A control chart pattern recognition system using a ... - Semantic Scholar
Computers & Industrial Engineering 48 (2005) 205–221 www.elsevier.com/locate/dsw
A control chart pattern recognition system using a statistical correlation coefficient method* Jenn-Hwai Yang, Miin-Shen Yang* Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan 32023, ROC
Abstract This paper presents a control chart pattern recognition system using a statistical correlation coefficient method. Pattern recognition techniques have been widely applied to identify unnatural patterns in control charts. Most of them are capable of recognizing a single unnatural pattern for different abnormal types. However, before an unnatural pattern occurs, a change point from normal to abnormal may appear at any point in control charts for most practical cases. Moreover, concurrent patterns where two unnatural patterns simultaneously exist may also occur in a control chart pattern recognition system. Our statistical correlation coefficient approach is a simple mechanism for recognizing these unnatural control chart patterns with good performance. This approach is also an effective method for the control chart pattern recognition without a tedious training process. q 2005 Elsevier Ltd. All rights reserved. Keywords: Control chart; Pattern recognition; Statistical correlation coefficient; Recognition rate; Change point; Concurrent pattern
1. Introduction Due to an increasing competition in products, consumers have become more critical in choosing products. The quality of products has become more important. Statistical process control (SPC) is usually used to improve the quality of products and reduce rework and scrap so that the quality expectation can be met (Grant & Leavenworth, 1996). Shewhart control charts (1931) are the most popular charts widely used in industry to detect abnormal process behavior. The most typical form of control charts consists of a central line and two control limits representing the specifications of the product and the variant range limits. This provides a useful method *
This manuscript was processed by Area Editor E.A. Elsayed. * Corresponding author. Tel.: C886 3 2653100; fax: C886 3 2653160. E-mail address: [email protected] (M.-S. Yang).
0360-8352/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2005.01.008
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for monitoring variations in the product manufacturing process. However, these control charts do not provide pattern-related information because they focus only on the latest plotted data points. In other words, only the last point on the chart is examined rather than the trend in the process over time. The value of average run length (ARL) is usually used to measure the frequency before the process is out-of-control. ARL is generally defined as the average number of points in a control chart until an out-ofcontrol signal occurs. Under a normal distribution N(m,s2) assumption with its mean m and variance s2, it is called out-of-control when its value is outside the range of G3s from the mean m. In general, it is unavoidable due to a random noise even though in a regular condition. For example, when we have a production process that the quality of products is under the N(0,1) assumption, the probability P(X%K3 or XR3)Z0.0027 such that 1/0.0027z370.4. Thus, ARL under the N(0,1) assumption is around 370.4. Similarly, for mZ0.5 with s2Z1, ARL is around 155.2. The ARL value for other mean shift m with s2Z1 is shown in Table 1. We can see that ARL will be decreasing when a mean shift m is increasing. We know that the smaller ARL exhibits out-of-control signal more often. In this case, it is not good for detecting abnormal process behavior with a Shewhart control chart because it is variable due to a mean shift m. Therefore, the control chart pattern recognition is important for recognizing unnatural patterns in control charts. Control chart pattern recognition has the capability to recognize unnatural patterns. In general, there are six unnatural patterns in control charts. They are upward trend, downward trend, upward shift, downward shift, cycle and systematic variation. These patterns present the long-term behavior of a process. If any of these patterns appears, something has occurred in the process such as a change of raw materials, equipment damage or the state of workers. If we can have such information on hand and then make decisions on the process, it will help us prevent failures appearing in the future. There have been many studies on control chart pattern recognition (see Al-Ghanim & Ludeman, 1997; Cheng, 1997; Guh & Hsieh, 1999; Pham & Oztemel, 1994; Yang & Yang, 2002). Recently, Guh (2003) proposed a hybrid artificial intelligence technique that consists of three major sub-systems so that it can be well used to build a real time SPC system. Most of these studies are considered for recognizing a single unnatural control chart pattern. However, in most real control chart applications, normal points may appear before abnormal points so that a change point from normal to abnormal may occur at any point in control charts. If we do not have a mechanism for recognizing such change patterns, we may incorrectly obtain the classification results. Therefore, under these circumstances, even though we have a good classification method for the control chart pattern recognition, we may still make an incorrect decision. Table 1 Average run length with a mean shift m m
ARL
0 0.5 1 1.5 2 2.5 3
370.4 155.2 43.9 15 6.3 3.2 2
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Another practical situation is concurrent patterns where two unnatural patterns may exist together. For recognizing concurrent unnatural patterns, Guh & Tannock (1999) used a back-propagation network (BPN) learning to perform the recognition for these concurrent control chart patterns. Although this BPN has a learning structure with a good network construction, the approach to the control chart pattern recognition is slow and complicated with a learning process. In this paper we propose a simple mechanism to recognize these unnatural patterns. The statistical correlation coefficient approach is used to construct a control chart pattern recognition system. By adding a threshold criterion, we can recognize these unnatural patterns even though they may change from normal to abnormal at any point in control charts. The remainder of this paper is organized as follows. In Section 2 we explain why a statistical correlation coefficient method is proposed and then present the proposed approach with the training and classification algorithms. Some simulation results and their comparisons are presented in Section 3. In Section 4, we present the performance for recognizing concurrent control chart patterns. Conclusions will be stated in Section 5.
2. The proposed control chart pattern recognition system Since control chart pattern recognition is an important step for industrial production processes, many researchers have made efforts toward finding various efficient methods for recognizing unnatural patterns in control charts. Hwarng and Hubele (1993) used a back-propagation neural network technique for detecting X-bar control charts. Cheng (1997); Guh and Hsieh (1999) used a neural network approach for recognizing an unnatural control chart pattern. Pham and Oztemel (1994) constructed a pattern recogniser using a learning vector quantization (LVQ) network. Yang and Yang (2002) created a fuzzy-soft LVQ to promote the recognition rate for control chart patterns. In order to decrease the number of reference vectors for each pattern used to identify different variation quantities, Al-Ghanim and Ludeman (1997) constructed a so-called ‘matched filter’ using a correlation analysis technique based on an inner pattern vector product. For concurrent control chart pattern recognition, Guh and Tannock (1999) proposed a back-propagation approach. In this section, we will propose a statistical correlation coefficient approach for the recognition of unnatural control chart patterns. The so-called matched filter approach involves obtaining a prototype for each pattern, called a ‘reference vector’ after the training algorithm is finished. Al-Ghanim and Ludeman (1997) evaluated the correlation between reference vectors and the input vector using the inner product. Let x and y be two random vectors with x 0 Z[x1.xn] and y 0 Z[y1.yn]. Then the inner product between x and y is defined as X xi yi : (1) hx; yi Z x 0 y Z Their inner-product correlation technique got good results in recognizing trend, cycle and systematic patterns. However, this method is poor in recognizing shift patterns. This is because the inner product presents the consistency in positive and negative components values in two vectors, but each component at the same position with an opposite sign will counteract the final inner product value. For example, if there are two vectors with nearly positive components, the inner-product correlation will become worse.
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Fig. 1. Illustration of the patterns x, y and z.
Let us consider the following three patterns with x 0 Z ½ K2 K2
0
2
2 ; y 0 Z ½ K2 K2
0
2
2 ; and z 0 Z ½ K1
0
1
2 2 :
The graph of the patterns x, y and z is shown in Fig. 1. Obviously, the similarity between x and y is larger than x and z. Calculating their correlation on the basis of inner product, we have hx; yi Z 16 and hx; yi Z 10: A larger correlation value represents higher similarity. The result is no doubt that x is more similar to y than to z. Now, let us shift the patterns x, y and z with a value of 2. That is, xs0 Z ½ 0
0
2 4
4 ; ys0 Z ½ 0
0
2
4
4 ; and zs0 Z ½ 1
2
3
4
4
A graph of the shifted patterns xs, ys and zs is shown in Fig. 2. We also calculate their inner product values with hxs ; ys i Z 36 and hxs ; zs i Z 38: The results show that x is more similar to z than to y that is obviously incorrect. This is because all of the components are nonnegative in xs, ys and zs. Under that condition, a larger component value determines a larger correlation value. Thus, the shift pattern will be difficult to be recognized on the basis of the inner-product approach. As we analyze the characteristic with upward shift and upward trend patterns, shown in Fig. 3, we note that both of their components are larger in the later part and tend to be positive. However, the trend toward large values in the upward trend pattern is faster than in the upward shift pattern. Thus, a shift pattern is always classified as a trend pattern because it has
Fig. 2. Illustration of the patterns xs, ys and zs.
