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Multivariate and multiscale monitoring of wastewater treatment operation
PII: S0043-1354(01)00069-0
Wat. Res. Vol. 35, No. 14, pp. 3402–3410, 2001 # 2001 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0043-1354/01/$ - see front matter
MULTIVARIATE AND MULTISCALE MONITORING OF WASTEWATER TREATMENT OPERATION C. ROSEN1* and J. A. LENNOX2 1
Department of Industrial Electrical Engineering and Automation, Lund University, Box 118, SE-221 00 Lund, Sweden and 2 Advanced Wastewater Management Centre, The University of Queensland, St Lucia, 4072 QLD, Australia (First received 25 October 1999; accepted in revised form 23 January 2001)
Abstract}In this work extensions to principal component analysis (PCA) for wastewater treatment (WWT) process monitoring are discussed. Conventional PCA has some limitations when used for WWT monitoring. Firstly, PCA assumes that data are stationary, which is normally not the case in WWT monitoring. Secondly, PCA is most suitable for monitoring data that display events in one time-scale. However, in WWT operation, disturbances and events occur in different time-scales. These two limitations make conventional PCA unsuitable for WWT monitoring. The first limitation can be overcome by use of adaptive PCA. In adaptive PCA, the PCA model is continuously updated using an exponential memory function. Variable mean, variance and co-variance are thus adapted to the changing conditions. The second problem can be solved by time-scale decomposition of data prior to analysis. The time-scale decomposition methodology involves wavelets and multiresolution analysis (MRA) in combination with PCA. MRA provides a tool for investigation and monitoring of process measurement at different timescales by decomposing measurement data into separate frequency bands. Time-scale decomposition increases the sensitivity of the monitoring, which makes it possible to detect small but significant events in data displaying large variations. Moreover, time-scale information is sometimes important in the interpretation of a disturbance to determine its physical cause. Also, by decomposing data, the problem of changing process conditions is partly solved. All the presented methods are illustrated with examples using real WWT process data. # 2001 Elsevier Science Ltd. All rights reserved Key words}adaptive PCA, monitoring, multiresolution, multiscale, wastewater, wavelets
INTRODUCTION
Process operation monitoring is carried out to ensure that the process operation complies with requirements on product quality, process safety and efficient use of resources. In most industrial applications, including wastewater treatment (WWT), process performance and operation are measured continuously. Often, the number of measured variables is high, demanding a structured approach to monitoring and analysis of the process. Multivariate statistics (MVS) provides a methodology to extract and structure information from large amounts of data. MVS has been used to monitor industrial processes for several decades (Wise and Gallagher, 1996; Kresta et al., 1991). It has also been applied to WWT operation (Rosen and Olsson, 1998; Rosen, 1998; Mujunen et al., 1998). Principal component analysis (PCA) is a fundamental and one of the most popular MVS-based monitoring methods. However, it is clear that the method, at least in its basic configuration, has some *Author to whom all correspondence should be addressed. E-mail: [email protected]
shortcomings. One of these is that PCA is not suited for monitoring processes that display non-stationary behaviour as PCA-based methods assume that the mean of the data is approximately constant. Moreover, variable variance cannot deviate too much from the data used to identify the model without losing the option to use statistically based detection or alarm limits. Furthermore, since the model describes the relationship between several variables, the co-variance structure must not change over time. Fortunately, there are ways to circumvent this problem. Straightforward ways are to identify new monitoring models as the process conditions change or to use a number of models and choose among them using a suitable criterion. A somewhat more sophisticated way is to constantly and automatically update the model}make the model adaptive. This can be achieved by using a moving window including historical data or by calculating the model recursively (Helland et al., 1991; Wold, 1994; Dayal and MacGregor, 1997). A second way to handle changing process conditions is based on wavelets. Wavelet analysis is a relatively recent technique for simultaneous analysis of the time and frequency contents of a signal.
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Wavelets have proven useful in many different fields, from image processing to model identification (see e.g. Strang and Nguyen, 1996; Alsberg et al., 1997). Wavelets have also been used for monitoring purposes in water and wastewater applications (Whitfield and Dohan, 1997; Lennox and Rosen, 2000). For the purpose of monitoring and detection, wavelets can be used to decompose a signal into different scales of decreasing levels of detail or resolution. This is sometimes referred to as multiresolution analysis (MRA). The separation of data into multiple time-scales implies that the faster scales will have an approximately constant mean and only the slower or slowest scale will display trends or longterm variations. Consequently, by omitting the slower/slowest scale from the monitoring, the problem of monitoring data from changing process conditions is partly solved in a way comparable with adaptive scaling of the mean. Multiple time-scale monitoring also provides a solution to another shortcoming of uniscale PCA-based monitoring. Since only one scale is monitored, uniscale PCA is most appropriate when the data contain events occurring at one scale, i.e. in a narrow frequency band. This is not the case in most industrial applications, and certainly not in WWT, where both fast and slow deviations occur. Correlations between variables are typically different at different scales. Ignoring this degrades the sensitivity and, consequently, the ability to detect small, but significant, changes in data. Combinations of MRA and PCA have been proposed by Kosanovich and Piovoso (1997) and Bakshi (1998). In this paper we apply and further develop this methodology to adapt it to wastewater treatment process monitoring. The paper is organised as follows. First a short introduction to multivariate monitoring and PCA is given, followed by a discussion on how to extend basic PCA to adaptive PCA. MRA is introduced and an integration of MRA and PCA is presented. A number of examples that serve as illustrations of the methods presented and that highlight the differences between the various approaches are discussed. The examples are based on real WWT process data.
PCA MONITORING
PCA can be described as a method to project a high-dimensional measurement space onto a space with significantly fewer dimensions. Often, for industrial process data, many variables are highly correlated, since they reflect relatively few underlying mechanisms that drive the process. In PCA, this correlation between the variables is used to find principal components (PCs) that represent the underlying mechanisms. The original variables are projected onto the PCs as scores. To compare variables with different amplitudes and variability, data are normally mean centred and scaled prior to analysis.
