144 8. MVS Applied to Real On-line Measurements
Variable Measured
number entity
1 in uent temperature
2 sludge concentration in line 1 3 sludge concentration in line 2
4 air valve position of blower 1 in line 1 5 air valve position of blower 2 in line 1 6 air valve position of blower 1 in line 2 7 air valve position of blower 2 in line 2 8 in uent conductivity
9 in uent ammonia 10 in uent pH 11 in uent ow rate
Table 8.2 Variables used in the PCA model.
a period of 2500 samples,i.e. approximately 9 days, and the variables used are given in Table 8.2. During the period from which the training data are collected, the process demonstrates a normal and desired behaviour. Thus, the process mode described by the PCA model can be said to be a normal operational mode. In the PCA model, six principal components are retained, capturing approximately 90% of the variability of the original data matrix (Table 8.3). The SPE and T2 measures of the new data are shown in Figure 8.2 and it is obvious that everything appears relatively normal until sample 731. At this time, both SPE and T2 increase suddenly, and exceed the con dence limits by a wide margin. The mode can be classi ed as abnormal.
The change is rapid, which suggests that the problem is not process related. Instead, the change may have been caused by a sensor failure.
From Figure 8.3 it is concluded that the contribution to the deviation in SPE at sample 731 is mainly caused by a change in variable 8, i.e. in uent conductivity. There are also signi cant contributions from variables 1 and 2, in uent temperature and sludge concentration, respectively. If the contribution to the changes along the PCs between sample 730, where the SPE and T2 measures are still small, and sample 731 is studied, the picture becomes clear (Figure 8.4). It is obvious that there is a change in variable 8 (in uent conductivity) and that this has caused theT2measure to exceed its limits. It can be seen in Figure 8.2 that the residuals decrease at sample 1070 as suddenly as they increased at sample 731. This is caused by a corrective action
8.2. Monitoring General Operational Modes 145
Principal Eigenvalue % Variance % Variance
component of captured captured
number cov(X) this PC total
1 4.70e+00 42.74 42.74
2 1.89e+00 17.18 59.93
3 1.32e+00 11.97 71.89
4 9.26e-01 8.42 80.31
5 6.65e-01 6.05 86.36
6 4.28e-01 3.89 90.25
7 3.40e-01 3.09 93.34
8 3.00e-01 2.73 96.06
9 2.52e-01 2.29 98.35
10 1.23e-01 1.12 99.47
11 5.78e-02 0.53 100.00
Table 8.3 Percent variance captured by the PCA model.
0 200 400 600 800 1000 1200
0 10 20 30 40 50 60
SPE, with 99% confidence limit.
0 200 400 600 800 1000 1200
0 50 100 150 200
T2, with 99% confidence limit.
time, (samples)
Figure 8.2 The SPE and T2 measures for new data.
146 8. MVS Applied to Real On-line Measurements
1 2 3 4 5 6 7 8 9 10 11
−4
−2 0 2 4
Prediction Error
Process Variable no
Figure 8.3 The contribution to the large SPE in sample 731.
1 2 3 4 5 6 7 8 9 10 11
−3
−2
−1 0 1
Process Variable no p j,1∆x j
1 2 3 4 5 6 7 8 9 10 11
−1 0 1 2 3 4
p j,2∆x j
Process Variable no
Figure 8.4 Variables contributing to change along PC no 1 (left) and PC no 2 (right).
8.2. Monitoring General Operational Modes 147
Principal Eigenvalue % Variance % Variance
component of captured captured
number cov(X) this PC total
1 3.96e+00 33.03 33.03
2 2.43e+00 20.21 53.24
3 1.84e+00 15.37 68.61
4 1.31e+00 10.96 79.57
5 8.14e-01 6.79 86.36
6 6.89e-01 5.74 92.10
7 4.09e-01 3.41 95.50
8 2.31e-01 1.92 97.43
9 1.72e-01 1.43 98.86
10 7.61e-02 0.63 99.50
11 6.02e-02 0.50 100.00
12 1.14e-14 0.00 100.00
Table 8.4 Percent variance captured by the PCA model.
performed on the failing sensor. The signi cant increase in both SPE and T2 makes it dicult to draw any conclusions of how the process is operating during the sensor failure. A small but signi cant change of the process behaviour would "drown" in the residuals caused by the sensor failure. Thus, it is important that the misleading data from the erroneous sensor are replaced by data not distorting the monitoring.