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Fig. 3. Upward shift and upward trend patterns.
a maximum inner-product value. This produces an incorrect recognition. We can use the statistical correlation coefficient method to solve this problem as follows. The statistical correlation coefficient between two random vectors x and y is defined as P i K yÞ ðxi K xÞðy ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r Z pP P 2 2 ðxi K xÞ ðyi K yÞ
(2)
The main idea of the correlation coefficient (2) is to measure the normalized consistency in the signs of the remainders of two random vectors about their sample means. This is different from the inner product that is about zero without normalization. Using the statistical correlation coefficient for the same above data vectors x, y and z, we obtain rxyZ1 and rxzZ0.9587. After shifting the vectors x, y and z with xs, ys and zs, the correlations are still the same with rxs ys Z 1 and rxs zs Z 0:9587. This means that the similarity measures using the statistical correlation coefficient were not changed even through shifting two random vectors. There are some relations between the inner product and statistical correlation coefficient. Assume that x~ 0 Z ½x~1 .x~n and y~ 0 Z ½y~1 .y~n are two normalized random vectors of x and y with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P P 2 xi ðxi K xÞ yi xi K x yi K y ; y~i Z where x Z where y Z x~i Z ; Sx Z ; Sx Sy n n K1 n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2 ðyi K yÞ : Sy Z n K1 1 ~ yi. ~ In this paper, we propose a control chart pattern recognition system We then have rxy Z nK1 hx; using the statistical correlation coefficient. Here we have six different unnatural patterns named as upward shift, downward shift, upward trend (or increasing trend), downward trend (or decreasing trend), cycle and systematic patterns (see Fig. 4). Before recognition, we implement a training algorithm to get N samples where each pattern sample has a length n. The pattern sample generators are defined as follows:
(a) Normal pattern xðtÞ Z nðtÞ; where nðtÞ follows a normal distribution Nð0; 1Þ:
(3)
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Fig. 4. Unnatural reference patterns.
(b) Upward and downward shift patterns ( 0 before shifting ; xðtÞ Z nðtÞ C ud u Z 1 after shifting
(4)
where d is the shift quantity randomly taken from 1 to 2.5 for upward shift and from K1 to K2.5 for downward shift. (c) Upward and downward trend patterns xðtÞ Z nðtÞGdt
(5)
where d is the trend slope randomly selected from 0.05 to 0.12 for upward trend and from K0.05 to K0.12 for downward trend. (d) Cyclic pattern 2pt xðtÞ Z nðtÞ C d sin (6) U where d is the amplitude randomly selected from 0.5 to 2.5 and U is the cycle length taken as UZ8 here.
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(e) Systematic pattern xðtÞ Z nðtÞ C ðK1Þt d
(7)
where d is the amplitude randomly selected from 0.5 to 2.5. As mentioned in Section 1, larger mean shifts decrease ARL values and may lose the essence of doing the control chart pattern recognition because of out-of-control signals appearing sooner and often. Thus, a principle for choosing abnormal disturbance level d (i.e. shift, slop and amplitude) is to keep the mean not more than 3s. Moreover, the pattern length n used in the generated samples for each pattern is important. We will discuss and demonstrate latter. Thus, the training algorithm is described as follows:
Training Algorithm Step 1: Determine the pattern length n and the sample number N. Step 2: First, select a pattern sample generator and a disturbance level d from (4)–(7). Generate a pattern vector x1 with a pattern length n. Second, change the disturbance level d until N samples x2,.xN are generated from the chosen pattern. Step 3: Estimate the reference pattern vector by using PN EðxÞ Z
iZ1 xi
N
:
We repeat the training algorithm until all six reference pattern vectors are generated for upward shift, downward shift, upward trend, downward trend, cyclic and systematic patterns. Our final generated reference pattern vectors are shown in Fig. 4. We would regard these six generated reference patterns as the prototypes to recognize input patterns. The purpose of our training algorithm is to obtain six prototypes for recognizing input control chart patterns. The Step 3 in the training algorithm with a mean vector E(x) of N samples for each pattern is used to obtain a reference vector as a representation of the unnatural pattern. After we finish the training stage, it is necessary to use Eqs. (3)–(7) to generate samples for testing. A threshold, h, is created for qualifying the winner whether it matches enough or not. If the similarity measure is smaller than h, the winner is not similar enough. We will classify it as a normal pattern. The mechanism will further help us to identify these normal conditions and continue until an unnatural pattern is recognized. Thus, a classification algorithm is created as follows:
Classification Algorithm Step 1: Determine a threshold h. Step 2: A processing data sequence containing recent n points is regarded as the pattern size to be recognized. Step 3: Input the data sequence to the recogniser and calculate its statistical correlation coefficient using (2) with six reference vectors to obtain outputs ru.s., rd.s. ru.t., rd.t., rcyc, rsys for upward shift, downward shift, upward trend, downward trend, cyclic and systematic patterns, respectively.
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Step 4: Choose the maximum value among all outputs to determine which pattern is a winner and then classified. However, if the maximum value is smaller than the threshold h, we classify it as a normal pattern. We continue the classification algorithm for input control chart points until an unnatural pattern is recognized or no point is inputted. The problem in our classification algorithm is about the selection of a threshold h and a pattern length n. If a normal pattern is presented, then a false alarm will occur (i.e. type I error) when it is recognized as an unnatural pattern. We use the type I error to discuss how to select a pattern length n and a threshold h for our proposed correlation coefficient method. To demonstrate a better choice of h and n, we take 1000 samples from the normal control chart pattern with different pattern lengths nZ8, 16, 20, 30, 40 and 50. We then repeat 10 times with simulations for each given pattern length. We calculate the average type I error for different pattern length with respect to different threshold setting. The results are shown in Fig. 5. It is known that a type I error with the value 0.05 is generally a common choice. Since type I error is the measure of false alarm that we classify it to be an unnatural pattern when the data is normal, type II error is used to measure the capability of classification for unnatural patterns. Clearly, a larger threshold h will decrease a type I error but increase a type II error. Therefore, too large or too small h is not suitable for recognition. According to the results in Fig. 5, the threshold hZ0.5 is a good choice where the type I 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 pattern lengths nZ20, 30, 40 and 50 are reasonable choice in our simulations. On the other hand, we find that the pattern with a pattern length nZ50 in Fig. 6 has a threshold hZ0.3 when it has a type I error 0.05. This smaller threshold 0.3 (compared to a threshold hZ0.5) may have larger type II error so that nZ50 is not a good choice. In fact, some expert commentators had explained that too large pattern length is not practical. Moreover, ARL will decrease when it has more shift mean as shown in Table 1 so that a large pattern length will lose the spirit of preventing frequent out-of control occurrence in control charts. The trend pattern signifies that the mean is increasing (decreasing) with a slope so that a larger pattern length will cause later sample points to have large mean shift. Under these circumstances, we recommend a better choice of a pattern length with nZ30 and a threshold with hZ0.5 for our proposed correlation coefficient method.
Fig. 5. Type I error to threshold for different pattern length.
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Fig. 6. Two control chart patterns at different period with fixed length.
3. Simulation results In this section, we use the proposed correlation coefficient method to simulate the recognition of control chart patterns. We first illustrate control chart patterns using the chart shown in Fig. 6. The control chart in Fig. 6 is assumed to have an upward trend which occurs at the point tZ21 (i.e. the change point from normal to abnormal pattern at tZ21). According to the recommended pattern length with nZ30 in Section 2, we use a moving window with a window size 30 to mark the points. The pattern in the window B contains all the points of an upward trend pattern from tZ21 up to tZ50. The pattern in the window A contains 30 points from tZ1 up to tZ30 where there are 20 normal points from tZ1 to tZ20 and 10 abnormal points from tZ21 to tZ30. Most of existing control chart pattern recognisers are not considered for these kinds of control charts. This is because it is difficult to predict when a change point from normal to abnormal will appear in control charts. In most practical cases, normal points may often appear before abnormal points in control charts such as shown in Fig. 6. When we use a moving window for a control chart pattern recognition shown in Fig. 6, we find there are overlapping points between windows A and B with both normal and abnormal points from tZ21 up to tZ30. Under these circumstances, one may correctly classify the pattern in the window B but have an incorrect classification for the pattern in the window A. That is, when the pattern in the window A appears first, those recognisers may classify it as another unnatural pattern. For example, the window A in Fig. 6 contains half parts of normal and the other half of trend so that it looks like a shift pattern. Therefore, creating a mechanism to be able to suspend a recogniser making a classification when it has no enough evidence to be seen as an unnatural pattern shall be an important consideration. For solving this problem, we use a threshold h in our proposed correlation coefficient method.