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For a more comprehensive description of PCA see e.g. Jackson (1980) or Wold et al. (1987). On-line monitoring using PCA Monitoring of the scores can be carried out using either conventional univariate statistical process control (SPC) techniques (see e.g. Bissel, 1994) or scatter plots (MacGregor and Kourti, 1995). In addition to monitoring the scores, the statistical fit of the model can be monitored. Two commonly used measures of fit are sum of squared prediction error (SPE) and Hotelling’s T 2 . SPE is a measure of the distance from the model plane to an observation, i.e. the model error. T 2 is a measure of the distance within the model plane from an observation to the origin. SPE and T 2 are convenient as they summarise the multivariate process information of the variables by single values and are normally monitored using conventional SPC charts with in-control limits defining the normal or desired operational region (Jackson and Mudholkar, 1979; Kresta et al., 1991). Robust limits can be determined empirically using, for instance, percentiles (Rosen, 1998). There is more information to be gained from a deeper investigation of the PCA model. When a disturbance is detected in score plots or in the SPE and T 2 plots, a physical interpretation can be found by transforming the model output back into the original process space. So-called contribution plots can be used to determine the variables that caused the deviation (MacGregor et al., 1994). Adaptive PCA PCA monitoring, as described above, assumes that the process conditions do not change significantly. This is rarely the case for WWT processes. Diurnal and seasonal changes affect the mean, variability and correlation of the variables. Small but important events tend to be obscured within the residuals related to the normal variation. A method to reduce the problem of changing conditions is to make the monitoring model adaptive, i.e. update the model on an adequate time-scale. If the co-variance structure of the monitored variables is believed to be static, i.e. the relations between the variables stay the same, regardless of their mean values and variability, the adaptation can be limited to data pre-processing. This may involve highpass filtering or updating of the scaling parameters. A simple way of updating the scaling parameters is to calculate them from a moving time window of appropriate length. A more sophisticated method is to update the mean and variance for each variable recursively. If the co-variance structure is believed to change over time, the model itself must be updated. In analogy with the updating of the scaling parameters, the model can be updated using a moving window. However, recursive updating can be applied as
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(Dayal and MacGregor, 1997): XT XðkÞ ¼ aXT Xðk 1Þ þ xT ðkÞxðkÞ where XT XðkÞ is the co-variance structure at time k, xðkÞ is the last sample row and a is a forgetting factor. The value of a may vary significantly depending on the aim of the monitoring. Adaptation of the co-variance matrix introduces some difficulties. Firstly, if no precautions are taken, the model will adapt to disturbances and failures atypical of the normal process behaviour. Secondly, if there is not sufficient excitation in the data, process information will be lost as old data are discounted. It may therefore be wise to test the information content of the data before it is used to update the model, especially if the forgetting factor is small, i.e. the effective memory is short (Wold, 1994; Dayal and MacGregor, 1997). A third problem caused by an adapting co-variance matrix is that the co-ordinate system defined by the PCs will rotate. This makes it hard to use scatter plots for interpretation of the process performance, as the co-ordinate system differs for each sample.
MULTISCALE MONITORING
For the purpose of monitoring and detection, wavelets can be used to decompose a signal into different time-scales with decreasing level of detail or resolution, i.e. MRA. A signal is filtered with a highpass filter and a lowpass filter, respectively. The result is one set of coefficients that describe the details of the signal and another set describing the approximation. The decomposition can be carried out to a desired number of scales by recursively applying the highpass and lowpass filters to the approximation coefficients of the previous level. When a filter has been applied, the result is downsampled. This means that the total number of coefficients in the wavelet domain is the same as the number of samples in the original signal. The coefficients still contain all the information carried by the original signal. For both the highpass and the lowpass filters, there are corresponding reconstruction filters. Using these filters for the reconstruction, together with upsampling, will result in a perfect reconstruction of the original signal in one scale. Alternatively, it is possible to separately reconstruct the components corresponding to each scale. The number of coefficients in the time domain will then be the original number of samples multiplied by the number of scales. MRA monitoring For monitoring purposes MRA is applied on-line to a moving time window. Univariate SPC techniques can be utilised to localise the scale in which a
disturbance is first detected. Fast disturbances are detected in the higher scales whilst slow disturbances are detected in the lower scales. The way a disturbance appears across scales can reveal information on the disturbance characteristics. For instance, a spike will be strong in the higher scales but barely visible in the lower scales. A step will appear clearly in all scales, whereas a ramp will appear most strongly in the lower scales. Monitoring of multivariate data is somewhat more problematic since increased sensitivity leads to more detections which have to be evaluated. Moreover, events that occur in several scales will trigger detections in more than one scale, even though they originate from the same event. This, together with the fact that all variables need to be monitored on several scales, call for multivariate methods that are capable of both reducing the dimension of the problem and handling collinear variables. A natural choice is PCA. Methodologies combining MRA with PCA have been proposed by Kosanovich and Piovoso (1997) and Bakshi (1998). PCA models are identified for each scale from data representing normal or desired behaviour. This is done by calculating the wavelet coefficients on each scale by applying a moving window the length of the longest filter to the training data. Monitoring new data involves decomposition and projection of data onto the PCA models on each scale. A window is used in the same manner as when the models are identified. The transformed data from the window on a certain scale are calculated and projected onto the PCA model at that scale. The scores are monitored using SPC techniques or scatter plots. The SPE and T 2 can be calculated and compared to the predefined confidence limits for that particular scale model. Time-scale decomposition increases the number of data points used for detection. This increased number is, however, based on only the number of original measurement points. If a certain overall confidence limit for all the scales together is to be obtained, the confidence limits on each scale must be adjusted (Bakshi, 1998). Recombination of scales A difficulty with the multiscale method described above is that when the number of scales increases, the number of entities to monitor increases as well. PCA will reduce the number, but with a high number of scales we might end up with more variables than in the original data set. A possible solution is to recombine some of the scales so that the effective number of scales to monitor decreases (Fig. 1). This enables us to both reduce the number of entities to monitor and to give physical meaning to the recombined scales by choosing them to correspond approximately to meaningful dynamic time-scales in the process. For instance, the first scales can be recombined to a ‘fast’ scale (e.g. hydraulic dynamics), the middle scales can be combined to constitute
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Fig. 1. Recombination of scales into fewer and more physically interpretable scales.
