Fault Detection and Identification Based on the Neighborhood

Article pubs.acs.org/IECR

Fault Detection and Identification Based on the Neighborhood Standardized Local Outlier Factor Method Hehe Ma, Yi Hu, and Hongbo Shi* Key Laboratory of Advanced Control and Optimization for Chemical Processes of the Ministry of Education, East China University of Science and Technology, Shanghai 200237, P.R. China S Supporting Information *

ABSTRACT: Complex chemical processes often have multiple operating modes to meet changes in production conditions. At the same time, the within-mode process data usually follow a complex combination of Gaussian and non-Gaussian distributions. The multimodality and the within-mode distribution uncertainty in multimode operating data make conventional multivariate statistical process monitoring (MSPM) methods unsuitable for practical complex processes. In this work, a novel method called neighborhood standardized local outlier factor (NSLOF) method is proposed. The local outlier factor of each sample, which means the degree of being an outlier, is used as a monitoring statistic. A new normalized Euclidean distance based on the local neighborhood standardization strategy is employed during the calculation of the monitoring index. Then, a contribution-based fault identification method is developed. Instead of building multiple monitoring models for complex chemical processes with different operating conditions, the proposed NSLOF method builds only one global model to monitor a multimode process without needing a priori process knowledge. Finally, the validity and effectiveness of the NSLOF approach are illustrated through a numerical example and the Tennessee Eastman process.

1. INTRODUCTION Modern chemical processes have become increasingly integrated and complex with the developments of automatic control techniques. Because of the interactions among different manufacturing plants, a small fault might cause a series of upsets and even an unwanted halt during operation. To ensure process safety and higher product quality, fault detection and identification of chemical processes have gained tremendous attention over the past few years. Because process historical data can be easily collected and stored, multivariate statistical process monitoring (MSPM) methods such as principal component analysis (PCA) and independent component analysis (ICA) have been widely applied and treated as one of the most essential types of tools for chemical process monitoring.1 The dramatic increases in plant scale and operation complexity pose many challenges for fault detection in chemical processes. To overcome specific monitoring challenges such as the dynamic and nonlinear characteristics in practical processes, various extensions of the MSPM methods or combined approaches have been proposed.2−8 However, most conventional MSPM methods often rely on some assumptions such as the assumption of a unimodal or Gaussian distribution.9,10 The monitoring performance of these methods might be unsatisfactory when the data distribution is complicated, which is a common situation in complex chemical processes. The main problems caused by the complicated data distribution can be summarized as follows: First, the multimodality of the data distribution will make traditional MSPM methods inappropriate for the multimode process monitoring. The operating conditions often shift because of changes in market demands, feedstocks, and/or manufacturing strategy.9,11 For most MSPM methods, the assumption that the process contains only one nominal operating region becomes invalid in multimode processes, which will lead to a high miss © 2013 American Chemical Society

rate, or missed detection rate. To address multimode process monitoring, the multiple-model strategy has been intensively researched recently. According to the multiple-model strategy, multiple local models need to be built and made to fit each mode individually.12 However, it is difficult to obtain a priori knowledge of how to separate the historical process data into multiple subsets corresponding to multiple modes. Usually, some automatic clustering techniques need to be employed in the offline modeling phase.13 At the same time, in the online monitoring phase, more efforts need to be made to identify the real state of the process from the monitoring results of multiple local models. If only one local model is selected for monitoring, specific pattern-matching rules should be determined to select the most suitable model for every new sample. For example, as extensions of conventional PCA/partial least-squares (PLS) methods for multimode process monitoring, the multiple-PCA-/ PLS-model methods proposed by Zhao et al.14,15 adopt the specific local model producing the minimum squared prediction error (SPE). Natarajan et al.16 chose the PCA model whose training data subset center was closest to the new sample as the best-fit local model. Alternatively, if all of the local monitoring models are used to monitor the new sample, a decision rule needs to be designed to integrate the detection results of the local models. Because Bayesian inference can easily transfer a traditional monitoring statistic into a fault probability, it has been widely used to combine the monitoring results of different local models.13,17 Received: Revised: Accepted: Published: 2389