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Fig. 7. Diagram of the correlation coefficient recognition system.
On the other hand, unnatural patterns, such as upward (downward) shift and upward (downward) trend, etc. may incur out-of-control signals during the pattern length nZ30. We can see this phenomenon from the ARL values shown in Table 1 of Section 1. The ARL values will drop down from 370 to 6 when the mean m increases from 0 to 2. We know that the smaller ARL exhibits out-of-control 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 0.12. According to ARL shown in Table 1, it is quite possible for Shewhart control chart to have out-ofcontrol signals for upward shift and trend patterns. These out-of-control signals are actually occurred at nZ46, 48 and 49 in Fig. 6. In fact, some out-of-control signals will be also occurred in Fig. 8. Of course, if an out-of-control signal is issued in Shewhart control chart, we stop the process and check the causes and then remove them. If an unnatural cause can be detected and removed, such as upward shift or trend, the model will become x(t)Zn(t), i.e. a normal pattern. However, it may be difficult for detecting such unnatural patterns using the Shewhart control chart because ARL will decrease when a mean m increases. This is the point that a control chart pattern recogniser is useful for recognizing unnatural patterns in control charts. Thus, the main idea of the proposed correlation coefficient method is that, if we do not have enough evidence to classify an input pattern as an unnatural pattern, even though it has a maximum correlation coefficient value to reference patterns, we still wait and take it as a normal pattern. The diagram of our recognition system is shown in Fig. 7. Because the statistical correlation coefficient r has a constraint with the range in the closed interval [K1,1], the threshold h can be used as a criterion what the similarity degree is enough for the recognition of an unnatural pattern. The pattern length for generating an unnatural control chart pattern and sample numbers in our simulations are chosen with nZ30 and NZ200. The moving window for the recognition of control charts uses a window size 30. For the normal control chart pattern, we take the sample number with NZ1000. Based on a threshold h, unless there exists sufficiently unnatural pattern evidence, we regard it as a normal and wait for more pattern information. According to the recommendation in Section 2, we choose hZ0.5 in our simulations. We implement 1000 runs for each pattern and then calculate its average accuracy as shown in Table 2. We mention that these testing samples are also simulated using the inner-product technique proposed by Al-Ghanim and Ludeman (1997). There are almost the same average accuracy as our approach when hZ0.4 in Table 2 for recognizing upward trend, downward trend, cyclic and systematic patterns. However, if we add upward shift and downward shift patterns, then their recognition average accuracy
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Table 2 Average accuracy for control chart patterns
hZ0.3 hZ0.4 hZ0.5 hZ0.6
u.s.
d.s.
u.t.
d.t.
cyc.
sys.
Normal
96.64 96.23 94.1 87.37
95.03 94.62 92.5 85.97
96.34 95.25 93.46 90.15
95.12 94.19 92.39 86.75
98.84 97.72 94.27 89.67
98.84 97.84 94.76 87.7
75.47 93.11 98.77 99.89
becomes worse. This is because the inner-product technique is confused the shift pattern with the trend pattern. This phenomenon had been explained in Section 2. To analyze the feasibility of a change point in control charts, we begin with 20 normal points in testing patterns. Thus, the first 20 moving windows started from tZ1 to tZ20 with a fix window size 30 contain both normal and abnormal points. The window started at tZ21 (i.e. the window B as shown in Fig. 6) contains the whole upward trend pattern as shown in Fig. 6. We consider for recognizing upward shift, (or downward shift), upward trend (or downward trend), cycle and systematic patterns in our simulations. All of these testing patterns are shown as the above figures in Fig. 8(a)–(d). We calculate the statistical correlation coefficient for each testing pattern with all six unnatural reference patterns as shown in Fig. 4. The correlation coefficients results are shown as the below figures in Fig. 8(a)–(d). When the input point of the control chart does not reach the window size 30, a zero correlation coefficient is presented. That is, the correlation coefficient will be 0 from tZ1 up to tZ29. When tZ30, the first correlation coefficient value is calculated. The window is moved ahead to calculate the second correlation coefficient value when tZ31, and so on. Since we set up a threshold h, the recogniser does not yet make a decision and still waits for more control chart points until a correlation coefficient value is over the h value. In Fig. 8(a), the given testing pattern is upward shift. There are six correlation coefficient curves between the given testing pattern and all six reference patterns. If we choose hZ0.5, then we can recognize the testing pattern is an upward shift (u.s.) pattern when tZ48. Similarly, the upward trend pattern is recognized when tZ37 as shown in Fig. 8(b). We see, even though there is much similar between the upward trend and upward shift patterns so that both curves of correlation coefficients in Fig. 8(a) and (b) have the same trend, it is still distinguishable after some points. For the cyclic and systematic patterns shown in Fig. 8(c) and (d), they are quite distinguishable. We note that a correlation coefficient curve shown in Fig. 8(c) is also a cyclic pattern. This is because the cycle length is taken as UZ8 so that different starting points induce a cyclic correlation coefficient curve. We mention that there are out-of-control signals in Fig. 8(a)–(c) that are similar to those in Fig. 6. The explanation and process procedure for Fig. 6 can also be used for those in Fig. 8(a)–(c).
4. Performance for recognizing concurrent patterns A concurrent pattern recognition in control charts was first investigated by Guh and Tannock (1999). They applied a back propagation network (BPN) for recognizing these concurrent patterns where two unnatural patterns may exist simultaneously. However, the training process of the BPN-based system tends to be relatively slow (see Guh & Tannock, 1999). On the other hand, the BPN construction is also complicated for a control chart pattern recognition 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 simulate the recognition of the concurrent control chart patterns with shift and trend, shift and cycle and trend and cycle as shown in Fig. 9. In our simulations, we use 200 testing samples for each unnatural pattern and 1000 testing samples for the normal pattern. The recognition results for different threshold h are shown in Tables 3.
Fig. 8. (a) Upward shift pattern and its correlation. (b) Upward trend pattern and its correlation. (c) Cyclic pattern and its correlation. (d) Systematic pattern and its correlation.
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Fig. 8 (continued )
We know that a threshold h give the capability of the recognition for unnatural patterns where a change point from normal to abnormal may appear at any point in control charts. Thus, we can treat these cases including single and concurrent patterns where an unnatural point may appear at any point in control charts. Comparing the results in Tables 2 and 3, the recognition rate for these single
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Fig. 9. Concurrent patterns.