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Fig. 2. The MSPCA procedure.
a ‘medium fast’ scale (e.g. concentration dynamics) and the lower and/or lowest scale (e.g. population dynamics) can represent the ‘slow’ changes. Multiscale PCA Multiscale PCA (MSPCA) was originally proposed by Bakshi (1998). The idea is that the PCA models on each scale are used to determine whether a scale contains significant information or not at a certain point in time. If a scale is significant, the data on that scale are used in the reconstruction of a uniscale estimate of the original data. The reconstructed data are then monitored using a unifying uniscale PCA, which is based on the scales included in the reconstruction. The recombination of scales reduces the number of entities to monitor to the level of normal or adaptive PCA, while the prior scale PCA detection acts as a multiscale and multivariate feature extractor. Data from a period of normal or desired process behaviour are decomposed into scales. A PCA model is identified on each scale. New data are decomposed and projected onto the scale models. If a model residual on any scale exceeds its confidence limit, the scale is said to be significant, and is used to reconstruct the scale data into uniscale set of data. The reconstruction is monitored using a uniscale PCA, which is based on training data of significant scales (Fig. 2). RESULTS AND DISCUSSION
In this section we demonstrate and discuss the methods presented earlier. First an example is given where basic PCA monitoring is carried out. This example serves as a comparison to the examples when various extensions of the PCA have been used. The data used are actual WWT process data from the Ronneby plant in Sweden. The plant is operated as a biological nutrient removal plant with additional chemical treatment. The data-sampling period is 5 min. There are 14 available variables from the online measurement system, including flow rate, pH, ammonia, temperature and air valve positions at several points in the plant. The data span several months, from summer to late autumn implying
Fig. 3. No adaptation. The relative SPE and T 2 , i.e. the ratios between the residuals and their limits.
significant changes in operating conditions over the period. In the examples using time-scale decomposed data, data are decomposed into six different scales using the Haar wavelet (Strang and Nguyen, 1996). Note that isolation of the variables is not discussed here. However, in all examples, contribution plots and other techniques can be utilised to determine which variables cause the deviations. Static PCA A PCA model is identified from a 20-day period when the operation is considered to be normal except for a few brief upsets. New data are projected onto the model and the result can be seen in Fig. 3. It is obvious that the changing conditions cannot be covered by the static model. Almost from the beginning of the new period, both SPE and T 2 violate their limits and from day 40 the performance of the model deteriorates significantly. Adaptive scaling parameters A monitoring model with adaptive scaling parameters is identified. Six PCs are retained. The forgetting factor, a, is set to 0.9995, which corresponds to 2000 samples ( 21 days). This is a reasonable choice, allowing detection of slower
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changes as well as fast ones. In Fig. 4 the SPE and T 2 are shown. It can be said that the model is generally valid (i.e. the SPE is below its limit) during the whole period, indicating a relatively constant co-variance structure. However, there are some periods with high SPE, for instance, at days 20–21, 24–25, 42, 60–61 and 70. The T 2 measure hardly exceeds its limit, not even when the SPE assumes extreme values. The reason is possibly that the deviations are mainly caused by changes in the relations between the variables. Thus, the model does not capture the variation in a correct way, and most of the variation is outside the model plane. This indicates that the co-variance structure of the data is changing. For this reason, the information obtained from the scores must be used with caution. Adaptive PCA A model with adaptive scaling parameters as well as adaptive co-variance structure is used to monitor the same variables as in the example above. The result is shown in Fig. 5. The SPE is here mostly well
inside its confidence region, while T 2 violates its limit more markedly (compare with Fig. 4). This implies that the model captures the variation and that most of the variation will be manifested in the T 2 . The reason is that when the co-variance structure adapts to new conditions, most of the variation will be in the model plane (T 2 ) and not orthogonal to the model plane (SPE). Put simply, some of the variation from the SPE chart in Fig. 4 has now been transferred to the T 2 chart in Fig. 5. This highlights the importance of monitoring both SPE and T 2 as they provide complementary information. An example of the complementary relation between SPE and T 2 can be seen if we take a closer look at the event at days 41–42 (Fig. 6). Initially, there is a change in the relation between the variables (i.e. detection by SPE at day 41.4). As the variable relations change and/or the model adapts, the SPE decreases again at day 41.6. However, the T 2 is now large, indicating high variable magnitudes. It stays high until day 42.2, when SPE again becomes high, now indicating a severe disturbance at day 42.4. PCA monitoring of physically interpretable scales
Fig. 4. Adaptive scaling parameters. The relative SPE and T 2 , i.e. the ratios between the residuals and their limits.
Recombination of scales is done to simplify interpretation. Ideally, this is done in such a way that the resulting scales have a physical interpretation. Here, we will not focus on how to obtain the different time-scales present in a system. We define fast dynamic as the first three scales. This approximately corresponds to variations from 0 to 4 h. The medium time-scale consists of scales 4 to 6, which corresponds to 4 to 32 h. The slow dynamic is represented by the approximation scale, i.e. approximately everything slower than the diurnal variation. The SPE and T 2 charts show that the detail scale models (fast and medium) are valid, whereas the approximation scale (slow) model is not (Figs 7 and 8). This implies that the slow scale can only be used as an indication of the absolute distance from
Fig. 5. Adaptive PCA. The relative SPE and T 2 , i.e. the ratios between the residuals and their limits.
Fig. 6. Adaptive PCA. A closer investigation of SPE and T 2 at days 41–42.
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Fig. 7. The SPE residuals for the recombined scales defined as fast, medium and slow. The slow scale should only be used as an indication of the absolute distance from operation defined by training data.
Fig. 8. The T 2 residuals for the recombined scales defined as fast, medium and slow. The slow scale should only be used as an indication of the absolute distance from operation defined by training data.
normal operational conditions. Note that some of the disturbances occur both in the fast and the medium scale, which suggests that these disturbances have a prolonged duration. This information may be useful for interpretation purposes. However, it is interesting to note that at, for instance, day 118 the disturbance is detected in the medium scale first, i.e. the disturbance is slow. This can be seen in the original data as a relatively slow increase in flow rate and all air valve positions. The T 2 charts display similar behaviour. Most disturbances are fast and, thus, commenced by a rapid change in one or several variables. The disturbance at day 118 is most obvious
in the medium scale, which is consistent with the SPE chart. MSPCA monitoring The MSPCA method is applied to the data used in the previous examples. The number of scales is seven, i.e. six detail scales and one approximation scale. Both SPE and T 2 are used on each scale to determine whether a scale is significant. The limits for the model residuals are corrected to obtain an overall confidence of 99%. The resulting uniscale SPE and T 2 can be seen in Fig. 9. The confidence limits on the
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Fig. 9. SPE (top) and T 2 (bottom) from the MSPCA during a time period of 100 days. The confidence limits for both residuals are 99%.