Detection of External Disturbance
In the previous example the cause of the change of operational mode was caused by a sensor failure. This should have been resolved dur-ing the screendur-ing stage of the data. A more interestdur-ing situation is if there are many concurrent variables driving the process in a cer-tain direction. A PCA model is developed to monitor the process behaviour. In order to obtain a general model, more samples than in the previous example were included. The model is based on 6000 samples (approximately 21 days) collected during normal operational conditions. There are periods of rain and other minor disturbances, which make the model slightly more tolerant to such disturbances in the data. The data used for developing the model are the same as in the previous example, except that a variable describing the di erence
148 8. MVS Applied to Real On-line Measurements
between sludge concentration in line 1 and 2 is included as variable 12.
The captured variance is shown in Table 8.4. The number of retained principal components are chosen to four. This implies that the model is able to capture approximately 80% of the variance of the training data.
In Figure 8.5, the SPE (top) and T2 (bottom) measures are plotted.
First we focus on the SPE measure. The residual is increasing signi- cantly from sample 650 and forward. As mentioned before, the SPE is the Euclidian distance between the sample and the model plane (with four dimensions in this case). After sample 650 the process leaves the model region and the distance to the model plane increases.
A residual contribution plot (Figure 8.6, top) reveals what has caused the deviation. At sample 685 there is no single variable that can be pointed out as the main cause. Variables 1 (in uent temperature) and 9 (in uent pH), together with variables 5 (air valve position) and 11 (in uent ow rate) are slightly more dominant than the others. The conclusion is that the process has left the model region and several variables have contributed to this. In the residual contribution plot at sample 852 (Figure 8.6, bottom), it is possible to distinguish a few dominant variables contributing to the deviation. Variables 2 (sludge concentration, line 1), 10 (in uent pH) and 12 (di erence in sludge concentration between lines 1 and 2) appear to be the main contrib-utors. The disturbance in sample 852 is thus not exactly the same as in sample 685.
The T2 measure also increases around sample 650. The residual reaches a peak at sample 685, and the reason can be found in the contribution plots. Figure 8.7 shows the variables contributing to the change along the principal component between samples 640 and 685.
Here, variables 4, 5, 6, 7 (air valve positions) and 11 (in uent ow rate) are the most dominant variables. An increased in uent ow rate causing an increased demand of aeration seem to be the reason that the T2 measure exceeds its limits. The variables contributing to the change along the PCs between samples 640 and 852 are shown in Fig-ure 8.8. There are two dominant variables: 2 (sludge concentration in line 1) and 12 (di erence in sludge concentration). The conclusion is that there is a change in the sludge concentration in line 1. Hence, the disturbances at samples 685 and 852 are not the same.
8.2. Monitoring General Operational Modes 149
0 100 200 300 400 500 600 700 800 900 1000
0 50 100 150 200
685
852 SPE, with 99 % confidence limit
0 100 200 300 400 500 600 700 800 900 1000
0 20 40 60 80 100
852 685
T2, with 99 % confidence limit
time (samples)
Figure 8.5 The SPE and T2 measures for the new data.
1 2 3 4 5 6 7 8 9 10 11 12
−10
−5 0 5 10
variables
Contributing variables at sample 852.
1 2 3 4 5 6 7 8 9 10 11 12
−3
−2
−1 0 1 2
Contributing variables at sample 685.
Figure 8.6 Variable contribution to SPE at samples 685 (top) and 852 (bottom).
