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Computers & Industrial Engineering 48 (2005) 205–221<br />
www.elsevier.com/locate/dsw<br />
A <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> <strong>system</strong> <strong>using</strong> a <strong>statistical</strong><br />
correlation coefficient method *<br />
Jenn-Hwai Yang, Miin-Shen Yang*<br />
Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan 32023, ROC<br />
Abstract<br />
This paper presents a <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> <strong>system</strong> <strong>using</strong> a <strong>statistical</strong> correlation coefficient method.<br />
Pattern <strong>recognition</strong> techniques have been widely applied to identify unnatural <strong>pattern</strong>s in <strong>control</strong> <strong>chart</strong>s. Most of<br />
them are capable of recognizing a single unnatural <strong>pattern</strong> for different abnormal types. However, before an<br />
unnatural <strong>pattern</strong> occurs, a change point from normal to abnormal may appear at any point in <strong>control</strong> <strong>chart</strong>s for<br />
most practical cases. Moreover, concurrent <strong>pattern</strong>s where two unnatural <strong>pattern</strong>s simultaneously exist may also<br />
occur in a <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> <strong>system</strong>. Our <strong>statistical</strong> correlation coefficient approach is a simple<br />
mechanism for recognizing these unnatural <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s with good performance. This approach is also an<br />
effective method for the <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> without a tedious training process.<br />
q 2005 Elsevier Ltd. All rights reserved.<br />
Keywords: Control <strong>chart</strong>; Pattern <strong>recognition</strong>; Statistical correlation coefficient; Recognition rate; Change point; Concurrent<br />
<strong>pattern</strong><br />
1. Introduction<br />
Due to an increasing competition in products, consumers have become more critical in choosing<br />
products. The quality of products has become more important. Statistical process <strong>control</strong> (SPC) is<br />
usually used to improve the quality of products and reduce rework and scrap so that the quality<br />
expectation can be met (Grant & Leavenworth, 1996).<br />
Shewhart <strong>control</strong> <strong>chart</strong>s (1931) are the most popular <strong>chart</strong>s widely used in industry to detect abnormal<br />
process behavior. The most typical form of <strong>control</strong> <strong>chart</strong>s consists of a central line and two <strong>control</strong> limits<br />
representing the specifications of the product and the variant range limits. This provides a useful method<br />
* This manuscript was processed by Area Editor E.A. Elsayed.<br />
* Corresponding author. Tel.: C886 3 2653100; fax: C886 3 2653160.<br />
E-mail address: msyang@math.cycu.edu.tw (M.-S. Yang).<br />
0360-8352/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.<br />
doi:10.1016/j.cie.2005.01.008
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J.-H. Yang, M.-S. Yang / Computers & Industrial Engineering 48 (2005) 205–221<br />
for monitoring variations in the product manufacturing process. However, these <strong>control</strong> <strong>chart</strong>s do not<br />
provide <strong>pattern</strong>-related information because they focus only on the latest plotted data points. In other<br />
words, only the last point on the <strong>chart</strong> is examined rather than the trend in the process over time.<br />
The value of average run length (ARL) is usually used to measure the frequency before the process is<br />
out-of-<strong>control</strong>. ARL is generally defined as the average number of points in a <strong>control</strong> <strong>chart</strong> until an out-of<strong>control</strong><br />
signal occurs. Under a normal distribution N(m,s 2 ) assumption with its mean m and variance s 2 ,it<br />
is called out-of-<strong>control</strong> when its value is outside the range of G3s from the mean m. In general, it is<br />
unavoidable due to a random noise even though in a regular condition. For example, when we have a<br />
production process that the quality of products is under the N(0,1) assumption, the probability P(X%K3<br />
or XR3)Z0.0027 such that 1/0.0027z370.4. Thus, ARL under the N(0,1) assumption is around 370.4.<br />
Similarly, for mZ0.5 with s 2 Z1, ARL is around 155.2. The ARL value for other mean shift m with s 2 Z1<br />
is shown in Table 1. We can see that ARL will be decreasing when a mean shift m is increasing. We know<br />
that the smaller ARL exhibits out-of-<strong>control</strong> signal more often. In this case, it is not good for detecting<br />
abnormal process behavior with a Shewhart <strong>control</strong> <strong>chart</strong> because it is variable due to a mean shift m.<br />
Therefore, the <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> is important for recognizing unnatural <strong>pattern</strong>s in <strong>control</strong><br />
<strong>chart</strong>s.<br />
Control <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> has the capability to recognize unnatural <strong>pattern</strong>s. In general,<br />
there are six unnatural <strong>pattern</strong>s in <strong>control</strong> <strong>chart</strong>s. They are upward trend, downward trend, upward<br />
shift, downward shift, cycle and <strong>system</strong>atic variation. These <strong>pattern</strong>s present the long-term behavior<br />
of a process. If any of these <strong>pattern</strong>s appears, something has occurred in the process such as a<br />
change of raw materials, equipment damage or the state of workers. If we can have such<br />
information on hand and then make decisions on the process, it will help us prevent failures<br />
appearing in the future.<br />
There have been many studies on <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> (see Al-Ghanim & Ludeman,<br />
1997; Cheng, 1997; Guh & Hsieh, 1999; Pham & Oztemel, 1994; Yang & Yang, 2002). Recently, Guh<br />
(2003) proposed a hybrid artificial intelligence technique that consists of three major sub-<strong>system</strong>s so that<br />
it can be well used to build a real time SPC <strong>system</strong>. Most of these studies are considered for recognizing<br />
a single unnatural <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>. However, in most real <strong>control</strong> <strong>chart</strong> applications, normal points<br />
may appear before abnormal points so that a change point from normal to abnormal may occur at any<br />
point in <strong>control</strong> <strong>chart</strong>s. If we do not have a mechanism for recognizing such change <strong>pattern</strong>s, we may<br />
incorrectly obtain the classification results. Therefore, under these circumstances, even though we have a<br />
