Int. J. Computer Aided Engineering and Technology, Vol. 3, Nos. 3/4, 2011
309
An improved scheme for online recognition of control chart patterns Adnan Hassan Department of Manufacturing and Industrial Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor Bahru, Malaysia E-mail:
[email protected] E-mail:
[email protected] Abstract: This paper proposes two alternative schemes for the online recognition of control chart patterns (CCPs), namely: 1 a scheme based on direct continuous recognition 2 a scheme based on ‘recognition only when necessary’. The study focuses on recognition of six CCPs plotted on the Shewhart X-bar chart, namely, random, shift-up, shift down, trend-up, trend-down and cyclic. The artificial neural network (ANN) recogniser used was based on multilayer perceptrons (MLPs) architecture. The performance of the schemes was evaluated based on percentage correct recognition, average run lengths (ARL) and average recognition attempts (ARA). The findings suggest that the online recognition should be made only when necessary. Continuous recognition is not only wasteful, but also results in poorer results. The methodology proposed in this study is a step forward in realising a truly automated and intelligent online statistical process control chart pattern recognition system. Keywords: online recognition; pattern recognition; statistical process control; SPC; control chart; artificial neural network; ANN. Reference to this paper should be made as follows: Hassan, A. (2011) ‘An improved scheme for online recognition of control chart patterns’, Int. J. Computer Aided Engineering and Technology, Vol. 3, Nos. 3/4, pp.309–321. Biographical notes: Adnan Hassan is an Associate Professor in the Department of Manufacturing and Industrial Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM). He received his BSc (Hons.) in Industrial Engineering from University of Miami, Florida, USA, MSc in Industrial Measurement Systems from Brunel University, UK and PhD from UTM. His current research is on pattern recognition, statistical process control and scheduling. He is a member of the Institute of Industrial Engineers, USA. He has been with UTM since 1989.
Copyright © 2011 Inderscience Enterprises Ltd.
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Introduction
Control charts were introduced by Shewhart in 1924 and remain one of the most important statistical process control (SPC) tools. The key feature of control charts is the provision of the method to differentiate between particular processes is operating within a statistically stable and an unstable state. Unstable processes may eventually produce distinct time series patterns as shown in Figure 1. Identification of these patterns coupled with engineering knowledge of the process would lead to focused diagnosis and troubleshooting. Figure 1
Examples of time series patterns on Shewhart X-bar control charts
Source: Cheng (1989)
Manual interpretation of CCPs often suffers from false alarms, causes delay and is limited to simple patterns. Developments in computing technology have motivated researchers to explore the use of artificial intelligence technologies such as expert systems, artificial neural network (ANN) and fuzzy sets to automatically and intelligently recognise control chart patterns (CCPs). Various classification schemes have been contributed over the last two decades. Most of the previous studies on CCP recognition were concerned with recognition of fully developed patterns as found in Hwarng and Hubele (1993), Hwarng and Chong (1995), Al-Ghanim and Jordan (1996), Anagun (1998), Guh and Hsieh (1999), Pham and Oztemel (1995), Pham and Chan (2001) and
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Hassan et al. (2003, 2006) among others. Their assumption that unstable processes were represented by fully developed patterns when presented to recognisers was too simplified. Such approach is useful only if the starting point of the unstable pattern is known. The instance when a stable process actually starts to deteriorate and reaches recognisable patterns is normally unknown. It is important that the recognition activity commences at the right moment. Unnecessary recognition of stable processes should be avoided. However, the emergence of any unstable processes should be identified and recognised as quickly as possible. Any excessive delay may be catastrophic for some processes. On the other hand, stopping the deteriorating processes too early may be too costly and increase the possibility of wrong diagnosis due to insufficient information. It is important that a functional online scheme be developed to balance between these two extremes. Researchers such as Tontini (1996, 1998) and Guh et al. (1999a, 1999b) have proposed online schemes within the context of the present discussion. Tontini (1996, 1998) sketchily discussed his scheme with the focus on cumulative online learning. Guh et al. (1999a, 1999b) implemented a dedicated recogniser (Module I) to continuously recognise the type of CCPs and another set of recognisers (Module II) to identify the pattern parameters. They reported poor results were obtained when implemented their methodology online. They argued that false recognition was largely attributed to the confusion with natural variation. To overcome poor discrimination performance in the earlier study, Guh et al. (1999b) proposed a discrimination algorithm. Since the algorithm was based on the concept of cascaded recognisers, the overall performance was heavily depending on the earlier general recogniser. There is a need for a better scheme for online recognition of CCPs where timely interpretation is critical. Online recognition is difficult since the running patterns are normally vague and having dynamic composition of data streams prior to becoming a fully developed pattern. The objective of this paper is to present two alternative schemes for online recognition of developing CCPs: 1
a scheme based on direct continuous recognition
2
a scheme based on ‘recognition only when necessary’.