150 8. MVS Applied to Real On-line Measurements
1 2 3 4 5 6 7 8 9 10 11 12
−1 0 1 2 3 4 5
Sample 640−685
Contr. along PC no 1
Variables
1 2 3 4 5 6 7 8 9 10 11 12
−2
−1 0 1 2
Sample 640−685
Contr. along PC no 2
Variables
1 2 3 4 5 6 7 8 9 10 11 12
−2
−1 0 1 2
Sample 640−685
Contr. along PC no 3
Variables
1 2 3 4 5 6 7 8 9 10 11 12
−1
−0.5 0 0.5 1 1.5 2
Sample 640−685
Contr. along PC no 4
Variables
Figure 8.7 Variables contributing to the change along the PC nos 1, 2, 3 and 4 between samples 640 and 685.
1 2 3 4 5 6 7 8 9 10 11 12
−1
−0.5 0 0.5 1 1.5 2
Sample 640−852
Contr. along PC no 1
Variables
1 2 3 4 5 6 7 8 9 10 11 12
−6
−4
−2 0 2 4
Sample 640−852
Contr. along PC no 2
Variables
1 2 3 4 5 6 7 8 9 10 11 12
−1 0 1 2 3 4 5
Sample 640−852
Contr. along PC no 3
Variables
1 2 3 4 5 6 7 8 9 10 11 12
−1 0 1 2 3 4 5
Sample 640−852
Contr. along PC no 4
Variables
Figure 8.8 Variables contributing to the change along the PC nos 1, 2, 3 and 4 between samples 640 and 852.
8.2. Monitoring General Operational Modes 151
The original variables are shown in Figures 8.9, 8.10 and 8.11. The limits are the approximate 95% con dence limits of the original train-ing data. It is obvious that there are disturbances in most variables somewhere between samples 600 and 700, and it is dicult to determ-ine which ones are responsible for the deviations inSPE and T2. The contribution plots make it easier to determine the most probable main contributors to the process deviation and, thus, to the change in mode from normal to abnormal operation.
0 100 200 300 400 500 600 700 800 900 1000
10 15 20
0 100 200 300 400 500 600 700 800 900 1000
0 2000 4000
0 100 200 300 400 500 600 700 800 900 1000
0 5000 10000
0 100 200 300 400 500 600 700 800 900 1000
0 20 40
time (samples)
Figure 8.9 The original (measured) variables 1 to 4, plotted in order from top to bottom. The dashed lines correspond to the approximate 95% con dence limits of the training data.
152 8. MVS Applied to Real On-line Measurements
0 100 200 300 400 500 600 700 800 900 1000
0 20 40
0 100 200 300 400 500 600 700 800 900 1000
0 50
0 100 200 300 400 500 600 700 800 900 1000
0 50
0 100 200 300 400 500 600 700 800 900 1000
0 1 2
time (samples)
Figure 8.10 The original (measured) variables 5 to 8, plotted in order from top to bottom. The dashed lines correspond to the approximate 95% con dence limits of the training data.
0 100 200 300 400 500 600 700 800 900 1000
−50 0 50
0 100 200 300 400 500 600 700 800 900 1000
6 8 10
0 100 200 300 400 500 600 700 800 900 1000
−1000 0 1000
0 100 200 300 400 500 600 700 800 900 1000
−2000
−1000 0 1000
time (samples)
Figure 8.11 The original (measured) variables 9 to 12, plotted in order from top to bottom. The dashed lines correspond to the approximate 95% con dence limits of the training data.
8.2. Monitoring General Operational Modes 153
Principal Eigenvalue % Variance % Variance
component of captured captured
number cov(X) this PC total
1 3.56e+00 35.55 35.55
2 1.43e+00 14.30 49.86
3 1.34e+00 13.40 63.26
4 1.08e+00 10.77 74.02
5 8.27e-01 8.27 82.29
6 6.90e-01 6.90 89.19
7 5.02e-01 5.02 94.21
8 3.56e-01 3.56 97.77
9 1.69e-01 1.69 99.45
10 5.46e-02 0.55 100.00
Table 8.5 Percent variance captured by the PCA model.
Detection Using Multiple Operational Classes
Monitoring the SPE and T2 residuals is e ective, but not especially transparent. As mentioned before, the nature of most industrial pro-cesses implies that only a few latent variables or principal components will cover most of the variability of the process space. By plotting, for instance, the rst score vector against the second score vector the process changes can be viewed as a point moving around in the plane as new samples are added. Points that cluster represent similar pro-cess behaviour and, consequently, deviating points indicate propro-cess changes. This makes the score plot useful for classi cation purposes.