good classification method for the <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong>, we may still make an incorrect<br />
decision.<br />
Table 1<br />
Average run length with a mean shift m<br />
m<br />
ARL<br />
0 370.4<br />
0.5 155.2<br />
1 43.9<br />
1.5 15<br />
2 6.3<br />
2.5 3.2<br />
3 2
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Another practical situation is concurrent <strong>pattern</strong>s where two unnatural <strong>pattern</strong>s may exist together. For<br />
recognizing concurrent unnatural <strong>pattern</strong>s, Guh & Tannock (1999) used a back-propagation network<br />
(BPN) learning to perform the <strong>recognition</strong> for these concurrent <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s. Although this BPN<br />
has a learning structure with a good network construction, the approach to the <strong>control</strong> <strong>chart</strong> <strong>pattern</strong><br />
<strong>recognition</strong> is slow and complicated with a learning process. In this paper we propose a simple<br />
mechanism to recognize these unnatural <strong>pattern</strong>s. The <strong>statistical</strong> correlation coefficient approach is used<br />
to construct a <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> <strong>system</strong>. By adding a threshold criterion, we can recognize<br />
these unnatural <strong>pattern</strong>s even though they may change from normal to abnormal at any point in <strong>control</strong><br />
<strong>chart</strong>s. The remainder of this paper is organized as follows. In Section 2 we explain why a <strong>statistical</strong><br />
correlation coefficient method is proposed and then present the proposed approach with the training and<br />
classification algorithms. Some simulation results and their comparisons are presented in Section 3. In<br />
Section 4, we present the performance for recognizing concurrent <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s. Conclusions<br />
will be stated in Section 5.<br />
2. The proposed <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> <strong>system</strong><br />
Since <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> is an important step for industrial production processes,<br />
many researchers have made efforts toward finding various efficient methods for recognizing<br />
unnatural <strong>pattern</strong>s in <strong>control</strong> <strong>chart</strong>s. Hwarng and Hubele (1993) used a back-propagation neural<br />
network technique for detecting X-bar <strong>control</strong> <strong>chart</strong>s. Cheng (1997); Guh and Hsieh (1999) used a<br />
neural network approach for recognizing an unnatural <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>. Pham and Oztemel<br />
(1994) constructed a <strong>pattern</strong> recogniser <strong>using</strong> a learning vector quantization (LVQ) network. Yang<br />
and Yang (2002) created a fuzzy-soft LVQ to promote the <strong>recognition</strong> rate for <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s.<br />
In order to decrease the number of reference vectors for each <strong>pattern</strong> used to identify different<br />
variation quantities, Al-Ghanim and Ludeman (1997) constructed a so-called ‘matched filter’ <strong>using</strong> a<br />
correlation analysis technique based on an inner <strong>pattern</strong> vector product. For concurrent <strong>control</strong> <strong>chart</strong><br />
<strong>pattern</strong> <strong>recognition</strong>, Guh and Tannock (1999) proposed a back-propagation approach. In this section,<br />
we will propose a <strong>statistical</strong> correlation coefficient approach for the <strong>recognition</strong> of unnatural <strong>control</strong><br />
<strong>chart</strong> <strong>pattern</strong>s.<br />
The so-called matched filter approach involves obtaining a prototype for each <strong>pattern</strong>, called a<br />
‘reference vector’ after the training algorithm is finished. Al-Ghanim and Ludeman (1997) evaluated the<br />
correlation between reference vectors and the input vector <strong>using</strong> the inner product.<br />
Let x and y be two random vectors with x 0 Z[x 1 .x n ] and y 0 Z[y 1 .y n ]. Then the inner product<br />
between x and y is defined as<br />
hx; yi Z x 0 y Z X x i y i : (1)<br />
Their inner-product correlation technique got good results in recognizing trend, cycle and <strong>system</strong>atic<br />
<strong>pattern</strong>s. However, this method is poor in recognizing shift <strong>pattern</strong>s. This is because the inner product<br />
presents the consistency in positive and negative components values in two vectors, but each component<br />
at the same position with an opposite sign will counteract the final inner product value. For example, if<br />
there are two vectors with nearly positive components, the inner-product correlation will become worse.
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Let us consider the following three <strong>pattern</strong>s with<br />
x 0 Z ½K2 K2 0 2 2Š; y 0 Z ½K2 K2 0 2 2Š; and z 0 Z ½K1 0 1 2 2Š:<br />
The graph of the <strong>pattern</strong>s x, y and z is shown in Fig. 1. Obviously, the similarity between x and y is<br />
larger than x and z. Calculating their correlation on the basis of inner product, we have<br />
hx; yi Z 16 and hx; yi Z 10:<br />
A larger correlation value represents higher similarity. The result is no doubt that x is more similar to<br />
y than to z.<br />
Now, let us shift the <strong>pattern</strong>s x, y and z with a value of 2. That is,<br />
x 0 s Z ½ 0 0 2 4 4Š; y 0 s Z ½ 0 0 2 4 4Š; and z 0 s Z ½ 1 2 3 4 4Š<br />
A graph of the shifted <strong>pattern</strong>s x s , y s and z s is shown in Fig. 2. We also calculate their inner product<br />
values with<br />
hx s ; y s i Z 36 and hx s ; z s i Z 38:<br />
Fig. 1. Illustration of the <strong>pattern</strong>s x, y and z.<br />
The results show that x is more similar to z than to y that is obviously incorrect. This is because all of<br />
the components are nonnegative in x s , y s and z s . Under that condition, a larger component value<br />
determines a larger correlation value. Thus, the shift <strong>pattern</strong> will be difficult to be recognized on the<br />
basis of the inner-product approach. As we analyze the characteristic with upward shift and upward<br />
trend <strong>pattern</strong>s, shown in Fig. 3, we note that both of their components are larger in the later part and tend<br />
to be positive. However, the trend toward large values in the upward trend <strong>pattern</strong> is faster than in the<br />
upward shift <strong>pattern</strong>. Thus, a shift <strong>pattern</strong> is always classified as a trend <strong>pattern</strong> because it has<br />
Fig. 2. Illustration of the <strong>pattern</strong>s x s , y s and z s .