July 31, 2012 December 2, 2012 January 11, 2013 January 12, 2013 dx.doi.org/10.1021/ie302042c | Ind. Eng. Chem. Res. 2013, 52, 2389−2402

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Different from the multiple-model strategy, Lane et al.18 developed an extension of PCA that can simultaneously monitor multiple product grades. However, it is based on the existence of a common eigenvector subspace for the sample variance/ covariance matrices of individual product grades. To take the transitions between different operating modes into account, Garcia-Alvarez et al.19 used a classical PCA model for each steady state and successfully employed a modification of the batch PCA approach for the transient states. In addition, the Gaussian mixture model (GMM) has been successfully applied in multimode processes.9,20−23 The GMM-based monitoring method can nicely handle multi-Gaussianity, and a probabilistic index can be developed for fault detection.21 Yu et al.9 proposed a novel multimode process monitoring approach based on the finite GMM and a Bayesian inference strategy. Its integrated global probabilistic index was demonstrated to be accurate and efficient for various types of faults in multimode processes. The hidden Markov model (HMM) was introduced to estimate data distributions from normal operations with multimodal characteristics.24,25 Several probabilistic fault detection and identification methods have also been proposed.10,26,27 However, it should be noted that most of these methods assume that the process data within each individual mode follow a multivariate Gaussian distribution. In fact, to obtain satisfactory monitoring performance, the effects of the non-Gaussian characteristics should also be taken into account.13,20 Second, the uncertainty of the data distribution within a single operating mode strongly influences the final monitoring performance. For example, PCA is a good dimension-reduction and information-extraction method that has no restrictions on data distribution.28 Nevertheless, because of the assumption of multivariate Gaussian distributions in its two monitoring statistics, the achievable performance of PCA is limited.20,28 In contrast to PCA, ICA is more suitable for non-Gaussian processes and involves higher-order statistics. However, even though ICA can extract more useful information from nonGaussian data than PCA, it does not outperform PCA if the process variables are normally distributed.6,8 In fact, the common situation in practical chemical processes is that the distribution of the operating data is uncertain. Some of the variables might be normally distributed, whereas others might be not. Most often, process data contain both Gaussian and non-Gaussian distributions at the same time. To extract the Gaussian and non-Gaussian information, several combinations of PCA and ICA have been proposed.6−8 Similarly, to obtain accurate monitoring results, the multimodality of a data distribution due to different operating conditions should be considered. To develop an efficient monitoring method for complex chemical processes with multimodality and within-mode uncertainty of data distribution, a novel neighborhood standardized local outlier factor (NSLOF) method is proposed in this work. First, the degree of isolation from normal data, called the local outlier factor (LOF), is calculated for each test sample and used as a monitoring statistic for fault detection. During the calculation of the LOF index, a new normalized Euclidean distance based on the local neighborhood standardization strategy is employed. Then, a contribution-based fault identification method in NSLOF is developed to determine the faulty variable. The focus of this work is fault detection and faulty-variable identification. Therefore, as a follow-up step of process monitoring, methods for classifying the type of current abnormal measurement values and diagnosing the fault type of the abnormal change are not included in this work. In contrast to

other traditional distance-based statistics, such as Hotelling’s T2 statistic and squared prediction error (SPE), the density-based LOF statistic is more accurate and robust, regardless of the data distribution. In NSLOF, the local structure of each variable is taken into account by employing the local normalized Euclidean distance to compute the value of the LOF, and the local density of the surrounding neighborhoods is also considered. In particular, in contrast to the multiple-model strategy, only a single model is built in the NSLOF method for multimode process monitoring, instead of a large number of local models. Furthermore, no a priori process knowledge is needed during the modeling and monitoring phases. The validity and effectiveness of the NSLOF approach are demonstrated through a numerical example and the Tennessee Eastman process. The remainder of this article is organized as follows: In section 2, the basic outline of traditional local outlier factor (LOF) method is briefly reviewed. The proposed NSLOF method is detailed in section 3.1, and the off-line and online monitoring procedures of the NSLOF approach are described in sections 3.2 and 3.3, respectively. The results and discussions of two illustrative examples are presented in section 4. Finally, some conclusions are summarized in section 5.

