Enhanced Fault Detection of an Air Quality Monitoring Network Majdi Mansouri and Hazem Nounou
Mohamed Faouzi Harkat and Mohamed Nounou
Electrical and Computer Engineering Program, Texas A&M University at Qatar, Doha, QATAR Email: {majdi.mansouri, hazem.nounou}@qatar.tamu.edu
Chemical Engineering Program, Texas A&M University at Qatar, Doha, QATAR Email:
[email protected]
Abstract—Environmental pollution has adverse consequences on human health and the ecosystem. Among the most dangerous types of pollution is air pollution in urban areas, which has been shown to be strongly linked to higher morbidity and mortality rates. Air pollution can be due to several factors, such as human activities (that produce pollutants such as nitrogen oxides, carbon oxides, and volatile organic compounds), photochemical reactions in the lower atmosphere (that produce ozone), or meteorological conditions that affect the concentrations of dust and particulate matter. The contamination levels of these pollutants need to be maintained below acceptable limits set by the world health organization (WHO) or air quality associations in various areas of the world in order to minimize the impact of these pollutants on humans and the environment. A detection of anomalies in measured air quality data is a crucial step towards improving the monitoring of air quality networks. Therefore, an enhanced fault detection technique of an air quality monitoring network using multiscale principal component analysis (MSPCA)-based on moving window generalized likelihood ratio test (MW-GLRT) is proposed. The presence of measurement noise in the data and model uncertainties degrade the quality of fault detection techniques by increasing the rate of false alarms. Thus, the objective of this paper is to enhance the fault detection of an air quality monitoring network by using wavelet-based multiscale representation of data, which is a powerful feature extraction tool to remove the noises from the data. Multiscale data representation has been used to enhance the fault detection abilities of principal component analysis. The results demonstrate the effectiveness of the MSPCA -based MW-GLRT method over the conventional MSPCA-based GLRT method and both of them provide a good performance compared with the conventional PCA and MSPCA methods.
I. I NTRODUCTION Maintaining high air quality is a major environmental concern that has a profound impact on human health and the ecosystem. Various industrial effluents, human activities, and meteorological factors contribute to the pollution of air by pollutants, such as carbon oxides, nitrogen oxides, ozone, and particulate matter. Air quality monitoring networks are usually used to monitor the quality of air, not only to make sure that air quality standards are maintained, but also to allow taking any necessary preventive or corrective measures to minimize the effect of possible undesirable changes in some of these pollutants. Proper operation of air quality networks is crucial to achieve their intended purpose. Therefore, the objective of this paper is to develop an enhanced fault detection technique that aim at enhancing the monitoring of air quality networks
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by developing a fault detection that can quickly detect sensor faults or serious anomalies in air quality data. Monitoring of air quality networks are crucial to ensure safety and protection of humans and the environment. In general, monitoring approaches [1] can be classified as: modelbased or data-driven approaches. Model-based monitoring approaches utilize predictions of process models to make decisions regarding the existence or absence of faults [2]. Hence, the effectiveness of such approaches is greatly influenced by the quality of the process models. In the case where the difference between the model prediction and process measurement is relatively small, this indicates that the process is operating normally and no fault exists. However, when such a difference is relatively large, this is an indication that a fault has occurred [2]. Data-based approached, on the other hand, assume that process measurements, when the process operates in normal (fault-free) conditions, are available [3]. Such fault-free process measurements are often used to construct empirical models that can be used for process monitoring. Several data-based monitoring approaches are available in the literature, such as latent variable regression (LVR) approaches, support vector machine (SVM) approaches [4], canonical variate analysis (CVA) approaches [5], pattern recognition approaches [6], as well as approaches that are based on fuzzy systems [7] and neural networks [8]. Monitoring approaches that are based on LVR models have been extensively used in practice to monitor various applications, such as air quality monitoring [9], water treatment [10], pharmacology [11], and health [12]. However, process data collected from most environmental and refinery