J.-H. Yang, M.-S. Yang / Computers & Industrial Engineering 48 (2005) 205–221 209<br />
Fig. 3. Upward shift and upward trend <strong>pattern</strong>s.<br />
a maximum inner-product value. This produces an incorrect <strong>recognition</strong>. We can use the <strong>statistical</strong><br />
correlation coefficient method to solve this problem as follows.<br />
The <strong>statistical</strong> correlation coefficient between two random vectors x and y is defined as<br />
P ðxi K xÞðy<br />
r Z<br />
i K yÞ<br />
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi<br />
P p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi<br />
ðxi K xÞ 2 P (2)<br />
ðyi K yÞ 2<br />
The main idea of the correlation coefficient (2) is to measure the normalized consistency in the signs<br />
of the remainders of two random vectors about their sample means. This is different from the inner<br />
product that is about zero without normalization. Using the <strong>statistical</strong> correlation coefficient for the same<br />
above data vectors x, y and z, we obtain r xy Z1 and r xz Z0.9587. After shifting the vectors x, y and z with<br />
x s , y s and z s , the correlations are still the same with r xs y s<br />
Z1 and r xs z s<br />
Z0:9587. This means that the<br />
similarity measures <strong>using</strong> the <strong>statistical</strong> correlation coefficient were not changed even through shifting<br />
two random vectors.<br />
There are some relations between the inner product and <strong>statistical</strong> correlation coefficient. Assume that<br />
~x 0 Z½~x 1 .~x n Š and ~y 0 Z½~y 1 .~y n Š are two normalized random vectors of x and y with<br />
~x i Z x P rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi<br />
P<br />
i K x<br />
xi<br />
where x Z<br />
S x<br />
n<br />
; S ðxi K xÞ 2<br />
x Z<br />
; ~y<br />
n K1 i Z y P<br />
i K y<br />
yi<br />
where y Z<br />
S y<br />
n ;<br />
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi<br />
P ðyi K yÞ 2<br />
S y Z<br />
:<br />
n K1<br />
We then have r xy Z 1<br />
nK1<br />
h ~x; ~yi. In this paper, we propose a <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> <strong>system</strong><br />
<strong>using</strong> the <strong>statistical</strong> correlation coefficient. Here we have six different unnatural <strong>pattern</strong>s named as<br />
upward shift, downward shift, upward trend (or increasing trend), downward trend (or decreasing trend),<br />
cycle and <strong>system</strong>atic <strong>pattern</strong>s (see Fig. 4). Before <strong>recognition</strong>, we implement a training algorithm to get<br />
N samples where each <strong>pattern</strong> sample has a length n. The <strong>pattern</strong> sample generators are defined as<br />
follows:<br />
(a) Normal <strong>pattern</strong><br />
xðtÞ Z nðtÞ; where nðtÞ follows a normal distribution Nð0; 1Þ: (3)
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J.-H. Yang, M.-S. Yang / Computers & Industrial Engineering 48 (2005) 205–221<br />
Fig. 4. Unnatural reference <strong>pattern</strong>s.<br />
(b) Upward and downward shift <strong>pattern</strong>s<br />
(<br />
0 before shifting<br />
xðtÞ Z nðtÞ Cud u Z<br />
1 after shifting ;<br />
(4)<br />
where d is the shift quantity randomly taken from 1 to 2.5 for upward shift and from K1toK2.5 for<br />
downward shift.<br />
(c) Upward and downward trend <strong>pattern</strong>s<br />
xðtÞ Z nðtÞGdt (5)<br />
where d is the trend slope randomly selected from 0.05 to 0.12 for upward trend and from K0.05<br />
to K0.12 for downward trend.<br />
(d) Cyclic <strong>pattern</strong><br />
<br />
xðtÞ Z nðtÞ Cd sin<br />
2pt <br />
(6)<br />
U<br />
where d is the amplitude randomly selected from 0.5 to 2.5 and U is the cycle length taken as UZ8<br />
here.
(e) Systematic <strong>pattern</strong><br />
J.-H. Yang, M.-S. Yang / Computers & Industrial Engineering 48 (2005) 205–221 211<br />
xðtÞ Z nðtÞ CðK1Þ t d (7)<br />
where d is the amplitude randomly selected from 0.5 to 2.5.<br />
As mentioned in Section 1, larger mean shifts decrease ARL values and may lose the essence of doing<br />
the <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> because of out-of-<strong>control</strong> signals appearing sooner and often. Thus,<br />
a principle for choosing abnormal disturbance level d (i.e. shift, slop and amplitude) is to keep the mean<br />
not more than 3s. Moreover, the <strong>pattern</strong> length n used in the generated samples for each <strong>pattern</strong> is<br />
important. We will discuss and demonstrate latter. Thus, the training algorithm is described as follows:<br />
Training Algorithm<br />
Step 1: Determine the <strong>pattern</strong> length n and the sample number N.<br />
Step 2: First, select a <strong>pattern</strong> sample generator and a disturbance level d from (4)–(7). Generate a <strong>pattern</strong><br />
vector x 1 with a <strong>pattern</strong> length n. Second, change the disturbance level d until N samples x 2 ,.x N<br />
are generated from the chosen <strong>pattern</strong>.<br />
Step 3: Estimate the reference <strong>pattern</strong> vector by <strong>using</strong><br />
P NiZ1<br />
x<br />
EðxÞ Z i<br />
:<br />
N<br />
We repeat the training algorithm until all six reference <strong>pattern</strong> vectors are generated for upward shift,<br />
downward shift, upward trend, downward trend, cyclic and <strong>system</strong>atic <strong>pattern</strong>s. Our final generated<br />
reference <strong>pattern</strong> vectors are shown in Fig. 4. We would regard these six generated reference <strong>pattern</strong>s as<br />
the prototypes to recognize input <strong>pattern</strong>s.<br />
The purpose of our training algorithm is to obtain six prototypes for recognizing input <strong>control</strong> <strong>chart</strong><br />
<strong>pattern</strong>s. The Step 3 in the training algorithm with a mean vector E(x) ofN samples for each <strong>pattern</strong> is<br />
used to obtain a reference vector as a representation of the unnatural <strong>pattern</strong>. After we finish the training<br />
stage, it is necessary to use Eqs. (3)–(7) to generate samples for testing. A threshold, h, is created for<br />
qualifying the winner whether it matches enough or not. If the similarity measure is smaller than h, the<br />
winner is not similar enough. We will classify it as a normal <strong>pattern</strong>. The mechanism will further help us<br />
to identify these normal conditions and continue until an unnatural <strong>pattern</strong> is recognized. Thus, a<br />
classification algorithm is created as follows:<br />
Classification Algorithm<br />
Step 1: Determine a threshold h.<br />
Step 2: A processing data sequence containing recent n points is regarded as the <strong>pattern</strong> size to be recognized.<br />
Step 3: Input the data sequence to the recogniser and calculate its <strong>statistical</strong> correlation coefficient <strong>using</strong><br />
(2) with six reference vectors to obtain outputs r u.s. , r d.s. r u.t. , r d.t. , r cyc , r sys for upward shift,<br />
downward shift, upward trend, downward trend, cyclic and <strong>system</strong>atic <strong>pattern</strong>s, respectively.