uniscale residuals are also 99%. The SPE chart indicates that the model is valid during the whole period. Consequently, the co-variance structure of the reconstructed data does not change significantly, mostly due to the fact that the approximation scale has been omitted from the analysis. There are some upsets in SPE, but the major deviations can be seen in the T 2 . There is an interesting and important feature of MSPCA that distinguishes it from the other two multiscale methods. Consider a step change in one or several variables at a certain point in time: all the methods will detect this step at the same time. However, if the variable or variables return to normal, both the method involving PCA on each scale and the method involving prior recombination of scales, will continue to detect a deviation due to the delay of the filters. This is generally not the case with MSPCA. When the variables return to normal, the number of significant scales will increase causing the confidence level to increase and generally compensate for the delay. An example of this can be seen after the disturbance at day 42, where the SPE quickly returns to values below its limits. Comparative remarks It is clear that basic PCA cannot be used with confidence when the process conditions change. Extending the basic PCA with adaptive scaling parameters improves the results significantly without losing the ability to use score plots. This is an advantage of a fixed co-variance structure since score plots are quite intuitive in the interpretation process. Unfortunately, the model with adaptive scaling parameters is not fully capable of representing the process. The model based on adaptive scaling
parameters and co-variance structure handles the changing conditions well. Qualitatively, the two multiscale methods presented here have similar performance and they basically detect the same events equally well. However, there are some differences. Recombination of scales implies more entities to monitor but if it is possible to discern some dominant time-scales of the studied process, the recombination of scales to physically interpretable scales ought to simplify the interpretation considerably. The capability of MSPCA to unify all the scales into one scale is desirable and provides a compact way of monitoring. Moreover, MSPCA has an important advantage in that it returns to normal as soon as a disturbance has ended. The examples show that both the adaptive and multiscale PCA are possible ways to overcome the difficulties associated with changing process conditions. Adaptive and multiscale PCA are similar in performance, but both methodologies have their specific advantages. Adaptive PCA involves no (or little) pre-processing and is relatively simple in its structure. The co-variance structure is updated, which means that new or changed relations between variables can be represented. However, the monitoring becomes relative in the sense that information about the distance from the operational point, on which the starting model is built, is lost. In multiscale PCA, this information is not lost, but represented by the lowest scale. Moreover, the sensitivity to small changes is higher. However, as opposed to adaptive PCA, the co-variance structure is static, which may introduce errors. A solution to this drawback could be to apply adaptive PCA on each scale.
Wastewater treatment process monitoring
It is the authors opinion that the simplest effective model should be used for monitoring. However, it is shown here that some extensions to basic PCA monitoring must be included. The multiscale approach includes far more degrees of freedom than the adaptive approach, but it provides a way to discern small changes in data with, for instance, large diurnal variations. Moreover, the multiscale approach provides information on the scale at which a disturbance occurs, which can be used for diagnostic purposes to find the physical cause. However, the advantages come at a price of higher complexity of the monitoring model. The choice of approach used depends on the complexity of the process. A simple process, or perhaps a section of a process, where most events occur in one scale, can be adequately monitored at one time-scale. In a more complex process, with different subprocesses and different time constants, a multiscale method may be more suitable. Further extensions A few important objections to PCA as a monitoring technique for industrial processes can be found in the literature. In addition to the ones raised on PCA and changing process conditions, it is often pointed out that PCA is a static technique, not suitable for dynamic processes. This is partly addressed by multiscale monitoring, since each scale can be seen as a ‘‘snapshot’’ of the process in a limited frequency range. However, other techniques are available. By including a time lag or history in the analysis, i.e. the old measurements are included in the X matrix, dynamic relationships between variables can be represented (e.g. Ku et al., 1995; Lou et al., 1999). This can further improve the monitoring model, but at a price of higher model complexity and a large number of variables in the X matrix. An interesting topic not discussed in this paper is related to integration of monitoring and control. Information on the operational state is important in control strategy design and implementation. In PCA monitoring the operational state can be determined from the process location in the PC space and loading plots can be used to determine suitable control actions (Rosen and Yuan, 2000). In multiscale monitoring one additional piece of information can be obtained: time-scale information. So, in addition to the information on the present operational state, information on how fast the state has changed can be used to determine the strength of corrective measures to drive the process back to its desired operation. Integration of time information in the control strategy is an interesting topic for further studies.
CONCLUSIONS
Adaptive and multiscale extensions of basic PCA monitoring can be used to monitor WWT process
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data during changing operational conditions. Two different approaches are presented. Adaptive PCA, in terms of updating mean, variance and co-variance structure, overcomes the problem of non-stationary process data. The monitoring model is recursively updated using an exponential memory function. When the relations between the variables are known to be constant, only scaling parameters need to be updated. The methodology is relatively simple and can easily be included in standard PCA-monitoring. However, the monitoring is relative and adaptive PCA need to be complemented by an absolute measure of the process state. This is not a difficulty with the second approach: multiscale monitoring. Here, the process data are split into several timescales. The higher scales represents faster changes and the lower scales represent slower ones. All but the lowest scale will have approximately zero mean, which means that the non-stationary problem is partly solved. The information about on which scale a disturbance occurs can be used to interpret the disturbance. When events in different scales are separated, the sensitivity of the monitoring is increased. The choice between the two approaches is a trade-off between complexity and information.
REFERENCES
Alsberg B. K., Woodward A. M. and Kell D. B. (1997) An introduction to wavelet transforms for chemometricians: a time-frequency approach. Chemometrics Intell. Lab. Syst. 37, 215–239. Bakshi B. R. (1998) Multiscale PCA with application to multivariate statistical process monitoring. AICHE J. 44(7), 1596–1610. Bissel D. (1994) Statistical Methods for SPC and TQM. Chapman & Hall, London, UK. Dayal B. S. and MacGregor J. F. (1997) Recursive exponentially weighted PLS and its application to adaptive control and prediction. J. Proc. Cont. 7(3), 169–179. Helland K., Berntsen H. E., Borgen O. S. and Martens H. (1991) Recursive algorithm for partial least squares regression. Chemometrics Intell. Lab. Syst. 14, 129–137. Jackson J. E. (1980) Principal components and factor analysis: Part I}principal components. J. Qual. Technol. 12(4), 201–213. Jackson J. E. and Mudholkar G. S. (1979) Control procedures for residual associated with principal component analysis. Technometrics 21(3), 341–349. Kosanovich K. A. and Piovoso M. J. (1997) PCA of wavelet transformed process data for monitoring. Intell. Data Anal. 1(2), http://wwweast.elsevier.com/ida/. Kresta J. V., MacGregor J. F. and Marlin T. E. (1991) Multivariate statistical monitoring of process operating performance. Can. J. Chem. Engng. 69, 35–47. Ku W., Storer R. H. and Georgakis C. (1995) Disturbance detection and isolation by dynamic principal component analysis. Chemometrics Intell. Lab. Syst. 30, 179–196. Lennox J. A. and Rosen C. (2000) Using wavelets to extract information from wastewater treatment process data. First World Water Congress of the International Water Association (IWA), Paris, France. Lou R., Misra M. and Himmelblau D. M. (1999) Sensor fault detection via multiscale analysis and dynamic PCA. Ind. Engng. Chem. Res. 38, 1489–1495.