and concurrent patterns as shown in Table 3(a)–(d) is actually lower than those single patterns as shown in Table 2. However, the recognition rates in Tables 3(a)–(d) are still impressive. From Tables 3(a)–(d), we find that concurrent patterns may be confused with relative single patterns. For example, the correct recognition number for the concurrent pattern usCut is 158 for hZ0.5 where 21 patterns are incorrectly recognized to be u.s. and 21 patterns are incorrectly recognized to be u.t. as shown in Table 3(c). All of these incorrect recognition patterns for u.s.Cu.t. come from u.s. Table 3a Performance for different patterns with threshold hZ0.3 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. d.s. u.t. (%) (%) (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
73.0 0.0 0.0 0.0 0.0 0.0 13.0 0.0 8.5 0.0 1.5 0.0 2.2
0.0 1.0 0.0 78.5 1.0 0.0 0.0 11.5 0.0 1.5 0.0 7.0 2.9
0.0 0.0 0.0 0.0 87.0 0.0 0.0 0.0 2.5 4.5 2.5 6.0 2.4
0.0 0.0 0.0 0.0 0.0 98.5 0.0 0.0 0.0 0.0 0.0 0.0 7.0
27.0 0.0 16.5 0.0 0.0 0.0 75.0 0.0 2.5 0.0 6.0 0.0 0.5
0.0 21.5 0.0 20.5 0.0 0.0 0.0 73.5 0.0 6.0 0.0 0.5 1.2
0.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 71.5 0.0 10.5 0.0 1.4
0.0 0.5 0.0 0.0 1.5 0.0 0.0 0.5 0.0 69.0 0.0 16.0 1.4
0.0 0.0 1.0 0.0 0.5 0.0 0.0 0.0 15.0 0.0 71.0 0.0 4.1
0.0 0.0 0.0 0.0 6.0 0.0 0.0 0.0 0.0 11.5 0.0 70.0 1.9
0.0 0.0 0.0 0.0 1.0 1.5 0.0 0.0 0.0 0.0 0.0 0.0 67.9
0.0 77.0 0.0 1.0 0.0 0.0 0.0 14.5 0.0 7.5 0.0 0.5 2.3
0.0 0.0 82.5 0.0 0.0 0.0 12.0 0.0 0.0 0.0 8.5 0.0 4.8
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Table 3b Performance for different patterns with threshold hZ0.4 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. (%)
d.s. (%)
u.t. (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
78.5 0.0 0.0 0.0 0.0 0.0 11.0 0.0 11.0 0.0 5.0 0.0 0.2
0.0 76.5 0.0 0.0 0.0 0.0 0.0 12.0 0.0 2.5 0.0 0.0 0.8
0.0 0.0 82.0 0.0 0.0 0.0 9.5 0.0 0.0 0.0 4.0 0.0 1.1
0.0 0.5 0.0 84.0 0.0 0.0 0.0 14.5 0.0 2.0 0.0 7.0 0.7
0.0 0.0 0.0 0.0 86.0 0.0 0.0 0.0 1.0 2.0 3.5 4.5 1.0
0.0 0.0 0.5 0.0 0.0 95.5 0.0 0.0 0.0 0.0 0.0 0.0 1.9
21.0 0.0 17.0 0.0 0.0 0.0 79.5 0.0 2.5 0.0 5.5 0.0 0.0
0.0 20.5 0.0 11.5 0.0 0.0 0.0 73.0 0.0 3.5 0.0 2.0 0.2
0.5 0.0 0.0 0.0 1.0 0.0 0.0 0.0 74.5 0.0 11.0 0.0 1.4
0.0 2.5 0.0 2.0 1.0 0.0 0.0 0.5 0.0 74.5 0.0 14.0 0.9
0.0 0.0 0.5 0.0 1.0 0.0 0.0 0.0 11.0 0.0 70.5 0.0 0.7
0.0 0.0 0.0 2.5 4.0 0.0 0.0 0.0 0.0 15.0 0.0 72.5 0.2
0.0 0.0 0.0 0.0 7.0 4.5 0.0 0.0 0.0 0.5 0.5 0.0 90.9
and u.t. This result is very reasonable. Totally, the results in Tables 3(a)–(d) show that our proposed statistical correlation coefficient method works well for recognizing single and concurrent control chart patterns with good performance even though a change point from normal to abnormal occurs in control charts.
Table 3c Performance for different patterns with threshold hZ0.5 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. d.s. u.t. (%) (%) (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
76.0 0.0 0.0 0.0 0.0 0.0 10.5 0.0 9.0 0.0 1.5 0.0 0.0
0.0 1.0 0.0 77.0 0.0 0.0 0.0 13.0 0.0 1.0 0.0 6.5 0.2
0.0 0.0 0.0 0.0 83.0 0.0 0.0 0.0 3.5 3.5 2.0 8.0 0.0
0.0 0.0 0.0 0.0 0.0 96.5 0.0 0.0 0.0 0.0 0.0 0.0 0.5
21.0 0.0 19.5 0.0 0.0 0.0 79.0 0.0 1.5 0.0 1.0 0.0 0.0
0.0 22.5 0.0 20.0 0.0 0.0 0.0 69.0 0.0 3.5 0.0 0.0 0.2
1.5 0.0 0.0 0.0 1.0 0.0 0.0 0.0 73.0 0.0 10.5 0.0 0.2
0.0 2.5 0.0 0.5 0.5 0.0 0.0 0.0 0.0 67.0 0.0 10.0 0.8
0.0 0.0 0.0 0.0 1.5 0.0 0.0 0.0 13.0 0.0 77.0 0.0 0.0
0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 11.0 0.0 72.5 0.0
1.0 1.0 0.5 1.5 13.0 3.5 0.0 0.0 0.0 2.5 5.0 3.0 98.1
0.0 73.0 0.0 0.0 0.0 0.0 0.0 18.0 0.0 11.5 0.0 0.0 0.0
0.5 0.0 80.0 0.0 0.0 0.0 10.5 0.0 0.0 0.0 3.0 0.0 0.0
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Table 3d Performance for different patterns with threshold hZ0.6 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. (%)
d.s. (%)
u.t. (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
70.5 0.0 0.0 0.0 0.0 0.0 10.5 0.0 6.5 0.0 0.5 0.0 0.0
0.0 73.5 0.0 0.0 0.0 0.0 0.0 15.5 0.0 4.5 0.0 0.0 0.0
0.0 0.0 70.0 0.0 0.0 0.0 14.0 0.0 0.0 0.0 2.5 0.0 0.0
0.0 1.0 0.0 71.0 0.0 0.0 0.0 12.5 0.0 1.0 0.0 5.0 0.0
0.0 0.0 0.0 0.0 72.5 0.0 0.0 0.0 2.0 2.0 3.0 4.0 0.0
0.0 0.0 0.0 0.0 0.0 86.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
24.0 0.0 17.5 0.0 0.0 0.0 75.0 0.0 0.5 0.0 4.0 0.0 0.0
0.0 20.0 0.0 14.5 0.0 0.0 0.0 72.0 0.0 3.5 0.0 0.0 0.0
1.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 74.5 0.0 8.0 0.0 0.0
0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0 69.0 0.0 10.5 0.0
0.0 0.0 0.5 0.0 1.0 0.0 0.0 0.0 9.5 0.0 65.0 0.0 0.0
0.0 0.0 0.0 2.0 1.5 0.0 0.0 0.0 0.0 12.0 0.0 70.0 0.0
4.0 5.5 12.0 12.0 25.0 14.0 0.5 0.0 7.0 8.0 17.0 10.5 100.0
5. Conclusions In this paper, we use a statistical correlation coefficient to create a simple mechanism for recognizing single and concurrent unnatural patterns where a change point from normal to abnormal may occur in control charts. By adding a threshold criterion, the proposed method becomes more powerful for practical applications. The selection of a pattern length and a threshold was also discussed in this paper. According to our simulations, the proposed statistical correlation coefficient method presents good results. Tedious learning training process is not necessary for our method. The proposed mechanism actually enhances the ability of recognition so that it helps us discover irregularities in a manufacturing process as early as possible to reduce flawed products. Overall, the proposed approach is a simple mechanism for recognizing unnatural patterns in control charts.
Acknowledgements The authors are grateful to the anonymous referees for their helpful comments and suggestions to improve the presentation of the paper. This work was supported in part by the National Science Council of Taiwan, ROC, under Grant NSC-89-2213-E-033-057.
References Al-Ghanim, A. M., & Ludeman, L. C. (1997). Automated unnatural pattern recognition on control charts using correlation analysis techniques. Computers and Industrial Engineering, 32(3), 679–690. Cheng, C. S. (1997). A neural network approach for the analysis of control chart patterns. International Journal of Production Research, 35, 667–697.
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Grant, E. E., & Leavenworth, R. S. (1996). Statistical quality control. New York: McGraw-Hill. Guh, R. S. (2003). Integrating artificial intelligence into on-line statistical process control. Quality and Reliability Engineering International, 19, 1–20. Guh, R. S., & Hsieh, Y. C. (1999). A neural network based model for abnormal pattern recognition of control charts. Computers and Industrial Engineering, 36, 97–108. Guh, R. S., & Tannock, J. D. T. (1999). Recognition of control chart concurrent patterns using a neural network approach. International Journal of Production Research, 37, 1743–1765. Hwarng, H., & Hubele, N. (1993). Backpropagation pattern recognisers for X-bar control charts: Methodology and performance. Computers and Industrial Engineering, 24(2), 219–235. Pham, D. T., & Oztemel, E. (1994). Control chart pattern recognition using learning vector quantization networks. International Journal of Production Research, 32, 721–729. Shewhart, W. A. (1931). Economic control quality manufactured products. New York: Van Nostrand. Yang, M. S., & Yang, J. H. (2002). A fuzzy-soft learning vector quantization for control chart pattern recognition. International Journal of Production Research, 40, 2721–2731.