2. LOCAL OUTLIER FACTOR The local outlier factor (LOF) method was first proposed for finding outliers in multidimensional data sets.29 The value of the local outlier factor indicates the extent to which a sample is an outlier.29,30 Because the LOF has been reported in detail and applied in the area of data mining,29,31,32 only several critical ideas and a brief summary are presented here. We first define five important concepts: (1) the local neighborhood of sample xi;29 (2) the k distance of sample xi;29 (3) the reachability distance of xi to its neighbor xfi, f = 1, 2, ..., k;29 (4) the local reachability density of sample xi;29 and (5) the local outlier factor of sample xi.29 (1) For a sample xi from X, where X ∈ Rn×m is the training data set with n samples and m variables, its neighbors can be sequenced according to the Euclidean distances between xi and other samples in X. Then, k nearest neighbors of xi are selected to compose the local neighborhood of sample xi. The parameter k, which is a positive integer, represents the size of its local neighborhood, N(xi). xfi is used to represent the f th-nearest neighbor of xi. d(xi,xfi) is the Euclidean distance between xi and xfi. (2) The k distance of sample xi is the radius of its local neighborhood N(xi). It is denoted as k_distance(xi) and represents the Euclidean distance between xi and its kth-nearest neighbor xki . (3) The reachability distance of xi to its neighbor xfi, f = 1, 2, ..., k, is defined as29 reachd(xi , xi f ) = max{k _distance(xi f ), d(xi , xi f )}, f = 1, 2, ..., k

(1)

The reason for utilizing the reachability distance for the calculation of LOF is that the statistical fluctuations of d(xi,xfi) can be significantly reduced in this way. The more neighbors chosen for each sample, the more similar the reachability distances for samples within the same neighborhood.29 (4) The local reachability density (lrd) of sample xi29 is given by 2390

dx.doi.org/10.1021/ie302042c | Ind. Eng. Chem. Res. 2013, 52, 2389−2402

Industrial & Engineering Chemistry Research lrd(xi) =

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k k ∑ f = 1 reach_d(xi ,

method is developed for fault identification. The details of the NSLOF method are reported next. In the modeling phase, the first step of the NSLOF method is to select the neighbors for each normal training sample according to the Euclidean distances. After the neighborhood of every sample is determined, a new squared Euclidean distance normalized using the local neighborhood standardization strategy is developed as follows

f

xi )

(2)

From eq 2, it can be intuitively found that the local reachability density of sample xi is the inverse of the average reachability distance between xi and its neighbor xfi. (5) The local outlier factor (LOF) of sample xi29 is given by LOF(xi) =

1 k

k

∑ f =1

lrd(xi f ) lrd(xi)

ds(xi , xi f ) =

{xi − me[N (xi f )]}{xi − me[N (xi f )]}T std2[N (xi f )]

(3)

ds(xi,xfi)

(4)

distance, me[N(xfi)] is and std[N(xfi)] is the

where is the squared Euclidean the mean vector of the data in N(xfi), standard deviation vector of N(xfi).12 After the additional step of using the neighborhood standardization strategy, the neighbors in N(xi) are reordered according to the normalized Euclidean distances. Then, the k distance of sample xi with the consideration of the neighborhood structures of its neighbors can be determined as

From the definitions of the reachability distance and local reachability density, it can easily be seen that the degree to which sample xi is an outlier is captured by its local outlier factor with respect to its surrounding neighborhood. Outliers can have large LOF values. For normal samples in the training data set, the LOF values are approximately 1, so these samples will not be labeled as outliers.29