models may contain high levels of noise, autocorrelation, and might deviate from normality, and this might adversely affect the performance of the conventional PCA monitoring chart in terms of modeling and fault detection. To address these issues, Bakshi [13] has developed a multiscale wavelet-based multivariate process monitoring technique through Multiscale Principal Component Analysis (MSPCA), in which, the MSPCA computes the PCA of the wavelet coefficients at each scale. Due to its multiscale nature, MSPCA has been shown to be more appropriate for modeling of data containing contributions from events whose behavior changes over time and frequency. The Generalized Likelihood Ratio Test (GLRT) chart has recently been incorporated with multivariate monitoring charts in order to improve the fault
detection performance of many multivariate process monitoring charts [14], [15], [16], [17]. In our previous studies, it is shown that the performance of the GLRT chart can be further improved through its implementation in a moving window of lagged residuals. The authors have successfully applied generalized likelihood ratio test (GLRT) for model-based fault detection. GLRT is composite hypothesis testing methods and is known to have better fault detection performance compared to conventional T 2 and Q statistics. In the moving window GLRT (MW-GLRT) method, the detection statistic equals the norm of the residuals in that window, which is equivalent to applying a mean filter on the squares of the residuals [18]. Therefore, this paper aims to extend the MSPCA technique through the utilization of the MW-GLRT chart in order to improve fault detection performance. The fault detection problem was addressed so that the data are first modeled using the MSPCA method and then the faults are detected using the MW-GLRT chart. The fault detection performances of the MSPCA-based MW-GLRT method are assessed and compared to existing techniques using air quality monitoring network data. The rest of the paper is organized as follows. Section II presents an introduction to multiscale wavelet-based data representation and a description of the Multiscale PCA algorithm. Section III introduces statistical hypothesis testing and generalized likelihood ratio charts. Then, Section IV aims to assess the performance of the developed MSPCA-based MWGLRT control chart using an air quality monitoring network data. Conclusions are then presented in section V. II. M ULTISCALE P RINCIPAL C OMPONENT A NALYSIS (MSPCA) DESCRIPTION The classical PCA is presented in details in [14], [16]. Next, the multiscale extension of PCA is presented. 1) Multiscale Wavelet-based Data Representation: Wavelet-based multiscale representation is a powerful dataanalysis tool that provides efficient separation of deterministic and stochastic features [19]. Given a time domain data set (signal), a coarser approximation of the signal (called a scaled signal) can be obtained by convoluting the original signal with a low pass filter (h), which is derived from a scaling basis function of the following form: [20]: Φij (t) =
√ 2−j Φ(2−j t − k),
(1)
where, k and j are discretized translation and dilation parameters, respectively. The detail signal, which is the difference between the original and the approximated signals, can be obtained by convoluting the original signal with a high pass filter (g), which is derived from a wavelet basis function of the following form [20]: Ψij (t) =
√ 2−j Ψ(2−j t − k).
(2)
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Fig. 1. Schematic illustration of MSPCA model.
After repeating these approximations, the original signal can be represented as the sum of the last scaled signal and all detail signals, i.e. [20]: x(t) =
−J n2
k=1
−J
aJk ΦJk +
J n2
ajk Ψjk (t),
(3)
j=1 k=1
where, n and J represent the length of the signal and the maximum possible decomposition depth, respectively. 2) MSPCA method description: The MSPCA model was developed by Bakshi [13] with the aim of combining the ability of PCA to extract cross-correlation between variables with the ability of orthonormal wavelets to separate feature from noise and approximately decorrelate autocorrelation between available measurements [13]. Figure 3 illustrates the MSPCA algorithm that was developed by Bakshi [13], and its key steps are highlighted in Algorithm 1. Algorithm 1: MSPCA algorithm Input: n × m data matrix X, confidence interval α. • For each column (i.e. process variable) in the data matrix compute the wavelet decomposition; • For each block (matrix) of scaled and detail coefficients at each scale, the covariance matrix is computed along with the number of principal components, as well as PCA loadings and scores of those wavelet coefficients; • Once the appropriate number of loadings is selected, wavelet coefficients larger than a certain threshold are selected; • For all scales together, PCA is carried out by after including only scales with significant events during reconstruction.