A control chart pattern recognition system using a statistical correlation coefficient method* Jenn-Hwai Yang, Miin-Shen Yang* Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan 32023, ROC
Abstract This paper presents a control chart pattern recognition system using a statistical correlation coefficient method. Pattern recognition techniques have been widely applied to identify unnatural patterns in control charts. Most of them are capable of recognizing a single unnatural pattern for different abnormal types. However, before an unnatural pattern occurs, a change point from normal to abnormal may appear at any point in control charts for most practical cases. Moreover, concurrent patterns where two unnatural patterns simultaneously exist may also occur in a control chart pattern recognition system. Our statistical correlation coefficient approach is a simple mechanism for recognizing these unnatural control chart patterns with good performance. This approach is also an effective method for the control chart pattern recognition without a tedious training process. q 2005 Elsevier Ltd. All rights reserved. Keywords: Control chart; Pattern recognition; Statistical correlation coefficient; Recognition rate; Change point; Concurrent pattern
1. Introduction Due to an increasing competition in products, consumers have become more critical in choosing products. The quality of products has become more important. Statistical process control (SPC) is usually used to improve the quality of products and reduce rework and scrap so that the quality expectation can be met (Grant & Leavenworth, 1996). Shewhart control charts (1931) are the most popular charts widely used in industry to detect abnormal process behavior. The most typical form of control charts consists of a central line and two control limits representing the specifications of the product and the variant range limits. This provides a useful method *
This manuscript was processed by Area Editor E.A. Elsayed. * Corresponding author. Tel.: C886 3 2653100; fax: C886 3 2653160. E-mail address: [email protected] (M.-S. Yang).
0360-8352/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2005.01.008
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for monitoring variations in the product manufacturing process. However, these control charts do not provide pattern-related information because they focus only on the latest plotted data points. In other words, only the last point on the chart is examined rather than the trend in the process over time. The value of average run length (ARL) is usually used to measure the frequency before the process is out-of-control. ARL is generally defined as the average number of points in a control chart until an out-ofcontrol signal occurs. Under a normal distribution N(m,s2) assumption with its mean m and variance s2, it is called out-of-control when its value is outside the range of G3s from the mean m. In general, it is unavoidable due to a random noise even though in a regular condition. For example, when we have a production process that the quality of products is under the N(0,1) assumption, the probability P(X%K3 or XR3)Z0.0027 such that 1/0.0027z370.4. Thus, ARL under the N(0,1) assumption is around 370.4. Similarly, for mZ0.5 with s2Z1, ARL is around 155.2. The ARL value for other mean shift m with s2Z1 is shown in Table 1. We can see that ARL will be decreasing when a mean shift m is increasing. We know that the smaller ARL exhibits out-of-control signal more often. In this case, it is not good for detecting abnormal process behavior with a Shewhart control chart because it is variable due to a mean shift m. Therefore, the control chart pattern recognition is important for recognizing unnatural patterns in control charts. Control chart pattern recognition has the capability to recognize unnatural patterns. In general, there are six unnatural patterns in control charts. They are upward trend, downward trend, upward shift, downward shift, cycle and systematic variation. These patterns present the long-term behavior of a process. If any of these patterns appears, something has occurred in the process such as a change of raw materials, equipment damage or the state of workers. If we can have such information on hand and then make decisions on the process, it will help us prevent failures appearing in the future. There have been many studies on control chart pattern recognition (see Al-Ghanim & Ludeman, 1997; Cheng, 1997; Guh & Hsieh, 1999; Pham & Oztemel, 1994; Yang & Yang, 2002). Recently, Guh (2003) proposed a hybrid artificial intelligence technique that consists of three major sub-systems so that it can be well used to build a real time SPC system. Most of these studies are considered for recognizing a single unnatural control chart pattern. However, in most real control chart applications, normal points may appear before abnormal points so that a change point from normal to abnormal may occur at any point in control charts. If we do not have a mechanism for recognizing such change patterns, we may incorrectly obtain the classification results. Therefore, under these circumstances, even though we have a good classification method for the control chart pattern recognition, we may still make an incorrect decision. Table 1 Average run length with a mean shift m m
ARL
0 0.5 1 1.5 2 2.5 3
370.4 155.2 43.9 15 6.3 3.2 2
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Another practical situation is concurrent patterns where two unnatural patterns may exist together. For recognizing concurrent unnatural patterns, Guh & Tannock (1999) used a back-propagation network (BPN) learning to perform the recognition for these concurrent control chart patterns. Although this BPN has a learning structure with a good network construction, the approach to the control chart pattern recognition is slow and complicated with a learning process. In this paper we propose a simple mechanism to recognize these unnatural patterns. The statistical correlation coefficient approach is used to construct a control chart pattern recognition system. By adding a threshold criterion, we can recognize these unnatural patterns even though they may change from normal to abnormal at any point in control charts. The remainder of this paper is organized as follows. In Section 2 we explain why a statistical correlation coefficient method is proposed and then present the proposed approach with the training and classification algorithms. Some simulation results and their comparisons are presented in Section 3. In Section 4, we present the performance for recognizing concurrent control chart patterns. Conclusions will be stated in Section 5.
2. The proposed control chart pattern recognition system Since control chart pattern recognition is an important step for industrial production processes, many researchers have made efforts toward finding various efficient methods for recognizing unnatural patterns in control charts. Hwarng and Hubele (1993) used a back-propagation neural network technique for detecting X-bar control charts. Cheng (1997); Guh and Hsieh (1999) used a neural network approach for recognizing an unnatural control chart pattern. Pham and Oztemel (1994) constructed a pattern recogniser using a learning vector quantization (LVQ) network. Yang and Yang (2002) created a fuzzy-soft LVQ to promote the recognition rate for control chart patterns. In order to decrease the number of reference vectors for each pattern used to identify different variation quantities, Al-Ghanim and Ludeman (1997) constructed a so-called ‘matched filter’ using a correlation analysis technique based on an inner pattern vector product. For concurrent control chart pattern recognition, Guh and Tannock (1999) proposed a back-propagation approach. In this section, we will propose a statistical correlation coefficient approach for the recognition of unnatural control chart patterns. The so-called matched filter approach involves obtaining a prototype for each pattern, called a ‘reference vector’ after the training algorithm is finished. Al-Ghanim and Ludeman (1997) evaluated the correlation between reference vectors and the input vector using the inner product. Let x and y be two random vectors with x 0 Z[x1.xn] and y 0 Z[y1.yn]. Then the inner product between x and y is defined as X xi yi : (1) hx; yi Z x 0 y Z Their inner-product correlation technique got good results in recognizing trend, cycle and systematic patterns. However, this method is poor in recognizing shift patterns. This is because the inner product presents the consistency in positive and negative components values in two vectors, but each component at the same position with an opposite sign will counteract the final inner product value. For example, if there are two vectors with nearly positive components, the inner-product correlation will become worse.
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Fig. 1. Illustration of the patterns x, y and z.
Let us consider the following three patterns with x 0 Z ½ K2 K2
0
2
2 ; y 0 Z ½ K2 K2
0
2
2 ; and z 0 Z ½ K1
0
1
2 2 :
The graph of the patterns x, y and z is shown in Fig. 1. Obviously, the similarity between x and y is larger than x and z. Calculating their correlation on the basis of inner product, we have hx; yi Z 16 and hx; yi Z 10: A larger correlation value represents higher similarity. The result is no doubt that x is more similar to y than to z. Now, let us shift the patterns x, y and z with a value of 2. That is, xs0 Z ½ 0
0
2 4
4 ; ys0 Z ½ 0
0
2
4
4 ; and zs0 Z ½ 1
2
3
4
4
A graph of the shifted patterns xs, ys and zs is shown in Fig. 2. We also calculate their inner product values with hxs ; ys i Z 36 and hxs ; zs i Z 38: The results show that x is more similar to z than to y that is obviously incorrect. This is because all of the components are nonnegative in xs, ys and zs. Under that condition, a larger component value determines a larger correlation value. Thus, the shift pattern will be difficult to be recognized on the basis of the inner-product approach. As we analyze the characteristic with upward shift and upward trend patterns, shown in Fig. 3, we note that both of their components are larger in the later part and tend to be positive. However, the trend toward large values in the upward trend pattern is faster than in the upward shift pattern. Thus, a shift pattern is always classified as a trend pattern because it has
Fig. 2. Illustration of the patterns xs, ys and zs.