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Step 4: Choose the maximum value among all outputs to determine which <strong>pattern</strong> is a winner and then<br />
classified. However, if the maximum value is smaller than the threshold h, we classify it as a<br />
normal <strong>pattern</strong>.<br />
We continue the classification algorithm for input <strong>control</strong> <strong>chart</strong> points until an unnatural <strong>pattern</strong> is<br />
recognized or no point is inputted.<br />
The problem in our classification algorithm is about the selection of a threshold h and a <strong>pattern</strong><br />
length n. If a normal <strong>pattern</strong> is presented, then a false alarm will occur (i.e. type I error) when it is<br />
recognized as an unnatural <strong>pattern</strong>. We use the type I error to discuss how to select a <strong>pattern</strong> length n<br />
and a threshold h for our proposed correlation coefficient method. To demonstrate a better choice of h<br />
and n, we take 1000 samples from the normal <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> with different <strong>pattern</strong> lengths nZ8,<br />
16, 20, 30, 40 and 50. We then repeat 10 times with simulations for each given <strong>pattern</strong> length. We<br />
calculate the average type I error for different <strong>pattern</strong> length with respect to different threshold setting.<br />
The results are shown in Fig. 5.<br />
It is known that a type I error with the value 0.05 is generally a common choice. Since type I error is<br />
the measure of false alarm that we classify it to be an unnatural <strong>pattern</strong> when the data is normal, type II<br />
error is used to measure the capability of classification for unnatural <strong>pattern</strong>s. Clearly, a larger threshold<br />
h will decrease a type I error but increase a type II error. Therefore, too large or too small h is not suitable<br />
for <strong>recognition</strong>. According to the results in Fig. 5, the threshold hZ0.5 is a good choice where the type I<br />
errors for nZ20, 30, 40 and 50 are less than 0.05 but those for nZ8 and 16 are larger than 0.05. Thus, the<br />
<strong>pattern</strong> lengths nZ20, 30, 40 and 50 are reasonable choice in our simulations. On the other hand, we find<br />
that the <strong>pattern</strong> with a <strong>pattern</strong> length nZ50 in Fig. 6 has a threshold hZ0.3 when it has a type I error<br />
0.05. This smaller threshold 0.3 (compared to a threshold hZ0.5) may have larger type II error so that<br />
nZ50 is not a good choice. In fact, some expert commentators had explained that too large <strong>pattern</strong><br />
length is not practical. Moreover, ARL will decrease when it has more shift mean as shown in Table 1 so<br />
that a large <strong>pattern</strong> length will lose the spirit of preventing frequent out-of <strong>control</strong> occurrence in <strong>control</strong><br />
<strong>chart</strong>s. The trend <strong>pattern</strong> signifies that the mean is increasing (decreasing) with a slope so that a larger<br />
<strong>pattern</strong> length will cause later sample points to have large mean shift. Under these circumstances, we<br />
recommend a better choice of a <strong>pattern</strong> length with nZ30 and a threshold with hZ0.5 for our proposed<br />
correlation coefficient method.<br />
Fig. 5. Type I error to threshold for different <strong>pattern</strong> length.
J.-H. Yang, M.-S. Yang / Computers & Industrial Engineering 48 (2005) 205–221 213<br />
Fig. 6. Two <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s at different period with fixed length.<br />
3. Simulation results<br />
In this section, we use the proposed correlation coefficient method to simulate the <strong>recognition</strong> of<br />
<strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s. We first illustrate <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s <strong>using</strong> the <strong>chart</strong> shown in Fig. 6. The<br />
<strong>control</strong> <strong>chart</strong> in Fig. 6 is assumed to have an upward trend which occurs at the point tZ21 (i.e. the<br />
change point from normal to abnormal <strong>pattern</strong> at tZ21). According to the recommended <strong>pattern</strong> length<br />
with nZ30 in Section 2, we use a moving window with a window size 30 to mark the points. The <strong>pattern</strong><br />
in the window B contains all the points of an upward trend <strong>pattern</strong> from tZ21 up to tZ50. The <strong>pattern</strong> in<br />
the window A contains 30 points from tZ1 uptotZ30 where there are 20 normal points from tZ1 to<br />
tZ20 and 10 abnormal points from tZ21 to tZ30.<br />
Most of existing <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> recognisers are not considered for these kinds of <strong>control</strong> <strong>chart</strong>s.<br />
This is because it is difficult to predict when a change point from normal to abnormal will appear in<br />
<strong>control</strong> <strong>chart</strong>s. In most practical cases, normal points may often appear before abnormal points in <strong>control</strong><br />
<strong>chart</strong>s such as shown in Fig. 6. When we use a moving window for a <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong><br />
shown in Fig. 6, we find there are overlapping points between windows A and B with both normal and<br />
abnormal points from tZ21 up to tZ30. Under these circumstances, one may correctly classify the<br />
<strong>pattern</strong> in the window B but have an incorrect classification for the <strong>pattern</strong> in the window A. That is,<br />
when the <strong>pattern</strong> in the window A appears first, those recognisers may classify it as another unnatural<br />
<strong>pattern</strong>. For example, the window A in Fig. 6 contains half parts of normal and the other half of trend so<br />
that it looks like a shift <strong>pattern</strong>. Therefore, creating a mechanism to be able to suspend a recogniser<br />
making a classification when it has no enough evidence to be seen as an unnatural <strong>pattern</strong> shall be an<br />
important consideration. For solving this problem, we use a threshold h in our proposed correlation<br />
coefficient method.