III. M OVING W INDOW GLRT (MW-GLRT) The GLRT is a hypothesis testing technique that has been implemented and utilized for model-based fault detection
purposes, and seeks to maximize the detection probability for a given false alarm rate [21]. Let y ∈ Rn is an observation vector formed by one of the two Gaussian distributions: N (0, σ 2 IN ) or N (θ = 0, σ 2 IN ), where θ is the mean vector (which is the value of the fault) and σ 2 > 0 is the variance that is assumed to be known in this problem. The hypothesis test can be formulated as follows [14], H0 = {y ∼ N (0, σ 2 IN )}, (null hypothesis), (4) H1 = {y ∼ N (θ, σ 2 IN )}, (alternative hypothesis). With the GLRT method, the unknown parameter, θ, is replaced by its maximum likelihood estimate. The GLRT decision function T(y) is expressed as follows: sup fθ (y)
T(y) = = = =
2 log
θ∈Rn
(5) f (y) θ=0 y22 y − θ22 2 log sup exp − / exp − 2σ 2 2σ 2 θ 1 min y − θ22 + y22 θ σ2 1 2 + y2 = 1 y2 , y − θ 2 2 2 2 2 σ σ
Therefore, in this work, we propose to combine the advantages of MSPCA model to MW-GLRT detection chart, and developing a MSPCA-based MW-GLRT technique in order to enhance the detection performance. Next, in Section IV, the performance of the developed MSPCA-based MW-GLRT technique is assessed and compared to the conventional PCA and MSPCA methods using an air quality monitoring network data. IV. MSPCA- BASED MW-GLRT
FAULT DETECTION AND
APPLICATION TO AN AIR QUALITY NETWORK
The idea behind a MSPCA-based MW-GLRT fault detection algorithm is to incorporate the advantages brought forward by MSPCA with the MW-GLRT fault detection chart. This can be accomplished through the fault detection algorithm illustrated in Figure 2.
; 063&$
where θ = arg miny − θ22 = y is the maximum likelihood θ
estimate of θ,the probability density function of Y is fθ (y) = 1 1 2 , and . 2 represents the exp − y − θ N 2 2σ2 n (2π)
2
5HVLGXDO
σ
Euclidean norm. To select the threshold for the GLRT statistic its distribution need to be determined. Since the noise is assumed to follow a Gaussian distribution, the test statistic leads to a chi-square ¯ is distributed as: distribution. The normalized residual E ¯ ∼ N (θ, σ 2 In ), E
χ2w .
)DXOW'HFODUHG EŽ
(6)
where θ = 0 under the null hypothesis (4). Then, the scaled test statistic is distributed as the non-central chi-square distribution as follows, 1 G = 2 y22 ∼ χ2n , (7) σ with n degrees of freedom. Although the GLRT chart does show improved performance it is important to note that the statistic is computed using only the current observation. Literature has shown [18] through many charts that the utilization of a technique with increased process many does further improve the fault detection performance, thus motivating the extension of the GLRT chart to one that incorporates a moving window, and the formulation of this statistic is described next. To define the threshold (M W Gα ), it is necessary to compute the distribution of the MW-GLRT statistic (M W G). Since the G(R) statistics are Chi-square distributed (see Equation (7)), thus, MW-GLRT statistic (M W G) will follow a Chi-square distribution with degree of freedom equal to the window length (WL), w [18], [17]., i.e., M W G(E) ∼
0:*/57
(8)
0:* 0:*Į
zĞƐ
1R)DXOWLQ3URFHVV
Fig. 2. Schematic illustration of proposed MSPCA-based MW-GLRT algorithm.
Next, the fault detection performance of the developed MSPCA-based MW-GLRT algorithm will be evaluated using three fault detection criteria: the missed detection (MD) rate, the false alarm (FA) rate, and the out-of-control average run length (ARL1 ) [22]. The missed detection rate is computed by calculating the percentage of observations that go undetected in the faulty region, while the false alarm rate is the computed by calculating the percentage of incorrect faulty declarations in the non-faulty region. ARL1 is the number of observations it takes for a particular technique to detect a fault in the faulty region after it has been introduced, i.e., speed of detection.
90
Missed Detection Rate (MDR) False Alarm Rate (FAR) Average Run Length (ARL )
Fault Detection (FD) Metric
80
1
70 60 50
Selected Solution : WL=4
40 30 20 10 0 0
5
10
15
20 25 30 Window Length (WL)
35
40
45
50
Fig. 3. The effect of the window length on the performance of the MW-GLRT in terms of FAR, MDR and ARL1 .