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Fig. 3. Upward shift and upward trend patterns.
a maximum inner-product value. This produces an incorrect recognition. We can use the statistical correlation coefficient method to solve this problem as follows. The statistical correlation coefficient between two random vectors x and y is defined as P i K yÞ ðxi K xÞðy ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r Z pP P 2 2 ðxi K xÞ ðyi K yÞ
(2)
The main idea of the correlation coefficient (2) is to measure the normalized consistency in the signs of the remainders of two random vectors about their sample means. This is different from the inner product that is about zero without normalization. Using the statistical correlation coefficient for the same above data vectors x, y and z, we obtain rxyZ1 and rxzZ0.9587. After shifting the vectors x, y and z with xs, ys and zs, the correlations are still the same with rxs ys Z 1 and rxs zs Z 0:9587. This means that the similarity measures using the statistical correlation coefficient were not changed even through shifting two random vectors. There are some relations between the inner product and statistical correlation coefficient. Assume that x~ 0 Z ½x~1 .x~n and y~ 0 Z ½y~1 .y~n are two normalized random vectors of x and y with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P P 2 xi ðxi K xÞ yi xi K x yi K y ; y~i Z where x Z where y Z x~i Z ; Sx Z ; Sx Sy n n K1 n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2 ðyi K yÞ : Sy Z n K1 1 ~ yi. ~ In this paper, we propose a control chart pattern recognition system We then have rxy Z nK1 hx; using the statistical correlation coefficient. Here we have six different unnatural patterns named as upward shift, downward shift, upward trend (or increasing trend), downward trend (or decreasing trend), cycle and systematic patterns (see Fig. 4). Before recognition, we implement a training algorithm to get N samples where each pattern sample has a length n. The pattern sample generators are defined as follows:
(a) Normal pattern xðtÞ Z nðtÞ; where nðtÞ follows a normal distribution Nð0; 1Þ:
(3)
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Fig. 4. Unnatural reference patterns.
(b) Upward and downward shift patterns ( 0 before shifting ; xðtÞ Z nðtÞ C ud u Z 1 after shifting
(4)
where d is the shift quantity randomly taken from 1 to 2.5 for upward shift and from K1 to K2.5 for downward shift. (c) Upward and downward trend patterns xðtÞ Z nðtÞGdt
(5)
where d is the trend slope randomly selected from 0.05 to 0.12 for upward trend and from K0.05 to K0.12 for downward trend. (d) Cyclic pattern 2pt xðtÞ Z nðtÞ C d sin (6) U where d is the amplitude randomly selected from 0.5 to 2.5 and U is the cycle length taken as UZ8 here.
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(e) Systematic pattern xðtÞ Z nðtÞ C ðK1Þt d
(7)
where d is the amplitude randomly selected from 0.5 to 2.5. As mentioned in Section 1, larger mean shifts decrease ARL values and may lose the essence of doing the control chart pattern recognition because of out-of-control signals appearing sooner and often. Thus, a principle for choosing abnormal disturbance level d (i.e. shift, slop and amplitude) is to keep the mean not more than 3s. Moreover, the pattern length n used in the generated samples for each pattern is important. We will discuss and demonstrate latter. Thus, the training algorithm is described as follows:
Training Algorithm Step 1: Determine the pattern length n and the sample number N. Step 2: First, select a pattern sample generator and a disturbance level d from (4)–(7). Generate a pattern vector x1 with a pattern length n. Second, change the disturbance level d until N samples x2,.xN are generated from the chosen pattern. Step 3: Estimate the reference pattern vector by using PN EðxÞ Z
iZ1 xi
N
:
We repeat the training algorithm until all six reference pattern vectors are generated for upward shift, downward shift, upward trend, downward trend, cyclic and systematic patterns. Our final generated reference pattern vectors are shown in Fig. 4. We would regard these six generated reference patterns as the prototypes to recognize input patterns. The purpose of our training algorithm is to obtain six prototypes for recognizing input control chart patterns. The Step 3 in the training algorithm with a mean vector E(x) of N samples for each pattern is used to obtain a reference vector as a representation of the unnatural pattern. After we finish the training stage, it is necessary to use Eqs. (3)–(7) to generate samples for testing. A threshold, h, is created for qualifying the winner whether it matches enough or not. If the similarity measure is smaller than h, the winner is not similar enough. We will classify it as a normal pattern. The mechanism will further help us to identify these normal conditions and continue until an unnatural pattern is recognized. Thus, a classification algorithm is created as follows:
Classification Algorithm Step 1: Determine a threshold h. Step 2: A processing data sequence containing recent n points is regarded as the pattern size to be recognized. Step 3: Input the data sequence to the recogniser and calculate its statistical correlation coefficient using (2) with six reference vectors to obtain outputs ru.s., rd.s. ru.t., rd.t., rcyc, rsys for upward shift, downward shift, upward trend, downward trend, cyclic and systematic patterns, respectively.
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Step 4: Choose the maximum value among all outputs to determine which pattern is a winner and then classified. However, if the maximum value is smaller than the threshold h, we classify it as a normal pattern. We continue the classification algorithm for input control chart points until an unnatural pattern is recognized or no point is inputted. The problem in our classification algorithm is about the selection of a threshold h and a pattern length n. If a normal pattern is presented, then a false alarm will occur (i.e. type I error) when it is recognized as an unnatural pattern. We use the type I error to discuss how to select a pattern length n and a threshold h for our proposed correlation coefficient method. To demonstrate a better choice of h and n, we take 1000 samples from the normal control chart pattern with different pattern lengths nZ8, 16, 20, 30, 40 and 50. We then repeat 10 times with simulations for each given pattern length. We calculate the average type I error for different pattern length with respect to different threshold setting. The results are shown in Fig. 5. It is known that a type I error with the value 0.05 is generally a common choice. Since type I error is the measure of false alarm that we classify it to be an unnatural pattern when the data is normal, type II error is used to measure the capability of classification for unnatural patterns. Clearly, a larger threshold h will decrease a type I error but increase a type II error. Therefore, too large or too small h is not suitable for recognition. According to the results in Fig. 5, the threshold hZ0.5 is a good choice where the type I 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 pattern lengths nZ20, 30, 40 and 50 are reasonable choice in our simulations. On the other hand, we find that the pattern with a pattern length nZ50 in Fig. 6 has a threshold hZ0.3 when it has a type I error 0.05. This smaller threshold 0.3 (compared to a threshold hZ0.5) may have larger type II error so that nZ50 is not a good choice. In fact, some expert commentators had explained that too large pattern length is not practical. Moreover, ARL will decrease when it has more shift mean as shown in Table 1 so that a large pattern length will lose the spirit of preventing frequent out-of control occurrence in control charts. The trend pattern signifies that the mean is increasing (decreasing) with a slope so that a larger pattern length will cause later sample points to have large mean shift. Under these circumstances, we recommend a better choice of a pattern length with nZ30 and a threshold with hZ0.5 for our proposed correlation coefficient method.
Fig. 5. Type I error to threshold for different pattern length.
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Fig. 6. Two control chart patterns at different period with fixed length.