Ns(xi) = {xi1 , xi2 , ..., xi f , ..., xik},

3. NEIGHBORHOOD STANDARDIZED LOCAL OUTLIER FACTOR METHOD FOR FAULT DETECTION AND IDENTIFICATION From the calculation of the local outlier factor, it can be seen that the LOF method is a density-based outlier-detection method. If sample xi is an outlier to data set X, there is a great possibility that xi is located far from the neighborhoods of its most neighbors. This means that, in most cases, the reachability distance of xi to its neighbor xfi equals the actual distance between xi and xfi. This actual distance is larger than the radius of the neighborhood of xfi. Therefore, sample xi will have a relatively small local reachability density. The LOF values of outliers will consequently be larger than those of normal samples. Because fault samples can be treated as outliers compared to the normal data, the LOF method offers a probable robust solution for fault detection. However, the traditional LOF method employs only Euclidean distances to determine the reachability distances. The common situation in practical complex chemical processes is that the scales of variables are significantly different. As analyzed in our previous study, a traditional data preprocessing method such as zscoring is not sufficient for data normalization when the process data contain multimode characteristics.12 To achieve satisfactory monitoring performance, the variance structures of the data, which also contain a great deal of important information, should also be considered. Therefore, it will be inadequate if the LOF approach is directly applied in fault detection of real processes. To obtain effective monitoring for complex chemical processes, a novel neighborhood standardized local outlier factor (NSLOF) method is proposed in this section. Then, the NSLOF procedure is reported. The details of the off-line modeling phase and the online monitoring phase are presented in sections 3.2 and 3.3, respectively. 3.1. Neighborhood Standardized Local Outlier Factor Method. Because the NSLOF approach is proposed for fault detection, the value of the local outlier factor is treated as a monitoring statistic. Compared to the conventional LOF approach, a local neighborhood standardization strategy is employed in the NSLOF method to extract the local information and consider the scales of variables. The normalized Euclidean distances are used to describe the relationships between the sample and its neighbors. To help the operators diagnose the possible root causes of the current fault, a contribution-based

f = 1, 2, ..., k

(5)

ds(xi , xi1) ≤ ds(xi , xi2) ≤ ··· ≤ ds(xi , xi f ) ≤ ··· ≤ ds(xi , xik),

f = 1, 2, ..., k

k _distances(xi) = ds(xi , xik)

(6) (7)

Then, the LOF value for each sample in the training data set can be calculated using eqs 1−3, with the normalized Euclidean distance calculated according to eq 4. After the local outlier factor of each normal training sample is calculated, the control limit of the monitoring statistic can be determined using kernel density estimation (KDE). In statistics, KDE is a nonparametric method for estimating the probability density function of a random variable.33 In the process monitoring area, KDE has been widely used to determine the control limits of monitoring statistics such as I2 and Ie2.34−36 A univariate kernel estimator is defined by the equation fĥ (y) =

1 nh

n

⎛ y − yi ⎞ ⎟ h ⎠

∑ K ⎜⎝ i=1

(8)

where fĥ (y) is the estimated probability density of y, y is the considered data point, yi is the observation, h is the smoothing parameter, and n is the number of samples. K is the kernel function, the most widely used one of which is the Gaussian kernel function. By using KDE, the control limit of the LOF can be determined by the percentage area such as the 99% density distribution. It should be noted that the training data in this article is composed of normal samples from different operating modes. It is possible that the historical data contain some faulty samples or outliers. If the training data are contaminated, the control limit of the monitoring model will be inaccuracy. To handle data contamination, some outlier-elimination methods or classifying methods can be employed as a data preprocessing step.37−40 For example, before the modeling phase, Lee et al.36 used the LOF for outlier detection to eliminate the outliers in a contaminated training data. Yu41,42 employed stationary testing and the Gaussian mixture model to remove nonstationarity and isolate the normal and multiple faulty clusters during the preprocessing steps. The proposed localized Fisher discriminant analysis 2391