A. Improved performance using a moving window GLRT The GLRT fault detection method has been shown to provide improved fault detection abilities over the conventional T 2 and Q statistics. Computing the statistic used in the GLRT, however, requires computing the norm of the evaluated residuals at every time instant. The authors in [18] have indicated that further improvement in the performance of the GLR test can be achieved by applying the GLRT statistic in a moving window, which improves the average run length (ARL) over the conventional GLRT. To investigate the impact of the choice of a window length on the MW-GLRT algorithm performance, the authors run the MSPCA-based MW-GLRT algorithm for different window length values and compute the fault detection metrics (missed detection rate, false alarm rate and ARL1 value) over the window length. The results of this study are illustrated in Figure 3. The results show that at larger window lengths, the missed detection rate and ARL1 decrease (which means faster and more effective detection), but the false alarm rate increases. This means that there is a trade-off between better detection and false alarms, which means that the window length can not be increased indefinitely and an optimum window length should be used. As shown in Figure 3, that the selected window length that provides a sub-optimal solution is equal to 4. The advantage of using a moving window GLRT statistic over a conventional GLRT statistic can be explained as follows. Computing the norm used in the GLRT statistic using multiple samples (when the window length is larger than one) provides a filtered or a smoother estimate of the GLRT statistic than what is obtained using the conventional GLRT method. Next, the developed MSPCA-based MW-GLRT fault detection algorithm is illustrated through its application on an air quality monitoring network. B. Application to Air Quality Monitoring Network During the study, the AIRLOR monitoring network consists of twenty stations located in rural, peri-urban and urban sites. Each monitoring station consists of a set of sensors for measuring concentrations of pollutants: carbon monoxide CO,
oxides of nitrogen (N O and N O2 ) measured by the same analyzer, the dioxide Sulfur SO2 and ozone O3 . Some stations also measure certain meteorological parameters. Ozone is a secondary pollutant produced by complex photochemical reactions between primary pollutants (i.e., N O, N O2 and V OC) emitted into the atmosphere. The pollutants’ concentrations depend mainly on the vertical and horizontal of the movements’ atmosphere that are related to the meteorological conditions. In the current work, only 6 neighbour measurement stations is considered. The data matrix X contains 18 state variables, v1 to v18 , which correspond respectively to ozone O3 and nitrogen oxides (N O2 and N O) gathered on these 6 stations. A sensor faults whose magnitude is approximately 20% of measurement for ozone [23] is simulated. Moreover, 1080 samples are used in simulation: the first samples 1 − 800 are used to compute the MSPCA model and the last samples 801 − 1080 are used to test the developed fault detection technique. According to MSPCA method, the wavelet basis function and the number of decomposition depth need set before wavelet decomposition. ’Haar’ wavelet is the orthogonal wavelet function. ’Haar’ wavelet was used for DWT and chosen to be the wavelet basis function. Best decomposition length is generally taken approximately as half of maximum decomposition length available as it is dyadic data set. It depends upon type of training data. In the current work, the decomposition length is fixed to 3. Next, the performance of the developed fault detection methods is illustrated. The comparison is assessed through one case study, in which, the sensor measuring the ozone (O3 ) is assumed to be faulty with an additive fault. C. Fault in the ozone (O3 ) Figures 4 and 5 show the fault detection results of the PCAbased Q, MSPCA-based Q, MSPCA-based GLRT, PCA-based MW-GLRT and MSPCA-based MW-GLRT techniques. It is shown from Figures 4.(a) and 4.(b), that MSPCA-based Q provides better results compared to the classical PCA-based Q technique with some false alarm rates. The FD results show also that, the MSPCA and PCA-based MW-GLRT techniques show a good detection rate for single fault in ozone O3 compared to the MSPCA-based GLRT techniques (see Figures 5.(a) and 5.(c), and all of them provide a good FD performance compared to the conventional PCA-based Q and PCA-based MW-GLRT techniques (see Figures 4 and 5 and Table I). Figures 4 and 5 show the fault detection results of the PCAbased Q, MSPCA-based Q, MSPCA-based GLRT, PCA-based MW-GLRT and MSPCA-based MW-GLRT techniques. It is shown from Figures 4.(a) and 4.(b) and Table I, that MSPCAbased Q provides better results compared to the classical PCA-based Q technique with some false alarm rates. The FD results show also that, the MSPCA-based GLRT and MWGLRT techniques show a good detection rate for single fault in ozone O3 compared to the conventional PCA-based GLRT technique (see Figure 5 and Table I).