A PCA model is identi ed from a set of data containing 10 000 samples (approximately 36 days) of process measurements. The variables in-cluded in the analysis is the same as the ones listed in Table 8.2.
However, the temperature measurement is excluded for reasons dis-cussed later. The model is able to capture 50% of the variability using two principal components (Table 8.5). In Figure 8.12, it is shown that the majority of the data is located in a cluster somewhat sym-metrically located around the origin. There is one more cluster below the main cluster. The clusters are demarcated by class boundaries.
These boundaries are determined empirically, by manually examining the plot. Four classes are de ned from the numerical characteristics of the data. Is it possible to nd a physical interpretation to the cluster appearances? Some of the variables of the data used to identify the
154 8. MVS Applied to Real On-line Measurements
−25 −20 −15 −10 −5 0 5 10
−8
−6
−4
−2 0 2 4
Scores on PC 1
Scores on PC 2
d
c
b a
Figure 8.12 Score plot with classes manually de ned from separate clusters and patterns in the plane.
PCA model are shown in Figure 8.13. There are some events occur-ring duoccur-ring the training period: immediately before sample 2000 there are a few peaks in the in uent ow rate (plot d) and in the air valve position (plot c), around samples 4400, 5800 and 6900 there are peaks in the suspended solids concentration (plots a and b) and from about samples 7000 to 7500 there is a period of rain increasing the in uent ow signi cantly (plot d). All these events are represented by the di erent class memberships in the score plot. Normal, dry-weather conditions are covered by class a. The peaks in ow rate together with the increased oxygen demand are represented by class d, longer periods of high ow rate fall into class b and upsets in the suspended solids concentrations belong to class c. As shown in Figure 8.12, di er-ent operating conditions appear as clusters or deviations in the score plot. Next step is to investigate how the PCA model performs when it is applied to new (on-line) data. In Figure 8.14, the PCA model and its classes, are used to classify new data (10000 samples). The majority of the data falls into the normal class (a), but there are some interesting deviations. Data are frequently classi ed as class b (high ow rate) and some times as class d (high oxygen demand). There are
8.2. Monitoring General Operational Modes 155
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 2000
3000 4000 5000
a
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 2000
3000 4000 5000
b
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20
40 c
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0
500 1000
time (samples) d
Figure 8.13 The original (measured) variables. Suspended solids concen-tration in line 1 (a) and in line 2 (b). Air valve position (c) and in uent ow rate (d).
also samples falling outside all the de ned classes, somewhere between and below classes b and d. In that case we have a situation not covered by the de ned classes, and the set of classes may have to be updated for future use. The data falling outside the de ned classes ranges from samples 1937 to 1972 in the rst event (the loop below class d) and from samples 3090 to 3100 in the second event (the loop below class b). By examining Figure 8.15, both events can be traced to an in-crease in both oxygen demand (plot c) and in uent ow rate (plot d).
This is not surprising as the events occur in between and below classes b and d in Figure 8.14. The samples inside class c can be interpreted as an increase in the suspended solids concentration around samples 2150 and 3460. As a matter of fact, some samples fall outside the axis of the score plot and are not included for clarity reasons. An example of an event outside the score plot can be seen in Figure 8.15 around sample 500. It is obvious that something is wrong at this point in time.
In this example it is demonstrated that PCA can be used for classi- cation of general operational modes, and that the number of classes
156 8. MVS Applied to Real On-line Measurements
−25 −20 −15 −10 −5 0 5 10
−8
−6
−4
−2 0 2 4
Scores on PC 1
Scores on PC 2
a
b
c d
Figure 8.14 Score plot with classes from Figure 8.14.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 2000
3000 4000 5000
a
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 2000
3000 4000 5000
b
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20
40 c
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0
500 1000
time (samples) d
Figure 8.15 The original (measured) variables. Suspended solids concen-tration in line 1 (a) and in line 2 (b). Air valve position (c) and in uent ow rate (d).