3. Simulation results In this section, we use the proposed correlation coefficient method to simulate the recognition of control chart patterns. We first illustrate control chart patterns using the chart shown in Fig. 6. The control chart in Fig. 6 is assumed to have an upward trend which occurs at the point tZ21 (i.e. the change point from normal to abnormal pattern at tZ21). According to the recommended pattern length with nZ30 in Section 2, we use a moving window with a window size 30 to mark the points. The pattern in the window B contains all the points of an upward trend pattern from tZ21 up to tZ50. The pattern in the window A contains 30 points from tZ1 up to tZ30 where there are 20 normal points from tZ1 to tZ20 and 10 abnormal points from tZ21 to tZ30. Most of existing control chart pattern recognisers are not considered for these kinds of control charts. This is because it is difficult to predict when a change point from normal to abnormal will appear in control charts. In most practical cases, normal points may often appear before abnormal points in control charts such as shown in Fig. 6. When we use a moving window for a control chart pattern recognition shown in Fig. 6, we find there are overlapping points between windows A and B with both normal and abnormal points from tZ21 up to tZ30. Under these circumstances, one may correctly classify the pattern in the window B but have an incorrect classification for the pattern in the window A. That is, when the pattern in the window A appears first, those recognisers may classify it as another unnatural pattern. For example, the window A in Fig. 6 contains half parts of normal and the other half of trend so that it looks like a shift pattern. Therefore, creating a mechanism to be able to suspend a recogniser making a classification when it has no enough evidence to be seen as an unnatural pattern shall be an important consideration. For solving this problem, we use a threshold h in our proposed correlation coefficient method.
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Fig. 7. Diagram of the correlation coefficient recognition system.
On the other hand, unnatural patterns, such as upward (downward) shift and upward (downward) trend, etc. may incur out-of-control signals during the pattern length nZ30. We can see this phenomenon from the ARL values shown in Table 1 of Section 1. The ARL values will drop down from 370 to 6 when the mean m increases from 0 to 2. We know that the smaller ARL exhibits out-of-control 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 0.12. According to ARL shown in Table 1, it is quite possible for Shewhart control chart to have out-ofcontrol signals for upward shift and trend patterns. These out-of-control signals are actually occurred at nZ46, 48 and 49 in Fig. 6. In fact, some out-of-control signals will be also occurred in Fig. 8. Of course, if an out-of-control signal is issued in Shewhart control chart, we stop the process and check the causes and then remove them. If an unnatural cause can be detected and removed, such as upward shift or trend, the model will become x(t)Zn(t), i.e. a normal pattern. However, it may be difficult for detecting such unnatural patterns using the Shewhart control chart because ARL will decrease when a mean m increases. This is the point that a control chart pattern recogniser is useful for recognizing unnatural patterns in control charts. Thus, the main idea of the proposed correlation coefficient method is that, if we do not have enough evidence to classify an input pattern as an unnatural pattern, even though it has a maximum correlation coefficient value to reference patterns, we still wait and take it as a normal pattern. The diagram of our recognition system is shown in Fig. 7. Because the statistical correlation coefficient r has a constraint with the range in the closed interval [K1,1], the threshold h can be used as a criterion what the similarity degree is enough for the recognition of an unnatural pattern. The pattern length for generating an unnatural control chart pattern and sample numbers in our simulations are chosen with nZ30 and NZ200. The moving window for the recognition of control charts uses a window size 30. For the normal control chart pattern, we take the sample number with NZ1000. Based on a threshold h, unless there exists sufficiently unnatural pattern evidence, we regard it as a normal and wait for more pattern information. According to the recommendation in Section 2, we choose hZ0.5 in our simulations. We implement 1000 runs for each pattern and then calculate its average accuracy as shown in Table 2. We mention that these testing samples are also simulated using the inner-product technique proposed by Al-Ghanim and Ludeman (1997). There are almost the same average accuracy as our approach when hZ0.4 in Table 2 for recognizing upward trend, downward trend, cyclic and systematic patterns. However, if we add upward shift and downward shift patterns, then their recognition average accuracy
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Table 2 Average accuracy for control chart patterns
hZ0.3 hZ0.4 hZ0.5 hZ0.6
u.s.
d.s.
u.t.
d.t.
cyc.
sys.
Normal
96.64 96.23 94.1 87.37
95.03 94.62 92.5 85.97
96.34 95.25 93.46 90.15
95.12 94.19 92.39 86.75
98.84 97.72 94.27 89.67
98.84 97.84 94.76 87.7
75.47 93.11 98.77 99.89
becomes worse. This is because the inner-product technique is confused the shift pattern with the trend pattern. This phenomenon had been explained in Section 2. To analyze the feasibility of a change point in control charts, we begin with 20 normal points in testing patterns. Thus, the first 20 moving windows started from tZ1 to tZ20 with a fix window size 30 contain both normal and abnormal points. The window started at tZ21 (i.e. the window B as shown in Fig. 6) contains the whole upward trend pattern as shown in Fig. 6. We consider for recognizing upward shift, (or downward shift), upward trend (or downward trend), cycle and systematic patterns in our simulations. All of these testing patterns are shown as the above figures in Fig. 8(a)–(d). We calculate the statistical correlation coefficient for each testing pattern with all six unnatural reference patterns as shown in Fig. 4. The correlation coefficients results are shown as the below figures in Fig. 8(a)–(d). When the input point of the control chart does not reach the window size 30, a zero correlation coefficient is presented. That is, the correlation coefficient will be 0 from tZ1 up to tZ29. When tZ30, the first correlation coefficient value is calculated. The window is moved ahead to calculate the second correlation coefficient value when tZ31, and so on. Since we set up a threshold h, the recogniser does not yet make a decision and still waits for more control chart points until a correlation coefficient value is over the h value. In Fig. 8(a), the given testing pattern is upward shift. There are six correlation coefficient curves between the given testing pattern and all six reference patterns. If we choose hZ0.5, then we can recognize the testing pattern is an upward shift (u.s.) pattern when tZ48. Similarly, the upward trend pattern is recognized when tZ37 as shown in Fig. 8(b). We see, even though there is much similar between the upward trend and upward shift patterns so that both curves of correlation coefficients in Fig. 8(a) and (b) have the same trend, it is still distinguishable after some points. For the cyclic and systematic patterns shown in Fig. 8(c) and (d), they are quite distinguishable. We note that a correlation coefficient curve shown in Fig. 8(c) is also a cyclic pattern. This is because the cycle length is taken as UZ8 so that different starting points induce a cyclic correlation coefficient curve. We mention that there are out-of-control signals in Fig. 8(a)–(c) that are similar to those in Fig. 6. The explanation and process procedure for Fig. 6 can also be used for those in Fig. 8(a)–(c).
4. Performance for recognizing concurrent patterns A concurrent pattern recognition in control charts was first investigated by Guh and Tannock (1999). They applied a back propagation network (BPN) for recognizing these concurrent patterns where two unnatural patterns may exist simultaneously. However, the training process of the BPN-based system tends to be relatively slow (see Guh & Tannock, 1999). On the other hand, the BPN construction is also complicated for a control chart pattern recognition 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 simulate the recognition of the concurrent control chart patterns with shift and trend, shift and cycle and trend and cycle as shown in Fig. 9. In our simulations, we use 200 testing samples for each unnatural pattern and 1000 testing samples for the normal pattern. The recognition results for different threshold h are shown in Tables 3.
Fig. 8. (a) Upward shift pattern and its correlation. (b) Upward trend pattern and its correlation. (c) Cyclic pattern and its correlation. (d) Systematic pattern and its correlation.