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MacGregor J. F. and Kourti T. (1995) Statistical process control of multivariate processes. Control Engng. Practice 3(3), 403–414. MacGregor J. F., Jaeckle C., Kiparissides C. and Koutoudi M. (1994) Process monitoring and diagnosis by multiblock PLS methods. AICHE J. 40(5), 826–838. Mujunen S.-P., Minkkinen P., Teppola P. and Wirkkala R.-S. (1998) Modeling of activated sludge plants treatment efficiency with PLSR: a process analytical case study. Chemometrics Intell. Lab. Syst. 41, 83–94. Rosen C. (1998) Monitoring wastewater treatment systems. Tech. Lic. thesis, Department of Industrial Electrical Engineering and Automation, Lund University, Lund, Sweden. Rosen C. and Olsson G. (1998) Disturbance detection in wastewater treatment plants. Water Sci. Technol. 37(12), 197–205. Rosen C. and Yuan Z. (2000) Supervisory control of wastewater treatment plants by combining principal
component analysis and fuzzy C-means clustering. Fifth International Symposium on System Analysis and Computing in Water Quality Management (WATERMATEX 2000), Gent, Belgium. Strang G. and Nguyen T. (1996) Wavelets and Filter Banks. Wellesley–Cambridge Press, Wellesley, MA, USA. Whitfield P. H. and Dohan K. (1997) Identification and characterization of water quality transient using wavelet analysis. II. application to electronic water quality data. Water Sci. Technol. 36(5), 337–348. Wise B. M. and Gallagher N. B. (1996) The process chemometrics approach to process monitoring and fault detection. J. Proc. Cont. 6(6), 329–348. Wold S. (1994) Exponentially weighted moving principal components analysis and projection to latent structures. Chemometrics Intell. Lab. Syst. 23, 149–161. Wold S., Esbensen K. and Geladi P. (1987) Principal component analysis. Chemometrics Intell. Lab. Syst. 2, 37–52.
Wat. Res. Vol. 35, No. 14, pp. 3402–3410, 2001 # 2001 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0043-1354/01/$ - see front matter
MULTIVARIATE AND MULTISCALE MONITORING OF WASTEWATER TREATMENT OPERATION C. ROSEN1* and J. A. LENNOX2 1
Department of Industrial Electrical Engineering and Automation, Lund University, Box 118, SE-221 00 Lund, Sweden and 2 Advanced Wastewater Management Centre, The University of Queensland, St Lucia, 4072 QLD, Australia (First received 25 October 1999; accepted in revised form 23 January 2001)
Abstract}In this work extensions to principal component analysis (PCA) for wastewater treatment (WWT) process monitoring are discussed. Conventional PCA has some limitations when used for WWT monitoring. Firstly, PCA assumes that data are stationary, which is normally not the case in WWT monitoring. Secondly, PCA is most suitable for monitoring data that display events in one time-scale. However, in WWT operation, disturbances and events occur in different time-scales. These two limitations make conventional PCA unsuitable for WWT monitoring. The first limitation can be overcome by use of adaptive PCA. In adaptive PCA, the PCA model is continuously updated using an exponential memory function. Variable mean, variance and co-variance are thus adapted to the changing conditions. The second problem can be solved by time-scale decomposition of data prior to analysis. The time-scale decomposition methodology involves wavelets and multiresolution analysis (MRA) in combination with PCA. MRA provides a tool for investigation and monitoring of process measurement at different timescales by decomposing measurement data into separate frequency bands. Time-scale decomposition increases the sensitivity of the monitoring, which makes it possible to detect small but significant events in data displaying large variations. Moreover, time-scale information is sometimes important in the interpretation of a disturbance to determine its physical cause. Also, by decomposing data, the problem of changing process conditions is partly solved. All the presented methods are illustrated with examples using real WWT process data. # 2001 Elsevier Science Ltd. All rights reserved Key words}adaptive PCA, monitoring, multiresolution, multiscale, wastewater, wavelets
INTRODUCTION
Process operation monitoring is carried out to ensure that the process operation complies with requirements on product quality, process safety and efficient use of resources. In most industrial applications, including wastewater treatment (WWT), process performance and operation are measured continuously. Often, the number of measured variables is high, demanding a structured approach to monitoring and analysis of the process. Multivariate statistics (MVS) provides a methodology to extract and structure information from large amounts of data. MVS has been used to monitor industrial processes for several decades (Wise and Gallagher, 1996; Kresta et al., 1991). It has also been applied to WWT operation (Rosen and Olsson, 1998; Rosen, 1998; Mujunen et al., 1998). Principal component analysis (PCA) is a fundamental and one of the most popular MVS-based monitoring methods. However, it is clear that the method, at least in its basic configuration, has some *Author to whom all correspondence should be addressed. E-mail: [email protected]
shortcomings. One of these is that PCA is not suited for monitoring processes that display non-stationary behaviour as PCA-based methods assume that the mean of the data is approximately constant. Moreover, variable variance cannot deviate too much from the data used to identify the model without losing the option to use statistically based detection or alarm limits. Furthermore, since the model describes the relationship between several variables, the co-variance structure must not change over time. Fortunately, there are ways to circumvent this problem. Straightforward ways are to identify new monitoring models as the process conditions change or to use a number of models and choose among them using a suitable criterion. A somewhat more sophisticated way is to constantly and automatically update the model}make the model adaptive. This can be achieved by using a moving window including historical data or by calculating the model recursively (Helland et al., 1991; Wold, 1994; Dayal and MacGregor, 1997). A second way to handle changing process conditions is based on wavelets. Wavelet analysis is a relatively recent technique for simultaneous analysis of the time and frequency contents of a signal.