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(LFDA) method was demonstrated to be accurate and sensitive in detecting and classifying multiple types of faults. In the monitoring phase, the first step is finding the k nearest neighbors for the new test sample xnew. It needs to be emphasized that all of the k nearest neighbors of xnew must be found in the normal training data set. Because all of the neighborhoods of the normal training samples have been determined, which means that the neighborhood of xfnew has been obtained, it is easy to calculate the normalized Euclidean distances between the neighborhoods of xnew and xfnew using eq 4. Then, the reachability distance of xnew to its neighbor xfnew can be determined by eq 9. This means that, if the sample xnew is located in the suprasphere f determined by the neighborhood of xnew , the reachability f distance equals the radius of the neighborhood of xnew . f Otherwise, the actual distance between xnew and xnew is set as the reachability distance. The local reachability density of sample xnew and the value of the LOF can be calculated using the equations

LOF(xnew ) =

lrd s(xnew ) =

f = 1, 2, ..., k

j LOF(vnew ) , j = 1, 2, ..., m LOF(xnew )

f lrd s(xnew ) lrd s(xnew )

⎛ v1 − μ 1 i i,f ds(xi , xi ) = ⎜⎜ 1 ⎝ std i , f ⎛ vim − μim, f ⎞2 ⎟⎟ , + ⎜⎜ m ⎝ std i , f ⎠

(9)

(11)

⎞2 ⎛ vj − μ j i,f ⎟ + ··· + ⎜ i ⎜ std j ⎟ i,f ⎝ ⎠ f = 1, 2, ..., k ;

⎞2 ⎟ + ··· ⎟ ⎠

j = 1, 2, ..., m (12)

The contribution of each variable to the final monitoring statistic can be obtained according to the equations

k

j contribution(vnew )=

f =1

f

k f ∑ f = 1 reach_ds(xnew , xnew )

k

∑

After the fault is detected by the control limit, the method must go a further step for faulty variable identification, which is important for operators to determine which variables are the possible root causes of the current fault. In this article, a contribution-based method is developed for fault identification in the NSLOF approach. It can be found that the squared Euclidean distance in eq 4 can be decomposed on the visual angle of each variable with the settings of xi = (v1i , ..., vji, ..., vmi ), me[N(xfi)] = (μi,f1 , ..., μji,f , ..., μmi,f ), and std[N(std1i,f, ..., stdi,fj , ..., stdmi,f )]

f f reach_ds(xnew , xnew ) = max{k _distances(xnew ), f ds(xnew , xnew )},

1 k

(10)

the neighbors of all of the new samples are determined in the original data space in the proposed NSLOF method, it might be infrequent for fault samples to find that their neighbors represent data from the wrong mode. Especially when the process data contain multiple variables and the operating modes are significantly different from each other, the mean differences of most variables might make it hard for fault samples to find neighbors from the wrong local cluster. Therefore, the contribution-based fault identification method might be helpful for monitoring practical processes and might have some realistic directive significance for process operators. In the proposed NSLOF method, the number of neighbors is

(14)

It should be noted that there is an extreme condition that the fault samples have neighbors belonging to other modes, which means that the fault samples are coincidentally located around data from the wrong mode. Under this condition, the decomposition of the local outlier factor in the NSLOF approach might have some problems for fault identification even though the fault samples can still be properly detected. This is because the variable weights during the calculation of the monitoring statistics are not accurate for the wrong mode. However, because

introduced as a parameter. To ensure a good estimation of the 2392



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Figure 1. Schematic diagram of the NSLOF method.