80
45 Fault region
Fault region Threhold
40 40
MSPCA−based GLRT Statistic
PCA−based Q Statistic
70 60 50 40 30 20 10 0 0
0 200
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(a) PCA based Q statistic in the presence of fault in O3 . 15
20
Fault region
Fault region MSPCA−based Q statistic Threshold
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400 500 600 700 Observation Number
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Fault region
GLRT statistic Threshold
18
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300
(a) MSPCA-based GLRT statistic in the presence of fault in O3 .
PCA−based MW−GLRT (WL = 4)
MSPCA−based Q Statistic
35 20 30
Fault region
GLRT statistic Threshold
16 10 14 12
0 0
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10 8 6 4 2
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900
0 0
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(b) MSPCA based Q statistic in the presence of fault in O3 . Fig. 4. The time evolution of the PCA and MSPCA-based Q statistics on a semi-logarithmic scale in the presence of fault in O3 . TABLE I S UMMARY OF M ISSED D ETECTION (%), FALSE A LARMS (%) AND ARL1 . Chart/Fault Detection Metric PCA-based Q MSPCA-based Q MSPCA-based GLRT PCA-based MW-GLRT(WL = 4) MSPCA-based MW-GLRT(WL = 4)
MDs (%) 81.15 3.0294 35.3993 74.13 0.7717
FAs (%) 68.26 0.8027 2.7765 0.519 0.7934
ARL1 4.5884 1.1560 1.5596 1 1
100
200
300
400 500 600 700 Observation Number
800
900
1000
(b) PCA-based MW-GLRT statistic (WL=4) in the presence of fault in O3 . 40 40
MSPCA−based MW−GLRT (WL = 4)
0 0
30 25
Fault region
GLRT statistic Threshold
35 20 0 0
500
1000
20 15 10 5 0 0
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400 500 600 700 Observation Number
800
900
1000
(c) MSPCA-based MW-GLRT statistic (WL=4) in the presence of fault in O3 .
V. C ONCLUSION In this paper, the problem of fault detection in air quality monitoring network using improved process monitoring technique was addressed. A multiscale principal component analysis (MSPCA)-based on moving window generalized likelihood ratio test (MW-GLRT) is proposed to enhance the fault detection of air quality monitoring network. The MSPCA model evaluates the PCA of the wavelet coefficients at each scale. Due to its multiscale nature, MSPCA is appropriate for modeling of data that contain contributions from events whose behavior changes over time and frequency. In the MWGLRT method, the detection statistic equals the norm of the residuals in that window, which is equivalent to applying a mean filter on the squares of the residuals. This means that a proper moving window length needs to be selected, which is similar to estimating the mean filter length in data filtering. The performance of the developed technique is assessed and compared to the conventional PCA-based MW-GLRT, PCA and MSPCA techniques in terms of false alarm rates, missed detection rates and ARL1 values.
Fig. 5. The time evolution of the MSPCA-based GLRT, PCA and MSPCAbased MW-GLRT statistics on a semi-logarithmic scale in the presence of fault in O3 .
ACKNOWLEDGMENT This work was supported by Qatar National Research Fund (a member of Qatar Foundation) under the NPRP grant NPRP8-836-2-353. R EFERENCES [1] V. Venkatasubramanian, R. Rengaswamy, and S. N. Kavuri, “A review of process fault detection and diagnosis: Part ii: Qualitative models and search strategies,” Computers & Chemical Engineering, vol. 27, no. 3, pp. 313–326, 2003. [2] M. Kinnaert, “Fault diagnosis based on analytical models for linear and nonlinear systems - a tutorial,” in Proceedings of the 15th International Workshop on Principles of Diagnosis, 2003, pp. 37–50. [3] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, “A review of process fault detection and diagnosis: Part iii: Process history based methods,” Computers & chemical engineering, vol. 27, no. 3, pp. 327–346, 2003.
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