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Wavelets have proven useful in many different fields, from image processing to model identification (see e.g. Strang and Nguyen, 1996; Alsberg et al., 1997). Wavelets have also been used for monitoring purposes in water and wastewater applications (Whitfield and Dohan, 1997; Lennox and Rosen, 2000). For the purpose of monitoring and detection, wavelets can be used to decompose a signal into different scales of decreasing levels of detail or resolution. This is sometimes referred to as multiresolution analysis (MRA). The separation of data into multiple time-scales implies that the faster scales will have an approximately constant mean and only the slower or slowest scale will display trends or longterm variations. Consequently, by omitting the slower/slowest scale from the monitoring, the problem of monitoring data from changing process conditions is partly solved in a way comparable with adaptive scaling of the mean. Multiple time-scale monitoring also provides a solution to another shortcoming of uniscale PCA-based monitoring. Since only one scale is monitored, uniscale PCA is most appropriate when the data contain events occurring at one scale, i.e. in a narrow frequency band. This is not the case in most industrial applications, and certainly not in WWT, where both fast and slow deviations occur. Correlations between variables are typically different at different scales. Ignoring this degrades the sensitivity and, consequently, the ability to detect small, but significant, changes in data. Combinations of MRA and PCA have been proposed by Kosanovich and Piovoso (1997) and Bakshi (1998). In this paper we apply and further develop this methodology to adapt it to wastewater treatment process monitoring. The paper is organised as follows. First a short introduction to multivariate monitoring and PCA is given, followed by a discussion on how to extend basic PCA to adaptive PCA. MRA is introduced and an integration of MRA and PCA is presented. A number of examples that serve as illustrations of the methods presented and that highlight the differences between the various approaches are discussed. The examples are based on real WWT process data.
PCA MONITORING
PCA can be described as a method to project a high-dimensional measurement space onto a space with significantly fewer dimensions. Often, for industrial process data, many variables are highly correlated, since they reflect relatively few underlying mechanisms that drive the process. In PCA, this correlation between the variables is used to find principal components (PCs) that represent the underlying mechanisms. The original variables are projected onto the PCs as scores. To compare variables with different amplitudes and variability, data are normally mean centred and scaled prior to analysis.
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For a more comprehensive description of PCA see e.g. Jackson (1980) or Wold et al. (1987). On-line monitoring using PCA Monitoring of the scores can be carried out using either conventional univariate statistical process control (SPC) techniques (see e.g. Bissel, 1994) or scatter plots (MacGregor and Kourti, 1995). In addition to monitoring the scores, the statistical fit of the model can be monitored. Two commonly used measures of fit are sum of squared prediction error (SPE) and Hotelling’s T 2 . SPE is a measure of the distance from the model plane to an observation, i.e. the model error. T 2 is a measure of the distance within the model plane from an observation to the origin. SPE and T 2 are convenient as they summarise the multivariate process information of the variables by single values and are normally monitored using conventional SPC charts with in-control limits defining the normal or desired operational region (Jackson and Mudholkar, 1979; Kresta et al., 1991). Robust limits can be determined empirically using, for instance, percentiles (Rosen, 1998). There is more information to be gained from a deeper investigation of the PCA model. When a disturbance is detected in score plots or in the SPE and T 2 plots, a physical interpretation can be found by transforming the model output back into the original process space. So-called contribution plots can be used to determine the variables that caused the deviation (MacGregor et al., 1994). Adaptive PCA PCA monitoring, as described above, assumes that the process conditions do not change significantly. This is rarely the case for WWT processes. Diurnal and seasonal changes affect the mean, variability and correlation of the variables. Small but important events tend to be obscured within the residuals related to the normal variation. A method to reduce the problem of changing conditions is to make the monitoring model adaptive, i.e. update the model on an adequate time-scale. If the co-variance structure of the monitored variables is believed to be static, i.e. the relations between the variables stay the same, regardless of their mean values and variability, the adaptation can be limited to data pre-processing. This may involve highpass filtering or updating of the scaling parameters. A simple way of updating the scaling parameters is to calculate them from a moving time window of appropriate length. A more sophisticated method is to update the mean and variance for each variable recursively. If the co-variance structure is believed to change over time, the model itself must be updated. In analogy with the updating of the scaling parameters, the model can be updated using a moving window. However, recursive updating can be applied as
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(Dayal and MacGregor, 1997): XT XðkÞ ¼ aXT Xðk 1Þ þ xT ðkÞxðkÞ where XT XðkÞ is the co-variance structure at time k, xðkÞ is the last sample row and a is a forgetting factor. The value of a may vary significantly depending on the aim of the monitoring. Adaptation of the co-variance matrix introduces some difficulties. Firstly, if no precautions are taken, the model will adapt to disturbances and failures atypical of the normal process behaviour. Secondly, if there is not sufficient excitation in the data, process information will be lost as old data are discounted. It may therefore be wise to test the information content of the data before it is used to update the model, especially if the forgetting factor is small, i.e. the effective memory is short (Wold, 1994; Dayal and MacGregor, 1997). A third problem caused by an adapting co-variance matrix is that the co-ordinate system defined by the PCs will rotate. This makes it hard to use scatter plots for interpretation of the process performance, as the co-ordinate system differs for each sample.
MULTISCALE MONITORING
For the purpose of monitoring and detection, wavelets can be used to decompose a signal into different time-scales with decreasing level of detail or resolution, i.e. MRA. A signal is filtered with a highpass filter and a lowpass filter, respectively. The result is one set of coefficients that describe the details of the signal and another set describing the approximation. The decomposition can be carried out to a desired number of scales by recursively applying the highpass and lowpass filters to the approximation coefficients of the previous level. When a filter has been applied, the result is downsampled. This means that the total number of coefficients in the wavelet domain is the same as the number of samples in the original signal. The coefficients still contain all the information carried by the original signal. For both the highpass and the lowpass filters, there are corresponding reconstruction filters. Using these filters for the reconstruction, together with upsampling, will result in a perfect reconstruction of the original signal in one scale. Alternatively, it is possible to separately reconstruct the components corresponding to each scale. The number of coefficients in the time domain will then be the original number of samples multiplied by the number of scales. MRA monitoring For monitoring purposes MRA is applied on-line to a moving time window. Univariate SPC techniques can be utilised to localise the scale in which a
disturbance is first detected. Fast disturbances are detected in the higher scales whilst slow disturbances are detected in the lower scales. The way a disturbance appears across scales can reveal information on the disturbance characteristics. For instance, a spike will be strong in the higher scales but barely visible in the lower scales. A step will appear clearly in all scales, whereas a ramp will appear most strongly in the lower scales. Monitoring of multivariate data is somewhat more problematic since increased sensitivity leads to more detections which have to be evaluated. Moreover, events that occur in several scales will trigger detections in more than one scale, even though they originate from the same event. This, together with the fact that all variables need to be monitored on several scales, call for multivariate methods that are capable of both reducing the dimension of the problem and handling collinear variables. A natural choice is PCA. Methodologies combining MRA with PCA have been proposed by Kosanovich and Piovoso (1997) and Bakshi (1998). PCA models are identified for each scale from data representing normal or desired behaviour. This is done by calculating the wavelet coefficients on each scale by applying a moving window the length of the longest filter to the training data. Monitoring new data involves decomposition and projection of data onto the PCA models on each scale. A window is used in the same manner as when the models are identified. The transformed data from the window on a certain scale are calculated and projected onto the PCA model at that scale. The scores are monitored using SPC techniques or scatter plots. The SPE and T 2 can be calculated and compared to the predefined confidence limits for that particular scale model. Time-scale decomposition increases the number of data points used for detection. This increased number is, however, based on only the number of original measurement points. If a certain overall confidence limit for all the scales together is to be obtained, the confidence limits on each scale must be adjusted (Bakshi, 1998). Recombination of scales A difficulty with the multiscale method described above is that when the number of scales increases, the number of entities to monitor increases as well. PCA will reduce the number, but with a high number of scales we might end up with more variables than in the original data set. A possible solution is to recombine some of the scales so that the effective number of scales to monitor decreases (Fig. 1). This enables us to both reduce the number of entities to monitor and to give physical meaning to the recombined scales by choosing them to correspond approximately to meaningful dynamic time-scales in the process. For instance, the first scales can be recombined to a ‘fast’ scale (e.g. hydraulic dynamics), the middle scales can be combined to constitute
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Fig. 1. Recombination of scales into fewer and more physically interpretable scales.