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Fig. 7. Diagram of the correlation coefficient <strong>recognition</strong> <strong>system</strong>.<br />
On the other hand, unnatural <strong>pattern</strong>s, such as upward (downward) shift and upward (downward)<br />
trend, etc. may incur out-of-<strong>control</strong> signals during the <strong>pattern</strong> length nZ30. We can see this<br />
phenomenon from the ARL values shown in Table 1 of Section 1. The ARL values will drop down from<br />
370 to 6 when the mean m increases from 0 to 2. We know that the smaller ARL exhibits out-of-<strong>control</strong><br />
signal more often, and Fig. 6 is created with x(t)Zn(t)Cdt of Eq. (5) where d is selected from 0.05 to<br />
0.12. According to ARL shown in Table 1, it is quite possible for Shewhart <strong>control</strong> <strong>chart</strong> to have out-of<strong>control</strong><br />
signals for upward shift and trend <strong>pattern</strong>s. These out-of-<strong>control</strong> signals are actually occurred at<br />
nZ46, 48 and 49 in Fig. 6. In fact, some out-of-<strong>control</strong> signals will be also occurred in Fig. 8. Of course,<br />
if an out-of-<strong>control</strong> signal is issued in Shewhart <strong>control</strong> <strong>chart</strong>, we stop the process and check the causes<br />
and then remove them. If an unnatural cause can be detected and removed, such as upward shift or trend,<br />
the model will become x(t)Zn(t), i.e. a normal <strong>pattern</strong>. However, it may be difficult for detecting such<br />
unnatural <strong>pattern</strong>s <strong>using</strong> the Shewhart <strong>control</strong> <strong>chart</strong> because ARL will decrease when a mean m increases.<br />
This is the point that a <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> recogniser is useful for recognizing unnatural <strong>pattern</strong>s in<br />
<strong>control</strong> <strong>chart</strong>s. Thus, the main idea of the proposed correlation coefficient method is that, if we do not<br />
have enough evidence to classify an input <strong>pattern</strong> as an unnatural <strong>pattern</strong>, even though it has a maximum<br />
correlation coefficient value to reference <strong>pattern</strong>s, we still wait and take it as a normal <strong>pattern</strong>. The<br />
diagram of our <strong>recognition</strong> <strong>system</strong> is shown in Fig. 7.<br />
Because the <strong>statistical</strong> correlation coefficient r has a constraint with the range in the closed interval<br />
[K1,1], the threshold h can be used as a criterion what the similarity degree is enough for the <strong>recognition</strong><br />
of an unnatural <strong>pattern</strong>. The <strong>pattern</strong> length for generating an unnatural <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> and sample<br />
numbers in our simulations are chosen with nZ30 and NZ200. The moving window for the <strong>recognition</strong><br />
of <strong>control</strong> <strong>chart</strong>s uses a window size 30. For the normal <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>, we take the sample number<br />
with NZ1000. Based on a threshold h, unless there exists sufficiently unnatural <strong>pattern</strong> evidence,<br />
we regard it as a normal and wait for more <strong>pattern</strong> information. According to the recommendation in<br />
Section 2, we choose hZ0.5 in our simulations. We implement 1000 runs for each <strong>pattern</strong> and then<br />
calculate its average accuracy as shown in Table 2.<br />
We mention that these testing samples are also simulated <strong>using</strong> the inner-product technique proposed<br />
by Al-Ghanim and Ludeman (1997). There are almost the same average accuracy as our approach when<br />
hZ0.4 in Table 2 for recognizing upward trend, downward trend, cyclic and <strong>system</strong>atic <strong>pattern</strong>s.<br />
However, if we add upward shift and downward shift <strong>pattern</strong>s, then their <strong>recognition</strong> average accuracy
J.-H. Yang, M.-S. Yang / Computers & Industrial Engineering 48 (2005) 205–221 215<br />
Table 2<br />
Average accuracy for <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s<br />
u.s. d.s. u.t. d.t. cyc. sys. Normal<br />
hZ0.3 96.64 95.03 96.34 95.12 98.84 98.84 75.47<br />
hZ0.4 96.23 94.62 95.25 94.19 97.72 97.84 93.11<br />
hZ0.5 94.1 92.5 93.46 92.39 94.27 94.76 98.77<br />
hZ0.6 87.37 85.97 90.15 86.75 89.67 87.7 99.89<br />
becomes worse. This is because the inner-product technique is confused the shift <strong>pattern</strong> with the trend<br />
