introduced.
4. Simulation Experiments
It is assumed that value of group size M, crossover
probability, Mutation probability, training error and gen-
eration gap is 100, 0.8, 0.05, 0.005 and 0.7 respectively.
To verify the performance of the above method, we make
a great deal of simulation experiments on the actual pro-
duction. The experiments are divided into three stages.
First, 500 “in-control” sample sets (mt=0) and 500
“out-of-control” sample sets (mt≠0), each which involves
200 data points and generated from an AMAR(1,1) noise
model, are selected to train the neural network. The 500
out-of-control samples sets perform a process which is
upset respectively by step-change with the step of
Table 1. R2P, AIC and CP values of all-possible-regression analyse
Variables combination P R2P AIC CP
Dt 2 0.9212 79.6391 100.9358
Dt-1 2 0.8473 82.5523 682.9437
Dt-2 2 0.5976 85.7604 1259.2257
(Dt,Dt-1) 3 0.8411 81.9853 55.8933
(Dt,Dt-2) 3 0.7952 78.0043 129.4748
(Dt-1,D t-2) 3 0.8319 86.8358 318.4751
(Dt, Dt-1, Dt-2) 4 0.9855 74.2201 3.9241
(Dt,Dt-Dt-1) 3 0.9381 88.9276 18.9937
(Dt,Dt-Dt-1,Dt-1-Dt-2) 4 0.9817 77.0256 5.8619
(Dt-1,D t-1-Dt-2) 3 0.8294 83.9967 53.7724
(Dt-1,D t-Dt-1 ,Dt-1-Dt-2 ) 4 0.9027 79.5720 142.9901
(Dt-2,D t-Dt-2 ) 3 0.8979 80.7342 46.7424
(Dt-2,D t-Dt-1 ,Dt-1-Dt-2 ) 4 0.9145 80.5612 290.0132
(Dt,Dt-1,Dt-1-Dt-2 , Dt-Dt-2) 5 0.9224 78.9267 4.5776
(Dt-1,D t-2,Dt-Dt-1 , Dt-Dt-2) 5 0.9225 78.9399 5.2481
(Dt,Dt-2,Dt-Dt-1, Dt-1 -Dt-2) 5 0.9224 78.9399 5.8932
(Dt,Dt-1,Dt-2,D t-Dt-1 , Dt-Dt-2) 6 0.9224 78.4516 6.9935

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 342
Table 2. Percentages of correct classification using integrated method with step change and process drift respectively
Step change Drift disturbance
0.5 1 2 3 5 0.5 1 1.5 2
Φ=0.8, θ=0.5 k
p
=0.5,kD=0.5, kI=-0.3 62% 82% 93% 96% 99% 62% 87% 91% 100%
Φ=0.8, θ=0.2 k
p
=0.5,kD=0.5, kI=-0.3 67% 89% 95% 99% 99% 67% 91% 94% 100%
Φ=0.8, θ=-0.5 k
p
=0.5,kD=0.5, kI=-0.3 37% 54% 77% 79% 82% 31% 53% 66% 72%
Φ=0.8, θ=0.5 k
p
=0.5,kD=0, kI=-0.3 53% 84% 90% 97% 100%
61% 88% 94% 99%
Φ=0.8, θ=0.5 k
p
=0,kD=0, kI=-0.3 45% 78% 88% 92% 97% 53% 85% 90% 98%
Table 3. Percentages of correct classification using Shewhart chart to detect disturbance arose by step change
Step change
0.5 1 2 3 5
Φ=0.8, θ=0.5 k
p
=0.5,kD=0.5,kI=-0.3 **** *** 57% ARL=97 72% ARL=85 86% ARL=4
Φ=0.8, θ=0.2 k
p
=0.5,kD=0.5,kI=-0.3 **** 13% ARL=100 64% ARL=92 83% ARL=77 89% ARL=3
Φ=0.8, θ=-0.5 k
p
=0.5,kD=0.5,kI=-0.3
**** **** **** 51% ARL=97 76% ARL=7
Φ=0.8, θ=0.5 k
p
=0.5,kD=0,kI=-0.3 ****
**** 54% ARL=98 70% ARL=89 92% ARL=4
Φ=0.8, θ=0.5 k
p
=0,kD=0,kI=-0.3 ****
**** 48% ARL=100 69% ARL=84 83% ARL=5
0.5/1/2/3/5 at data 50 and is eliminated quickly and com-
pletely at the data 150.Likewise, the 500 out-of-control
sample sets are generated with the process drift with the
slope of 0.25/0.5/1/2/3 at data 50 and are removed
quickly and completely at data 100.
Second, once output error is within the permitted
scope, objectives of training the neural network based on
genetic algorithm have been achieved successfully. The
neural network can be used to monitor the process dis-
turbance. 200 out-of-control sample sets, which are gen-
erated with the use of step=0.5, 1, 1.5, 2, 3, 5 at time t
and slope= 0.5, 1, 1.5, 2, 3 at time t, is given to verify the
performance of the above method. The result is shown in
Table 2.
At last, for comparison, Shewhart chart of Minitab
software is used to simulate the above sample sets with
step change. Result is shown in Table 3.
5. Result Analysis of Simulation Experiment
As seen from Figure 17 and 18, the alone neural net-
works need 1200 steps to converge at the error target
value. However, neural networks based on the genetic
algorithm only need 550 steps to converge at the error
target value. So neural networks based on the genetic
algorithm can reduced training time significantly. Its
training speed is faster. Furthermore, if alone neural
networks are used, error target value cannot gain when
step is small such as 1 and 0.8.
In actual manufacturing industry, parameters often
change with the change of environment. So we choose
five combinations of Φ, θ, kp, kD and kI in order to verify
Figure 17. Training error curve of neural networks
Figure 18. Training error curve of neural networks based
genetic algorithm

An Integrated Use of Advanced T2 Statistics and Neural Network and Genetic Algorithm in Monitoring Process Disturbance 343
the method and cover a reasonable range of the parame-
ter space. In terms of the Table 1, the value of parameter
Φ, θ has serious impact on the resolution capability of the
integrated method. It is very applicable to combine a
positive and large Φ with a positive and small θ. On the
contrary, the combinations of a positive Φ and a negative
θ worsen with the ability to identify the process distur-
bance accurately. There is no obvious correlation be-
tween change of the controller parameter kp , kD , kI and
monitoring ability. With respect to the drift disturbance,
step change is easier to be monitored.
According to Tables 3 and Table 4, the advantage of
the integrated method is significant. The neural network
requires only one sample to recognize the disturbance
and identify the disturbance type. But Shewhart chart
requires an average of 3 to 7 samples to recognize the
process disturbance with step 5. When step=2 and 3, an
average of 70 to 100 samples are required to detect the
disturbance. Even the disturbance with step=1 and 0.5
can not be monitored.
6. Acknowledgments
This paper was supported by the grants from National
Natural Science Foundation of China (No 70771102),
Aviation Science Foundation of China (No. 2007ZG
550050).
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