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Fig. 8 (continued )
We know that a threshold h give the capability of the recognition for unnatural patterns where a change point from normal to abnormal may appear at any point in control charts. Thus, we can treat these cases including single and concurrent patterns where an unnatural point may appear at any point in control charts. Comparing the results in Tables 2 and 3, the recognition rate for these single
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Fig. 9. Concurrent patterns.
and concurrent patterns as shown in Table 3(a)–(d) is actually lower than those single patterns as shown in Table 2. However, the recognition rates in Tables 3(a)–(d) are still impressive. From Tables 3(a)–(d), we find that concurrent patterns may be confused with relative single patterns. For example, the correct recognition number for the concurrent pattern usCut is 158 for hZ0.5 where 21 patterns are incorrectly recognized to be u.s. and 21 patterns are incorrectly recognized to be u.t. as shown in Table 3(c). All of these incorrect recognition patterns for u.s.Cu.t. come from u.s. Table 3a Performance for different patterns with threshold hZ0.3 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. d.s. u.t. (%) (%) (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
73.0 0.0 0.0 0.0 0.0 0.0 13.0 0.0 8.5 0.0 1.5 0.0 2.2
0.0 1.0 0.0 78.5 1.0 0.0 0.0 11.5 0.0 1.5 0.0 7.0 2.9
0.0 0.0 0.0 0.0 87.0 0.0 0.0 0.0 2.5 4.5 2.5 6.0 2.4
0.0 0.0 0.0 0.0 0.0 98.5 0.0 0.0 0.0 0.0 0.0 0.0 7.0
27.0 0.0 16.5 0.0 0.0 0.0 75.0 0.0 2.5 0.0 6.0 0.0 0.5
0.0 21.5 0.0 20.5 0.0 0.0 0.0 73.5 0.0 6.0 0.0 0.5 1.2
0.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 71.5 0.0 10.5 0.0 1.4
0.0 0.5 0.0 0.0 1.5 0.0 0.0 0.5 0.0 69.0 0.0 16.0 1.4
0.0 0.0 1.0 0.0 0.5 0.0 0.0 0.0 15.0 0.0 71.0 0.0 4.1
0.0 0.0 0.0 0.0 6.0 0.0 0.0 0.0 0.0 11.5 0.0 70.0 1.9
0.0 0.0 0.0 0.0 1.0 1.5 0.0 0.0 0.0 0.0 0.0 0.0 67.9
0.0 77.0 0.0 1.0 0.0 0.0 0.0 14.5 0.0 7.5 0.0 0.5 2.3
0.0 0.0 82.5 0.0 0.0 0.0 12.0 0.0 0.0 0.0 8.5 0.0 4.8
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Table 3b Performance for different patterns with threshold hZ0.4 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. (%)
d.s. (%)
u.t. (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
78.5 0.0 0.0 0.0 0.0 0.0 11.0 0.0 11.0 0.0 5.0 0.0 0.2
0.0 76.5 0.0 0.0 0.0 0.0 0.0 12.0 0.0 2.5 0.0 0.0 0.8
0.0 0.0 82.0 0.0 0.0 0.0 9.5 0.0 0.0 0.0 4.0 0.0 1.1
0.0 0.5 0.0 84.0 0.0 0.0 0.0 14.5 0.0 2.0 0.0 7.0 0.7
0.0 0.0 0.0 0.0 86.0 0.0 0.0 0.0 1.0 2.0 3.5 4.5 1.0
0.0 0.0 0.5 0.0 0.0 95.5 0.0 0.0 0.0 0.0 0.0 0.0 1.9
21.0 0.0 17.0 0.0 0.0 0.0 79.5 0.0 2.5 0.0 5.5 0.0 0.0
0.0 20.5 0.0 11.5 0.0 0.0 0.0 73.0 0.0 3.5 0.0 2.0 0.2
0.5 0.0 0.0 0.0 1.0 0.0 0.0 0.0 74.5 0.0 11.0 0.0 1.4
0.0 2.5 0.0 2.0 1.0 0.0 0.0 0.5 0.0 74.5 0.0 14.0 0.9
0.0 0.0 0.5 0.0 1.0 0.0 0.0 0.0 11.0 0.0 70.5 0.0 0.7
0.0 0.0 0.0 2.5 4.0 0.0 0.0 0.0 0.0 15.0 0.0 72.5 0.2
0.0 0.0 0.0 0.0 7.0 4.5 0.0 0.0 0.0 0.5 0.5 0.0 90.9
and u.t. This result is very reasonable. Totally, the results in Tables 3(a)–(d) show that our proposed statistical correlation coefficient method works well for recognizing single and concurrent control chart patterns with good performance even though a change point from normal to abnormal occurs in control charts.
Table 3c Performance for different patterns with threshold hZ0.5 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. d.s. u.t. (%) (%) (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
76.0 0.0 0.0 0.0 0.0 0.0 10.5 0.0 9.0 0.0 1.5 0.0 0.0
0.0 1.0 0.0 77.0 0.0 0.0 0.0 13.0 0.0 1.0 0.0 6.5 0.2
0.0 0.0 0.0 0.0 83.0 0.0 0.0 0.0 3.5 3.5 2.0 8.0 0.0
0.0 0.0 0.0 0.0 0.0 96.5 0.0 0.0 0.0 0.0 0.0 0.0 0.5
21.0 0.0 19.5 0.0 0.0 0.0 79.0 0.0 1.5 0.0 1.0 0.0 0.0
0.0 22.5 0.0 20.0 0.0 0.0 0.0 69.0 0.0 3.5 0.0 0.0 0.2
1.5 0.0 0.0 0.0 1.0 0.0 0.0 0.0 73.0 0.0 10.5 0.0 0.2
0.0 2.5 0.0 0.5 0.5 0.0 0.0 0.0 0.0 67.0 0.0 10.0 0.8
0.0 0.0 0.0 0.0 1.5 0.0 0.0 0.0 13.0 0.0 77.0 0.0 0.0
0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 11.0 0.0 72.5 0.0
1.0 1.0 0.5 1.5 13.0 3.5 0.0 0.0 0.0 2.5 5.0 3.0 98.1
0.0 73.0 0.0 0.0 0.0 0.0 0.0 18.0 0.0 11.5 0.0 0.0 0.0
0.5 0.0 80.0 0.0 0.0 0.0 10.5 0.0 0.0 0.0 3.0 0.0 0.0
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Table 3d Performance for different patterns with threshold hZ0.6 Input
u.s. d.s. u.t. d.t. cyc. sys. u.s.Cu.t. d.s.Cd.t. u.s.Ccyc. d.s.Ccyc. u.t.Ccyc. d.t.Ccyc. Normal
Recognized pattern u.s. (%)
d.s. (%)
u.t. (%)
d.t. (%)
cyc. (%)
sys. (%)
u.s.C u.t. (%)
d.s.C d.t. (%)
u.s.C cyc. (%)
d.s.C cyc. (%)
u.t.C cyc. (%)
d.t.C cyc. (%)
Normal (%)
70.5 0.0 0.0 0.0 0.0 0.0 10.5 0.0 6.5 0.0 0.5 0.0 0.0
0.0 73.5 0.0 0.0 0.0 0.0 0.0 15.5 0.0 4.5 0.0 0.0 0.0
0.0 0.0 70.0 0.0 0.0 0.0 14.0 0.0 0.0 0.0 2.5 0.0 0.0
0.0 1.0 0.0 71.0 0.0 0.0 0.0 12.5 0.0 1.0 0.0 5.0 0.0
0.0 0.0 0.0 0.0 72.5 0.0 0.0 0.0 2.0 2.0 3.0 4.0 0.0
0.0 0.0 0.0 0.0 0.0 86.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
24.0 0.0 17.5 0.0 0.0 0.0 75.0 0.0 0.5 0.0 4.0 0.0 0.0
0.0 20.0 0.0 14.5 0.0 0.0 0.0 72.0 0.0 3.5 0.0 0.0 0.0
1.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 74.5 0.0 8.0 0.0 0.0
0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0 69.0 0.0 10.5 0.0
0.0 0.0 0.5 0.0 1.0 0.0 0.0 0.0 9.5 0.0 65.0 0.0 0.0
0.0 0.0 0.0 2.0 1.5 0.0 0.0 0.0 0.0 12.0 0.0 70.0 0.0
4.0 5.5 12.0 12.0 25.0 14.0 0.5 0.0 7.0 8.0 17.0 10.5 100.0
5. Conclusions In this paper, we use a statistical correlation coefficient to create a simple mechanism for recognizing single and concurrent unnatural patterns where a change point from normal to abnormal may occur in control charts. By adding a threshold criterion, the proposed method becomes more powerful for practical applications. The selection of a pattern length and a threshold was also discussed in this paper. According to our simulations, the proposed statistical correlation coefficient method presents good results. Tedious learning training process is not necessary for our method. The proposed mechanism actually enhances the ability of recognition so that it helps us discover irregularities in a manufacturing process as early as possible to reduce flawed products. Overall, the proposed approach is a simple mechanism for recognizing unnatural patterns in control charts.
Acknowledgements The authors are grateful to the anonymous referees for their helpful comments and suggestions to improve the presentation of the paper. This work was supported in part by the National Science Council of Taiwan, ROC, under Grant NSC-89-2213-E-033-057.
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