<strong>pattern</strong>. This phenomenon had been explained in Section 2.<br />
To analyze the feasibility of a change point in <strong>control</strong> <strong>chart</strong>s, we begin with 20 normal points in testing<br />
<strong>pattern</strong>s. Thus, the first 20 moving windows started from tZ1totZ20 with a fix window size 30 contain<br />
both normal and abnormal points. The window started at tZ21 (i.e. the window B as shown in Fig. 6)<br />
contains the whole upward trend <strong>pattern</strong> as shown in Fig. 6. We consider for recognizing upward shift, (or<br />
downward shift), upward trend (or downward trend), cycle and <strong>system</strong>atic <strong>pattern</strong>s in our simulations. All<br />
of these testing <strong>pattern</strong>s are shown as the above figures in Fig. 8(a)–(d). We calculate the <strong>statistical</strong><br />
correlation coefficient for each testing <strong>pattern</strong> with all six unnatural reference <strong>pattern</strong>s as shown in Fig. 4.<br />
The correlation coefficients results are shown as the below figures in Fig. 8(a)–(d). When the input point of<br />
the <strong>control</strong> <strong>chart</strong> does not reach the window size 30, a zero correlation coefficient is presented. That is, the<br />
correlation coefficient will be 0 from tZ1uptotZ29. When tZ30, the first correlation coefficient value is<br />
calculated. The window is moved ahead to calculate the second correlation coefficient value when tZ31,<br />
and so on. Since we set up a threshold h, the recogniser does not yet make a decision and still waits for more<br />
<strong>control</strong> <strong>chart</strong> points until a correlation coefficient value is over the h value.<br />
In Fig. 8(a), the given testing <strong>pattern</strong> is upward shift. There are six correlation coefficient curves<br />
between the given testing <strong>pattern</strong> and all six reference <strong>pattern</strong>s. If we choose hZ0.5, then we can<br />
recognize the testing <strong>pattern</strong> is an upward shift (u.s.) <strong>pattern</strong> when tZ48. Similarly, the upward trend<br />
<strong>pattern</strong> is recognized when tZ37 as shown in Fig. 8(b). We see, even though there is much similar<br />
between the upward trend and upward shift <strong>pattern</strong>s so that both curves of correlation coefficients in<br />
Fig. 8(a) and (b) have the same trend, it is still distinguishable after some points. For the cyclic and<br />
<strong>system</strong>atic <strong>pattern</strong>s shown in Fig. 8(c) and (d), they are quite distinguishable. We note that a correlation<br />
coefficient curve shown in Fig. 8(c) is also a cyclic <strong>pattern</strong>. This is because the cycle length is taken as<br />
UZ8 so that different starting points induce a cyclic correlation coefficient curve. We mention that there<br />
are out-of-<strong>control</strong> signals in Fig. 8(a)–(c) that are similar to those in Fig. 6. The explanation and process<br />
procedure for Fig. 6 can also be used for those in Fig. 8(a)–(c).<br />
4. Performance for recognizing concurrent <strong>pattern</strong>s<br />
A concurrent <strong>pattern</strong> <strong>recognition</strong> in <strong>control</strong> <strong>chart</strong>s was first investigated by Guh and Tannock (1999).<br />
They applied a back propagation network (BPN) for recognizing these concurrent <strong>pattern</strong>s where two<br />
unnatural <strong>pattern</strong>s may exist simultaneously. However, the training process of the BPN-based <strong>system</strong><br />
tends to be relatively slow (see Guh & Tannock, 1999). On the other hand, the BPN construction is also<br />
complicated for a <strong>control</strong> <strong>chart</strong> <strong>pattern</strong> <strong>recognition</strong> where its performance may heavily depend on
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the number of neurons and layers. In this section, we use the proposed correlation coefficient method to<br />
simulate the <strong>recognition</strong> of the concurrent <strong>control</strong> <strong>chart</strong> <strong>pattern</strong>s with shift and trend, shift and cycle and<br />
trend and cycle as shown in Fig. 9. In our simulations, we use 200 testing samples for each unnatural<br />
<strong>pattern</strong> and 1000 testing samples for the normal <strong>pattern</strong>. The <strong>recognition</strong> results for different threshold h<br />
are shown in Tables 3.<br />