The challenge here is to provide a balance between timely recognition of unstable patterns as they develop and minimise unnecessary recognition of stable processes. The rest of this paper is organised as follows. Section 2 discusses the input data representation and recogniser design. Section 3 presents the scheme for continuous online recognition, while Section 4 discusses the scheme based on ‘recognition only when necessary’ approach. Sections 5 and 6 provide the performance of the schemes and finally Section 7 concludes the paper.
2
Input data representation and recogniser design
2.1 Input data representation This research focuses on classification of six developing CCPs plotted on the Shewhart X-bar chart, namely, random, shift-up, shift down, trend-up, trend-down and cyclic. Since a large amount of samples were required for the classifiers’ training and they were not
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economically available, simulated data was used. This study adopted Swift’s (1987) methodology to simulate individual process data since the methodology has been widely accepted by other researchers. The parameters used in this study for simulating the data streams are given in Table 1. The values of these parameters were varied randomly in a uniform manner between the lowest and the highest values. Random noise of 1/3 sigma was added to all the unstable patterns. These parameters were chosen to keep the data streams within the control limits ( ±3σ ) since, for preventive purposes, the status of a process should be identified while it is operating within these limits. Table 1
Parameters for simulating process variation data streams
Pattern types
Parameters (σ)
Linear trend-up
Gradient: 0.015 to 0.025
Linear trend-down
Gradient: –0.025 to –0.015
Sudden shift-up
Shift magnitude: 0.7 to 2.5
Sudden shift-down
Shift magnitude: –2.5 to –0.7
Cyclic
Amplitude: 0.5 to 2.5; period = 10
Stable process
Mean = 10.0; standard deviation = 0.5 Standardised: N (0,1)
Besides the random data streams, the ANN recognisers were also trained and tested using data partially developed (75%–90%) unstable data streams. Each data stream was presented to the monitoring window as standardised subgroup averages of time sequenced data. To facilitate the ANN training and to improve its performance, each of the raw data stream was then represented as a set of minimal statistical features, namely, mean, standard deviation, cusum, slope, average autocorrelation and mean-square-value. The mathematical expressions for extracting statistical features are as the following: a
Cusum The Cusum statistics provides accumulated deviation from μo that are above target with statistic C+ and accumulated deviations from μo below target with C– statistic (Montgomery, 2001): Ci+ = max ⎡ 0, xi − ( μ 0 + K ) + Ci+−1 ⎤ ⎣ ⎦
(1)
Ci− = max ⎡ 0, ( μ 0 − K ) − xi + Ci−−1 ⎤ ⎣ ⎦
(2)
The last cusum statistic for each data stream was taken as the representative feature. The reference value, K was set to 0.5 shift magnitude. b
Slope The slope ( βˆ1 ) for a data stream can be obtained from the best-fitting straight line, Yˆ = βˆ + βˆ X (Neter et al., 1996) where, o
1
An improved scheme for online recognition of control chart patterns n
βˆ1 =
⎛
∑ ⎜⎝ X
i
i =1
n
_ ⎞⎛ _⎞ − X ⎟ ⎜ Yi − Y ⎟ ⎠⎝ ⎠
_ ⎞ ⎛ ⎜ Xi − X ⎟ ⎠ i =1 ⎝
∑
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(3)
2
Y denotes the mean of Yi and X denotes mean of Xi of the (Xi, Yi) pairs of sample observations.