3.3. Online Monitoring Phase of the NSLOF Method. The procedure for the online monitoring phase of the NSLOF method is as follows: (1) Obtain a new sample xnew. (2) Find its k nearest neighbors in the training data set X. (3) Calculate the squared Euclidean distances between the sample xnew and the mean of xfnew’s neighbors according to eq 4. (4) Determine the k distance of the new sample xnew using the squared Euclidean distances. (5) Calculate the LOF statistic for xnew according to eqs 9−11. (6) Monitor whether the statistic exceeds its control limit and determine whether the process is normal. (7) If the sample xnew is faulty, the final step is to diagnosis the possible root causes according the contribution-based method in eq 14; if the current sample is a normal one, go back to step 1 and start the monitoring procedure for the next new sample. A schematic diagram of the NSLOF method is presented in Figure 1.

normal data, the parameter k cannot be too small. If k is smaller than 10, the statistical fluctuations of different neighborhoods will significantly affect the value of the LOF. The statistical fluctuations in reachability distances are weakened as the k value increases.29 Thus, the first guideline for determining the value of k is that it should be larger than 10 to remove unwanted statistical fluctuations. Because the normal training data used for process monitoring might come from multiple operating modes and the neighborhoods must be local, the value of k cannot be too large. For example, if k is close to or even larger than the size of the subset of data representing a single mode, the neighbors of xi might overlap multiple modes. The superiority of neighborhood mean vectors and standard deviation vector in eq 4 will decrease in this case. On the other hand, k should be small in consideration of the computation complexity. The distance calculation will become computationally intensive if the size of the training set and the number of neighbors are too large. The impacts of the size of k are studied in detail in section 4. 3.2. Off-Line Modeling Phase of the NSLOF Method. The procedure for the off-line modeling phase of the NSLOF method is as follows: (1) Acquire the normal operating data set X ∈ Rn×m. (2) For each sample xi in X, determine its local neighborhood N(xi), which consists of its k nearest neighbors. (3) Calculate the squared Euclidean distances between the sample xi and the mean of xfi ’s neighbors according to eq 4. (4) Reorder the neighbors in N(xi) and determine the k distance of sample xi. (5) Calculate the reachability distances and the local reachability density of sample xi. (6) Determine the LOF statistic for each normal sample and the control limit of the LOF statistic by using KDE method.

4. CASE STUDIES In this section, two cases are studied to demonstrate the process monitoring performance of the proposed method. First, a numerical example, whose data distribution is not exactly Gaussian and has multimode characteristics, is employed in section 4.1. Then, the monitoring results of the widely used Tennessee Eastman process are presented in section 4.2. At the same time, the effects of different k values in the NSLOF method are also analyzed in section 4.2. Because PCA is one of the most popular monitoring methods with distance-based monitoring statistics, it is employed as a representative to compare with the density-based LOF method. The corresponding discussions are 2393



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data in fault case 1 show that presented in Figure 2. Figure 2 shows that the training data set contains two mode subsets,

used to illustrate the superiority of density-based monitoring statistics when operating data have a complex distribution. A methodology comparison between the LOF approach and the proposed density-based NSLOF method is used to demonstrate the improvements of the NSLOF method. The monitoring results of PCA, the traditional LOF approach, and the proposed NSLOF method in the numerical example and the Tennessee Eastman (TE) process are analyzed in detail. To erase the effects of the variable scales, the most widely used z-score method was employed in the PCA and LOF methods to standardize the process data. Because the normal data contain multiple distributions, the empirical thresholds instead of the theoretical thresholds are employed in PCA for fair comparison with the LOF and NSLOF methods. All of the control limits in the three methods were determined using KDE with the 99% confidence level. 4.1. Illustrative Numerical Example. A numerical example that was first suggested by Ge and Song13 is redesigned to fit the specific test purpose in this work. The example contains five variables that are driven by two factors, s1 and s2. The simulation data can be generated from the system of equations

Figure 2. Scatter plots of training data and test data in fault case 1 in the numerical example.