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Fig. 2. The MSPCA procedure.
a ‘medium fast’ scale (e.g. concentration dynamics) and the lower and/or lowest scale (e.g. population dynamics) can represent the ‘slow’ changes. Multiscale PCA Multiscale PCA (MSPCA) was originally proposed by Bakshi (1998). The idea is that the PCA models on each scale are used to determine whether a scale contains significant information or not at a certain point in time. If a scale is significant, the data on that scale are used in the reconstruction of a uniscale estimate of the original data. The reconstructed data are then monitored using a unifying uniscale PCA, which is based on the scales included in the reconstruction. The recombination of scales reduces the number of entities to monitor to the level of normal or adaptive PCA, while the prior scale PCA detection acts as a multiscale and multivariate feature extractor. Data from a period of normal or desired process behaviour are decomposed into scales. A PCA model is identified on each scale. New data are decomposed and projected onto the scale models. If a model residual on any scale exceeds its confidence limit, the scale is said to be significant, and is used to reconstruct the scale data into uniscale set of data. The reconstruction is monitored using a uniscale PCA, which is based on training data of significant scales (Fig. 2). RESULTS AND DISCUSSION
In this section we demonstrate and discuss the methods presented earlier. First an example is given where basic PCA monitoring is carried out. This example serves as a comparison to the examples when various extensions of the PCA have been used. The data used are actual WWT process data from the Ronneby plant in Sweden. The plant is operated as a biological nutrient removal plant with additional chemical treatment. The data-sampling period is 5 min. There are 14 available variables from the online measurement system, including flow rate, pH, ammonia, temperature and air valve positions at several points in the plant. The data span several months, from summer to late autumn implying
Fig. 3. No adaptation. The relative SPE and T 2 , i.e. the ratios between the residuals and their limits.
significant changes in operating conditions over the period. In the examples using time-scale decomposed data, data are decomposed into six different scales using the Haar wavelet (Strang and Nguyen, 1996). Note that isolation of the variables is not discussed here. However, in all examples, contribution plots and other techniques can be utilised to determine which variables cause the deviations. Static PCA A PCA model is identified from a 20-day period when the operation is considered to be normal except for a few brief upsets. New data are projected onto the model and the result can be seen in Fig. 3. It is obvious that the changing conditions cannot be covered by the static model. Almost from the beginning of the new period, both SPE and T 2 violate their limits and from day 40 the performance of the model deteriorates significantly. Adaptive scaling parameters A monitoring model with adaptive scaling parameters is identified. Six PCs are retained. The forgetting factor, a, is set to 0.9995, which corresponds to 2000 samples ( 21 days). This is a reasonable choice, allowing detection of slower
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changes as well as fast ones. In Fig. 4 the SPE and T 2 are shown. It can be said that the model is generally valid (i.e. the SPE is below its limit) during the whole period, indicating a relatively constant co-variance structure. However, there are some periods with high SPE, for instance, at days 20–21, 24–25, 42, 60–61 and 70. The T 2 measure hardly exceeds its limit, not even when the SPE assumes extreme values. The reason is possibly that the deviations are mainly caused by changes in the relations between the variables. Thus, the model does not capture the variation in a correct way, and most of the variation is outside the model plane. This indicates that the co-variance structure of the data is changing. For this reason, the information obtained from the scores must be used with caution. Adaptive PCA A model with adaptive scaling parameters as well as adaptive co-variance structure is used to monitor the same variables as in the example above. The result is shown in Fig. 5. The SPE is here mostly well
inside its confidence region, while T 2 violates its limit more markedly (compare with Fig. 4). This implies that the model captures the variation and that most of the variation will be manifested in the T 2 . The reason is that when the co-variance structure adapts to new conditions, most of the variation will be in the model plane (T 2 ) and not orthogonal to the model plane (SPE). Put simply, some of the variation from the SPE chart in Fig. 4 has now been transferred to the T 2 chart in Fig. 5. This highlights the importance of monitoring both SPE and T 2 as they provide complementary information. An example of the complementary relation between SPE and T 2 can be seen if we take a closer look at the event at days 41–42 (Fig. 6). Initially, there is a change in the relation between the variables (i.e. detection by SPE at day 41.4). As the variable relations change and/or the model adapts, the SPE decreases again at day 41.6. However, the T 2 is now large, indicating high variable magnitudes. It stays high until day 42.2, when SPE again becomes high, now indicating a severe disturbance at day 42.4. PCA monitoring of physically interpretable scales
Fig. 4. Adaptive scaling parameters. The relative SPE and T 2 , i.e. the ratios between the residuals and their limits.
Recombination of scales is done to simplify interpretation. Ideally, this is done in such a way that the resulting scales have a physical interpretation. Here, we will not focus on how to obtain the different time-scales present in a system. We define fast dynamic as the first three scales. This approximately corresponds to variations from 0 to 4 h. The medium time-scale consists of scales 4 to 6, which corresponds to 4 to 32 h. The slow dynamic is represented by the approximation scale, i.e. approximately everything slower than the diurnal variation. The SPE and T 2 charts show that the detail scale models (fast and medium) are valid, whereas the approximation scale (slow) model is not (Figs 7 and 8). This implies that the slow scale can only be used as an indication of the absolute distance from
Fig. 5. Adaptive PCA. The relative SPE and T 2 , i.e. the ratios between the residuals and their limits.
Fig. 6. Adaptive PCA. A closer investigation of SPE and T 2 at days 41–42.
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Fig. 7. The SPE residuals for the recombined scales defined as fast, medium and slow. The slow scale should only be used as an indication of the absolute distance from operation defined by training data.