c
Average autocorrelation Autocorrelation measures the dependence of a data at one instant in time with another data at another instant in time. The autocorrelation feature was extracted using the following expression (Brook and Wynne, 1988): Rxx [ k ] ≅
1 ⎡ xo xk + x1 x1+ k + ...xN − k xN ⎤⎦ N +1− k ⎣
(4)
where N is the number of observations and k is the lag. The average values for autocorrelation at lag 1 and 2 were used as the feature. These lags were chosen based on preliminary simulation runs. d
Mean-square-value (Brook and Wynne, 1988) x% 2 =
N
x02 + x12 + x22 + ... + xN2 1 xi2 = N +1 N + 1 i =0
∑
(5)
The statistical features for mean and standard deviation are as commonly found in the statistical text books. The above minimal feature set was selected through extensive simulation experimental runs using fractional factorial experimental design. The extracted statistical features were then normalised using the equation (6) such that their values would fall within [–1, 1]. This normalisation is necessary since ANN training can be made more efficient when the input vector falls within a certain range (Demuth and Beale, 1998). Pn = 2 ×
( P − Pmn ) −1 ( Pmx − Pmn )
where: P = observed feature values Pn = normalised observed feature values Pmn = minimum feature value for the feature Pmx = maximum feature value for the feature.
(6)
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2.2 ANN recogniser design The recogniser was developed based on multilayer perceptrons (MLP) architecture with basic structure comprises one input layer, one hidden layer and one output layer as shown in Figure 2. The design and training parameters were selected based on trial and error and factorial experimental designs. The selected ANN structure was 6 × 10 × 6. The training algorithm used was BFGS quasi-Newton (Demuth and Beale, 1998). Details of design procedures for such MLP are widely available, for example in Patterson (1996) and Haykin (1999). Before the relative merit between the use of continuous and the recognition only when necessary approaches could be evaluated, the recogniser had to be properly trained. Procedures for the recogniser training as in Hassan et al. (2003) were adopted in this study. The training samples were 3,888 (648 for each pattern type) and the testing samples were 1,296 (216 for each type). The trained recogniser was then embedded into a monitoring framework with either continuous direct classification or recognition only when necessary approach. Figure 2 MLP neural network structure
Source: Hassan et al. (2003)
3
Scheme for direct continuous online recognition
Figure 3 shows the generalised framework that integrates the building blocks for the scheme with direct continuous online recognition (Scheme A). In this design, the recognition activities are attempted continuously for all data streams as they appear in the monitoring window. The use of the statistical features presented in Section 2 has reduced the size of input vector from 20 (raw data) to merely six.
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Figure 3 Scheme for direct continuous online recognition (Scheme A) Manufacturing Process
Resume process
Process data streams
Continue monitoring
SPC monitoring window
Continue monitoring (Get more data)
Input data representation (minimal feature set)
ANN-based pattern recogniser
Maximum recognition attempts exceeded?
No
Recognition output conclusive?
No (Output
≤Threshold) Yes (Output > Threshold)
Yes Unable to predict the type of pattern (Reject class)
Predicted pattern type (class)
Yes
Random pattern?
No
This bounding box indicates the scope of the scheme
Process diagnosis, adustment and remedial action as deemed necessary
The streams of process data were simulated to begin in a stable state and then they either remained stable throughout the monitoring period or progressively deteriorated into any one of the five possible unstable types, namely, shift-up, shift down, trend-up, trend-down and cyclic. The predicted class for the data stream was recorded whenever the maximum output exceeded the classification threshold value, Oc = 0.8. Generally, a low Oc leads to fewer rejections while high value of Oc increases the stringency for classification. If the maximum output is less than or equal the threshold
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( max{O1 , O2 ,...OM } ≤ Oc ), the monitoring process is allowed to proceed and more process data is obtained. This feature provides opportunity for more unstable process data to be included before making a decision. The maximum number of recognition attempts (Ac) was set to 20 corresponding to the size of the observation window. If the recognition result remained inconclusive at the 20th attempt, the pattern was classified as reject class (M+1). The predicted pattern is not necessarily always correct. It can be either similar to the true class or classified into some other (wrong) classes. The provision of a reject class provides flexibility in decision making, particularly to avoid being forced into making wrong classification.
4
Scheme for recognition only when necessary
The need to continuously recognise stable data stream is arguable. It is considered unnecessary in normal situations. However, the recognition of stable process becomes necessary when the system needs to recover from false alarms. Thus, this section proposes an alternative scheme by incorporating stability tests using runs rules and cusum as a procedure to determine the necessity to recognise prior to triggering the recognition process. This procedure was intended to avoid unnecessary recognition of stable processes and promote timely recognition of unstable processes.