Fig. 8. (a) Upward shift <strong>pattern</strong> and its correlation. (b) Upward trend <strong>pattern</strong> and its correlation. (c) Cyclic <strong>pattern</strong> and its<br />
correlation. (d) Systematic <strong>pattern</strong> and its correlation.
J.-H. Yang, M.-S. Yang / Computers & Industrial Engineering 48 (2005) 205–221 217<br />
Fig. 8 (continued)<br />
We know that a threshold h give the capability of the <strong>recognition</strong> for unnatural <strong>pattern</strong>s where a<br />
change point from normal to abnormal may appear at any point in <strong>control</strong> <strong>chart</strong>s. Thus, we can treat these<br />
cases including single and concurrent <strong>pattern</strong>s where an unnatural point may appear at any point in<br />
<strong>control</strong> <strong>chart</strong>s. Comparing the results in Tables 2 and 3, the <strong>recognition</strong> rate for these single
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Fig. 9. Concurrent <strong>pattern</strong>s.<br />
and concurrent <strong>pattern</strong>s as shown in Table 3(a)–(d) is actually lower than those single <strong>pattern</strong>s as shown<br />
in Table 2. However, the <strong>recognition</strong> rates in Tables 3(a)–(d) are still impressive.<br />
From Tables 3(a)–(d), we find that concurrent <strong>pattern</strong>s may be confused with relative single <strong>pattern</strong>s.<br />
For example, the correct <strong>recognition</strong> number for the concurrent <strong>pattern</strong> usCut is 158 for hZ0.5<br />
where 21 <strong>pattern</strong>s are incorrectly recognized to be u.s. and 21 <strong>pattern</strong>s are incorrectly recognized to be<br />
u.t. as shown in Table 3(c). All of these incorrect <strong>recognition</strong> <strong>pattern</strong>s for u.s.Cu.t. come from u.s.<br />
Table 3a<br />
Performance for different <strong>pattern</strong>s with threshold hZ0.3<br />
Input<br />
Recognized <strong>pattern</strong><br />
u.s.<br />
(%)<br />
d.s.<br />
(%)<br />
u.t.<br />
(%)<br />
d.t.<br />
(%)<br />
cyc.<br />
(%)<br />
sys.<br />
(%)<br />
u.s.C<br />
u.t.<br />
(%)<br />
d.s.C<br />
d.t.<br />
(%)<br />
u.s.C<br />
cyc.<br />
(%)<br />
d.s.C<br />
cyc.<br />
(%)<br />
u.t.C<br />
cyc.<br />
(%)<br />
d.t.C<br />
cyc.<br />
(%)<br />
Normal<br />
(%)<br />
u.s. 73.0 0.0 0.0 0.0 0.0 0.0 27.0 0.0 0.0 0.0 0.0 0.0 0.0<br />
d.s. 0.0 77.0 0.0 1.0 0.0 0.0 0.0 21.5 0.0 0.5 0.0 0.0 0.0<br />
u.t. 0.0 0.0 82.5 0.0 0.0 0.0 16.5 0.0 0.0 0.0 1.0 0.0 0.0<br />
d.t. 0.0 1.0 0.0 78.5 0.0 0.0 0.0 20.5 0.0 0.0 0.0 0.0 0.0<br />
cyc. 0.0 0.0 0.0 1.0 87.0 0.0 0.0 0.0 3.0 1.5 0.5 6.0 1.0<br />
sys. 0.0 0.0 0.0 0.0 0.0 98.5 0.0 0.0 0.0 0.0 0.0 0.0 1.5<br />
u.s.Cu.t. 13.0 0.0 12.0 0.0 0.0 0.0 75.0 0.0 0.0 0.0 0.0 0.0 0.0<br />
d.s.Cd.t. 0.0 14.5 0.0 11.5 0.0 0.0 0.0 73.5 0.0 0.5 0.0 0.0 0.0<br />
u.s.Ccyc. 8.5 0.0 0.0 0.0 2.5 0.0 2.5 0.0 71.5 0.0 15.0 0.0 0.0<br />
d.s.Ccyc. 0.0 7.5 0.0 1.5 4.5 0.0 0.0 6.0 0.0 69.0 0.0 11.5 0.0<br />
u.t.Ccyc. 1.5 0.0 8.5 0.0 2.5 0.0 6.0 0.0 10.5 0.0 71.0 0.0 0.0<br />
d.t.Ccyc. 0.0 0.5 0.0 7.0 6.0 0.0 0.0 0.5 0.0 16.0 0.0 70.0 0.0<br />
Normal 2.2 2.3 4.8 2.9 2.4 7.0 0.5 1.2 1.4 1.4 4.1 1.9 67.9
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Table 3b<br />
Performance for different <strong>pattern</strong>s with threshold hZ0.4<br />
Input<br />
Recognized <strong>pattern</strong><br />
u.s.<br />
(%)<br />
d.s.<br />
(%)<br />
u.t.<br />
(%)<br />
d.t.<br />
(%)<br />
cyc.<br />
(%)<br />
sys.<br />
(%)<br />
u.s.C<br />
u.t.<br />
(%)<br />
d.s.C<br />
d.t.<br />
(%)<br />
u.s.C<br />
cyc.<br />
(%)<br />
d.s.C<br />
cyc.<br />
(%)<br />
u.t.C<br />
cyc.<br />
(%)<br />
d.t.C<br />
cyc.<br />
(%)<br />
Normal<br />
(%)<br />
u.s. 78.5 0.0 0.0 0.0 0.0 0.0 21.0 0.0 0.5 0.0 0.0 0.0 0.0<br />
d.s. 0.0 76.5 0.0 0.5 0.0 0.0 0.0 20.5 0.0 2.5 0.0 0.0 0.0<br />
u.t. 0.0 0.0 82.0 0.0 0.0 0.5 17.0 0.0 0.0 0.0 0.5 0.0 0.0<br />
d.t. 0.0 0.0 0.0 84.0 0.0 0.0 0.0 11.5 0.0 2.0 0.0 2.5 0.0<br />
cyc. 0.0 0.0 0.0 0.0 86.0 0.0 0.0 0.0 1.0 1.0 1.0 4.0 7.0<br />
sys. 0.0 0.0 0.0 0.0 0.0 95.5 0.0 0.0 0.0 0.0 0.0 0.0 4.5<br />
u.s.Cu.t. 11.0 0.0 9.5 0.0 0.0 0.0 79.5 0.0 0.0 0.0 0.0 0.0 0.0<br />
d.s.Cd.t. 0.0 12.0 0.0 14.5 0.0 0.0 0.0 73.0 0.0 0.5 0.0 0.0 0.0<br />
u.s.Ccyc. 11.0 0.0 0.0 0.0 1.0 0.0 2.5 0.0 74.5 0.0 11.0 0.0 0.0<br />
d.s.Ccyc. 0.0 2.5 0.0 2.0 2.0 0.0 0.0 3.5 0.0 74.5 0.0 15.0 0.5<br />
u.t.Ccyc. 5.0 0.0 4.0 0.0 3.5 0.0 5.5 0.0 11.0 0.0 70.5 0.0 0.5<br />
d.t.Ccyc. 0.0 0.0 0.0 7.0 4.5 0.0 0.0 2.0 0.0 14.0 0.0 72.5 0.0<br />
Normal 0.2 0.8 1.1 0.7 1.0 1.9 0.0 0.2 1.4 0.9 0.7 0.2 90.9<br />
and u.t. This result is very reasonable. Totally, the results in Tables 3(a)–(d) show that our proposed<br />