Fig. 8. The T 2 residuals for the recombined scales defined as fast, medium and slow. The slow scale should only be used as an indication of the absolute distance from operation defined by training data.
normal operational conditions. Note that some of the disturbances occur both in the fast and the medium scale, which suggests that these disturbances have a prolonged duration. This information may be useful for interpretation purposes. However, it is interesting to note that at, for instance, day 118 the disturbance is detected in the medium scale first, i.e. the disturbance is slow. This can be seen in the original data as a relatively slow increase in flow rate and all air valve positions. The T 2 charts display similar behaviour. Most disturbances are fast and, thus, commenced by a rapid change in one or several variables. The disturbance at day 118 is most obvious
in the medium scale, which is consistent with the SPE chart. MSPCA monitoring The MSPCA method is applied to the data used in the previous examples. The number of scales is seven, i.e. six detail scales and one approximation scale. Both SPE and T 2 are used on each scale to determine whether a scale is significant. The limits for the model residuals are corrected to obtain an overall confidence of 99%. The resulting uniscale SPE and T 2 can be seen in Fig. 9. The confidence limits on the
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Fig. 9. SPE (top) and T 2 (bottom) from the MSPCA during a time period of 100 days. The confidence limits for both residuals are 99%.
uniscale residuals are also 99%. The SPE chart indicates that the model is valid during the whole period. Consequently, the co-variance structure of the reconstructed data does not change significantly, mostly due to the fact that the approximation scale has been omitted from the analysis. There are some upsets in SPE, but the major deviations can be seen in the T 2 . There is an interesting and important feature of MSPCA that distinguishes it from the other two multiscale methods. Consider a step change in one or several variables at a certain point in time: all the methods will detect this step at the same time. However, if the variable or variables return to normal, both the method involving PCA on each scale and the method involving prior recombination of scales, will continue to detect a deviation due to the delay of the filters. This is generally not the case with MSPCA. When the variables return to normal, the number of significant scales will increase causing the confidence level to increase and generally compensate for the delay. An example of this can be seen after the disturbance at day 42, where the SPE quickly returns to values below its limits. Comparative remarks It is clear that basic PCA cannot be used with confidence when the process conditions change. Extending the basic PCA with adaptive scaling parameters improves the results significantly without losing the ability to use score plots. This is an advantage of a fixed co-variance structure since score plots are quite intuitive in the interpretation process. Unfortunately, the model with adaptive scaling parameters is not fully capable of representing the process. The model based on adaptive scaling
parameters and co-variance structure handles the changing conditions well. Qualitatively, the two multiscale methods presented here have similar performance and they basically detect the same events equally well. However, there are some differences. Recombination of scales implies more entities to monitor but if it is possible to discern some dominant time-scales of the studied process, the recombination of scales to physically interpretable scales ought to simplify the interpretation considerably. The capability of MSPCA to unify all the scales into one scale is desirable and provides a compact way of monitoring. Moreover, MSPCA has an important advantage in that it returns to normal as soon as a disturbance has ended. The examples show that both the adaptive and multiscale PCA are possible ways to overcome the difficulties associated with changing process conditions. Adaptive and multiscale PCA are similar in performance, but both methodologies have their specific advantages. Adaptive PCA involves no (or little) pre-processing and is relatively simple in its structure. The co-variance structure is updated, which means that new or changed relations between variables can be represented. However, the monitoring becomes relative in the sense that information about the distance from the operational point, on which the starting model is built, is lost. In multiscale PCA, this information is not lost, but represented by the lowest scale. Moreover, the sensitivity to small changes is higher. However, as opposed to adaptive PCA, the co-variance structure is static, which may introduce errors. A solution to this drawback could be to apply adaptive PCA on each scale.
Wastewater treatment process monitoring
It is the authors opinion that the simplest effective model should be used for monitoring. However, it is shown here that some extensions to basic PCA monitoring must be included. The multiscale approach includes far more degrees of freedom than the adaptive approach, but it provides a way to discern small changes in data with, for instance, large diurnal variations. Moreover, the multiscale approach provides information on the scale at which a disturbance occurs, which can be used for diagnostic purposes to find the physical cause. However, the advantages come at a price of higher complexity of the monitoring model. The choice of approach used depends on the complexity of the process. A simple process, or perhaps a section of a process, where most events occur in one scale, can be adequately monitored at one time-scale. In a more complex process, with different subprocesses and different time constants, a multiscale method may be more suitable. Further extensions A few important objections to PCA as a monitoring technique for industrial processes can be found in the literature. In addition to the ones raised on PCA and changing process conditions, it is often pointed out that PCA is a static technique, not suitable for dynamic processes. This is partly addressed by multiscale monitoring, since each scale can be seen as a ‘‘snapshot’’ of the process in a limited frequency range. However, other techniques are available. By including a time lag or history in the analysis, i.e. the old measurements are included in the X matrix, dynamic relationships between variables can be represented (e.g. Ku et al., 1995; Lou et al., 1999). This can further improve the monitoring model, but at a price of higher model complexity and a large number of variables in the X matrix. An interesting topic not discussed in this paper is related to integration of monitoring and control. Information on the operational state is important in control strategy design and implementation. In PCA monitoring the operational state can be determined from the process location in the PC space and loading plots can be used to determine suitable control actions (Rosen and Yuan, 2000). In multiscale monitoring one additional piece of information can be obtained: time-scale information. So, in addition to the information on the present operational state, information on how fast the state has changed can be used to determine the strength of corrective measures to drive the process back to its desired operation. Integration of time information in the control strategy is an interesting topic for further studies.
CONCLUSIONS
Adaptive and multiscale extensions of basic PCA monitoring can be used to monitor WWT process
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data during changing operational conditions. Two different approaches are presented. Adaptive PCA, in terms of updating mean, variance and co-variance structure, overcomes the problem of non-stationary process data. The monitoring model is recursively updated using an exponential memory function. When the relations between the variables are known to be constant, only scaling parameters need to be updated. The methodology is relatively simple and can easily be included in standard PCA-monitoring. However, the monitoring is relative and adaptive PCA need to be complemented by an absolute measure of the process state. This is not a difficulty with the second approach: multiscale monitoring. Here, the process data are split into several timescales. The higher scales represents faster changes and the lower scales represent slower ones. All but the lowest scale will have approximately zero mean, which means that the non-stationary problem is partly solved. The information about on which scale a disturbance occurs can be used to interpret the disturbance. When events in different scales are separated, the sensitivity of the monitoring is increased. The choice between the two approaches is a trade-off between complexity and information.
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