4.1 Runs rules and cusum for stability tests Supplementary runs rules based on Nelson (1984, 1985) and cusum were adopted as stability tests. The details for the runs rules are given in Table 2. These runs rules are widely known in the area of SPC and are self-explanatory. The status of a process is continuously monitored by the above runs rules as the process progresses. The testing procedure begins with Test 1 and the other tests are applied sequentially if deemed necessary. Table 2
Nelson’s runs rules
No. 1
Description One or more than one point outside the control limit
2
Nine consecutive points are on the same side of the central line
3
Six points in a row steadily increasing or decreasing
4
14 points in a row alternating up and down
5
Two out of three consecutive points of the same side of the central line are between 2 sigma and the control limits
6
Four out of five consecutive points are between 1 sigma and the control limit on the same side of the central line.
7
15 consecutive points are between –1 sigma and +1 sigma.
8
Eight consecutive points are on both sides of centre line with none within –1 sigma and +1 sigma Source: Nelson (1984, 1985)
As noted above, this scheme also included a tabular cusum as stability tests. The statistics C+ and C– were obtained as described in equations (1) and (2). The design of tabular
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cusum also requires selection of the reference value (K) and the decision interval (H). The reference value was set to K = 0.75σ and the decision interval was set to H = 3.34σ, where sigma is the standard deviation. These values were selected based on preliminary investigation and reference to Hawkins (1993). Generally, a smaller K resulted in a looser monitoring and a higher K resulted in a tighter monitoring with many samples proceeding to the recognition stage. The last cusum statistic was taken as the measure to test the process stability. Attempts to classify the data streams commenced when the last cusum statistics were greater than the decision interval H. Figure 4 Scheme for recognition only when necessary (Scheme B) Manufacturing Process
Resume process
Process data streams
Continue monitoring SPC monitoring window
Continue the process Continue monitoring (Get more data)
Yes
Process stable? (runs rules, CUSUM) No Input data representation (minimal feature set)
ANN-based pattern recogniser
No
Maximum recognition attempts exceeded?
No (Output
≤ Threshold)
Recognition output conclusive? Yes (Output > Threshold)
Yes Unable to predict the type of pattern (reject class)
Predicted pattern type (class)
Yes
Random pattern? No
This bounding box indicates the scope of the scheme
Process diagnosis, adustment and remedial action as deemed necessary
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4.2 The scheme The detailed design for this scheme is shown in Figure 4 (Scheme B). All the components are the same as in Figure 3 except the shaded diamond which consists of Nelson’s runs-rules and cusum tests. With this configuration, all the process data streams will be tested for its stability prior to the recognition process. If a data stream is identified as coming from a stable process, then the process will continue without any recognition. As such, attempts to classify streams of process data are allowed only when they are identified as possibly coming from an unstable process. The rest of the procedure for decision-making is the same as described in Section 3.
5
Performance of the alternative schemes
Experiments were conducted for the above alternative scheme designs using 20 different data sets, each comprising a total of 3,600 developing control chart patterns (DCCPs) (600 for each type). The performance of the alternative schemes was compared based on percentage correctly recognised, average run length (ARL) and average recognition attempt (ARA).
5.1 Recognition performance for the scheme with continuous recognition The classification performance of the scheme with direct continues recognition is summarised in a confusion matrix, Table 3. The result suggests that a significant percentage of random patterns (stable process) presented to the recogniser were either wrongly classified as unstable patterns (24.5%) or as reject class (66.8%). Only 8.7% were correctly classified as random patterns. For non-random processes (unstable data streams), the highest correct classification was for cyclic patterns (87.8%), followed by trend-up (84.8%), trend-down (80.9%), shift-down (75.6%) and shift-up (64.7%). The aggregate percentage for correct recognition of unstable data streams was 78.8%. Table 3
Confusion matrix for the Scheme A Recognised class (%)
True class
RN
TU
TD
SU
SD
CY
RJ
RN
8.68
0.18
0.13
5.64
14.00
4.58
66.78
TU
2.21
84.77
0.00
1.29
2.54
5.00
4.19
TD
3.58
0.02
80.93
0.72
4.28
3.88
6.60
SU
2.70
16.76
0.03
64.68
4.00
1.63
10.20
SD
3.97
0.03
9.49
1.24
75.57
1.73
7.98
CY
5.35
0.05
0.00
1.32
2.39
87.80
3.09
Note: RN = random, TU = trend-up, TD = trend-down, SU = shift-up, SD = shift-down, CY = cyclic and RJ = reject.