<strong>statistical</strong> correlation coefficient method works well for recognizing single and concurrent <strong>control</strong> <strong>chart</strong><br />
<strong>pattern</strong>s with good performance even though a change point from normal to abnormal occurs in <strong>control</strong><br />
<strong>chart</strong>s.<br />
Table 3c<br />
Performance for different <strong>pattern</strong>s with threshold hZ0.5<br />
Input<br />
Recognized <strong>pattern</strong><br />
u.s.<br />
(%)<br />
d.s.<br />
(%)<br />
u.t.<br />
(%)<br />
d.t.<br />
(%)<br />
cyc.<br />
(%)<br />
sys.<br />
(%)<br />
u.s.C<br />
u.t.<br />
(%)<br />
d.s.C<br />
d.t.<br />
(%)<br />
u.s.C<br />
cyc.<br />
(%)<br />
d.s.C<br />
cyc.<br />
(%)<br />
u.t.C<br />
cyc.<br />
(%)<br />
d.t.C<br />
cyc.<br />
(%)<br />
Normal<br />
(%)<br />
u.s. 76.0 0.0 0.5 0.0 0.0 0.0 21.0 0.0 1.5 0.0 0.0 0.0 1.0<br />
d.s. 0.0 73.0 0.0 1.0 0.0 0.0 0.0 22.5 0.0 2.5 0.0 0.0 1.0<br />
u.t. 0.0 0.0 80.0 0.0 0.0 0.0 19.5 0.0 0.0 0.0 0.0 0.0 0.5<br />
d.t. 0.0 0.0 0.0 77.0 0.0 0.0 0.0 20.0 0.0 0.5 0.0 1.0 1.5<br />
cyc. 0.0 0.0 0.0 0.0 83.0 0.0 0.0 0.0 1.0 0.5 1.5 1.0 13.0<br />
sys. 0.0 0.0 0.0 0.0 0.0 96.5 0.0 0.0 0.0 0.0 0.0 0.0 3.5<br />
u.s.Cu.t. 10.5 0.0 10.5 0.0 0.0 0.0 79.0 0.0 0.0 0.0 0.0 0.0 0.0<br />
d.s.Cd.t. 0.0 18.0 0.0 13.0 0.0 0.0 0.0 69.0 0.0 0.0 0.0 0.0 0.0<br />
u.s.Ccyc. 9.0 0.0 0.0 0.0 3.5 0.0 1.5 0.0 73.0 0.0 13.0 0.0 0.0<br />
d.s.Ccyc. 0.0 11.5 0.0 1.0 3.5 0.0 0.0 3.5 0.0 67.0 0.0 11.0 2.5<br />
u.t.Ccyc. 1.5 0.0 3.0 0.0 2.0 0.0 1.0 0.0 10.5 0.0 77.0 0.0 5.0<br />
d.t.Ccyc. 0.0 0.0 0.0 6.5 8.0 0.0 0.0 0.0 0.0 10.0 0.0 72.5 3.0<br />
Normal 0.0 0.0 0.0 0.2 0.0 0.5 0.0 0.2 0.2 0.8 0.0 0.0 98.1
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Table 3d<br />
Performance for different <strong>pattern</strong>s with threshold hZ0.6<br />
Input<br />
Recognized <strong>pattern</strong><br />
u.s.<br />
(%)<br />
d.s.<br />
(%)<br />
u.t.<br />
(%)<br />
d.t.<br />
(%)<br />
cyc.<br />
(%)<br />
sys.<br />
(%)<br />
u.s.C<br />
u.t.<br />
(%)<br />
d.s.C<br />
d.t.<br />
(%)<br />
u.s.C<br />
cyc.<br />
(%)<br />
d.s.C<br />
cyc.<br />
(%)<br />
u.t.C<br />
cyc.<br />
(%)<br />
d.t.C<br />
cyc.<br />
(%)<br />
Normal<br />
(%)<br />
u.s. 70.5 0.0 0.0 0.0 0.0 0.0 24.0 0.0 1.5 0.0 0.0 0.0 4.0<br />
d.s. 0.0 73.5 0.0 1.0 0.0 0.0 0.0 20.0 0.0 0.0 0.0 0.0 5.5<br />
u.t. 0.0 0.0 70.0 0.0 0.0 0.0 17.5 0.0 0.0 0.0 0.5 0.0 12.0<br />
d.t. 0.0 0.0 0.0 71.0 0.0 0.0 0.0 14.5 0.0 0.5 0.0 2.0 12.0<br />
cyc. 0.0 0.0 0.0 0.0 72.5 0.0 0.0 0.0 0.0 0.0 1.0 1.5 25.0<br />
sys. 0.0 0.0 0.0 0.0 0.0 86.0 0.0 0.0 0.0 0.0 0.0 0.0 14.0<br />
u.s.Cu.t. 10.5 0.0 14.0 0.0 0.0 0.0 75.0 0.0 0.0 0.0 0.0 0.0 0.5<br />
d.s.Cd.t. 0.0 15.5 0.0 12.5 0.0 0.0 0.0 72.0 0.0 0.0 0.0 0.0 0.0<br />
u.s.Ccyc. 6.5 0.0 0.0 0.0 2.0 0.0 0.5 0.0 74.5 0.0 9.5 0.0 7.0<br />
d.s.Ccyc. 0.0 4.5 0.0 1.0 2.0 0.0 0.0 3.5 0.0 69.0 0.0 12.0 8.0<br />
u.t.Ccyc. 0.5 0.0 2.5 0.0 3.0 0.0 4.0 0.0 8.0 0.0 65.0 0.0 17.0<br />
d.t.Ccyc. 0.0 0.0 0.0 5.0 4.0 0.0 0.0 0.0 0.0 10.5 0.0 70.0 10.5<br />
Normal 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0<br />
5. Conclusions<br />
In this paper, we use a <strong>statistical</strong> correlation coefficient to create a simple mechanism for recognizing<br />
single and concurrent unnatural <strong>pattern</strong>s where a change point from normal to abnormal may occur in<br />
<strong>control</strong> <strong>chart</strong>s. By adding a threshold criterion, the proposed method becomes more powerful for<br />
practical applications. The selection of a <strong>pattern</strong> length and a threshold was also discussed in this paper.<br />
According to our simulations, the proposed <strong>statistical</strong> correlation coefficient method presents good<br />
results. Tedious learning training process is not necessary for our method. The proposed mechanism<br />
actually enhances the ability of <strong>recognition</strong> so that it helps us discover irregularities in a manufacturing<br />
process as early as possible to reduce flawed products. Overall, the proposed approach is a simple<br />
mechanism for recognizing unnatural <strong>pattern</strong>s in <strong>control</strong> <strong>chart</strong>s.<br />
Acknowledgements<br />
The authors are grateful to the anonymous referees for their helpful comments and suggestions to<br />
improve the presentation of the paper. This work was supported in part by the National Science Council<br />
of Taiwan, ROC, under Grant NSC-89-2213-E-033-057.<br />
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