The results suggest that random data streams are the most difficult to recognise compared to non-random data streams. The above findings are in agreement with Guh et al. (1999a) who reported poor performance when CCP was recognised in a real time mode.
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5.2 Recognition performance for the scheme with ‘recognition only when necessary’ Some improvements can be noticed from the results in Table 4. The percentage for correct classification of stable process has significantly improved (87.5%). This improvement was attributed to a significant reduction in unnecessary recognition and the ability of the Scheme B to recover from false alarms. For the non-random processes, the highest correct classification was maintained by cyclic patterns (85.0%), followed by trend-up (80.9%), trend-down (75.1%), shift-down (62.6%) and shift-up (51.9%). The aggregate percentage for correct classification of unstable process is 71%. Table 4
Confusion matrix for the Scheme B Recognised class (%)
True class
RN
TU
TD
SU
SD
CY
RJ
RN
87.52
0.11
0.07
3.24
6.43
1.21
1.43
TU
0.98
80.93
0.00
0.75
1.20
1.43
14.73
TD
1.30
0.00
75.08
0.43
2.77
0.80
19.63
SU
1.96
11.98
0.01
51.90
1.79
0.49
31.87
SD
1.70
0.03
7.77
0.68
62.55
0.68
26.59
CY
12.36
0.03
0.00
0.67
1.09
84.98
0.88
Note: RN = random, TU = trend-up, TD = trend-down, SU = shift-up, SD = shift-down, CY = cyclic and RJ = reject.
5.3 ARL and ARA performance for stable and unstable processes Basically, ARL0 measures the average run length of first occurrence of false alarm when tested for stable processes. The larger the value of ARL0 is the better for the scheme. On the other hand, for ARAcorrect that measures the average number of recognition attempts to correctly classify the pattern, the lower ARAcorrect value is the better. Table 5 shows that Scheme B resulted in significantly better ARL0 (212.0) compared to the Scheme A (89.3). The average number of recognition attempts (ARAcorrect) was also significantly reduced from 70.3 (Scheme A) to 3.6 attempts (Scheme B). The results in Table 5 are based on the averages of ten test data sets where each set comprised 500 streams of stable process. Each stream consisted of 600 data points (subgroup means). Table 5
ARL0 and ARA for stable processes
Scheme
Performance measures* ARL0
ARAcorrect
A Direct continuous recognition
89.3 (3.3)
70.3 (3.3)
B Recognition only when necessary
212.0 (6.1)
3.6 (0.1)
Note: *In brackets are standard deviations.
Table 6 compares the schemes performance in terms of ARLtarget and ARAcorrect for unstable processes. ARLtarget is the minimum number of observations required before the targeted unstable pattern is correctly recognised. The smaller the value of ARLtarget is the better for the scheme. The results show that there was a marginal deterioration in
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ARLtarget when using the Scheme B (increased from 10.76 to 11.09). However, the ARAcorrect significantly reduced from 11.76 to 4.82. Results in Table 5 were based on 20 test data sets, each consisted a total of 3,000 developing unstable patterns or 600 patterns for each type. Table 6
ARL and ARA for unstable processes
Scheme
Performance measures* ARLtarget
ARAcorrect
A Continuous recognition
10.76 (0.096)
11.76 (0.096)
B Recognition only when necessary
11.09 (0.069)
4.82 (0.092)
Note: *In brackets are standard deviations.
7
Conclusions
This paper investigated schemes for online recognition of CCPs based on continuous online recognition and ‘recognition only when necessary’ approaches. The findings indicate that the unnecessary recognition of stable patterns was significantly reduced when runs rules and cusum were implemented prior to the recognition process. The results also reveal that the stability tests significantly improve the ARL0. The proposed scheme has shown promising results for both stable and unstable processes. This design promotes timely recognition of unstable patterns and avoids unnecessary recognition of stable patterns. Currently, we are extending this work to further improve the overall recognition performance.
Acknowledgements The author thanks Research Management Center, Universiti Teknologi Malaysia (UTM) and the Ministry of Science, Technology and Innovation (MOSTI), Malaysia for IRPA research grant Vol. No. 74227.
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