x1 = 0.5768s1 + 0.3766s2 + e1 x 2 = 0.7382s12 + 0.0566s2 + e 2

which means the distribution of the training data has multimodality characteristics. In addition, the within-mode data cannot be described a one single standard distribution such as a Gaussian distribution. In fault case 1, the fault samples significantly deviate from the normal region of mode 1. The score plots of training data and test data for fault case 1 in the principal space of PCA are shown in Figure 3. Because the

x3 = 0.8291s1 + 0.4009s2 2 + e3 x4 = 0.6519s1s2 + 0.2070s2 + e4 x5 = 0.3972s1 + 0.8045s2 + e5

(15)

where e1, e2, e3, e4, and e5 are zero-mean white noises with a standard deviation of 0.01. The changes of the two data sources are used to reflect shifts in the operating conditions. In this work, two different operating modes were constructed from the following data sources mode 1:

mode 2:

s1

uniform( − 10, − 7)

s2

N ( − 15, 1)

s1

uniform(2, 5)

s2

N (7, 1)

First, 400 samples of each mode were simulated, and a total of 800 multimode samples was generated by eq 15 to constitute the normal training data set. In the off-line modeling phase, the number of principal components in PCA was determined by the cumulative percentage of explained variance (85%). Fifty neighbors were selected for each sample in the LOF and NSLOF methods. The monitoring results are calculated based on the 99% control limits. Because the monitoring procedure of the traditional PCA has been widely reported,5,43,44 the details are not shown in this article. To test the feasibility of the proposed NSLOF method, two fault cases are designed in the following way. In each test data set, the first 400 samples were normal ones, and the remaining 400 samples were faulty ones. For fault case 1, the system was initially run in mode 1; then, a step change of 4 was added to x5 starting from sample 401 to the end. This fault magnitude reached 25% of the normal level. For fault case 2, the system was initially run in mode 2; then a drifting error of 0.02(i − 400) was added to x1 at sample 401, where i indicates the sample number. To show the multimodal distribution of the data in the numerical example, scatter plots of the training data and the test

Figure 3. Score plots of PCA in fault case 1 in the numerical example.

training data show multimodality characteristics, the 99% probability ellipse of the normal distribution is inflated. It can be seen that faulty samples are contained in the ellipse and still mixed with the normal samples. For multimode process data, the principal component space still contains multiple data distributions. The decision boundaries of the two monitoring indices cannot encompass the normal region closely when the data contain multimodality characteristics.12 It is foreseeable that PCA has high missed detection rates. The stem plots for the LOF and NSLOF methods in fault case 1 are presented in Figures 4 and 5, respectively. In the stem plots, 2394



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statistics are distance-based, the missed detection rates of PCA are very high when the data contain multimodality characteristics and have a non-Gaussian distribution.12 The PCA model mistakenly labels many fault samples as normal ones. The monitoring results of PCA in Table 1 reconfirm this statement. For fault case 1, the LOF missed detection rate was 98.8%, which cannot indicate that the LOF method is better than PCA. However, from the monitoring plots in Figure 6b, it can be seen that the difference between normal samples and fault samples has been separated by the monitoring statistic in the LOF approach, although the monitoring statistic is below the control limit. The missed detection rate of LOF in fault case 2 is 24%, which is lower than that of T2 and SPE. As shown in Figure 7b, LOF can detect the drifting error earlier than PCA. Based on these discussions, it is reasonable to believe that the density-based monitoring statistic in LOF is more accurate than traditional distance-based statistics in PCA for multimode monitoring. In contrast to the PCA and LOF approaches, the NSLOF method has the lowest missed detection rates in the two fault cases. The step change in fault case 1 can be fully detected by the NSLOF method. For fault case 2, the NSLOF method provides the best monitoring performance, with an 11.5% missed detection rate. The fault in case 2 can be easily detected even when the fault magnitude is small. The reason is that the utilization of the local neighborhood standardization strategy can help the NSLOF method extract the local information more thoroughly than the LOF. At the same time, the superiority of the LOF approach is completely maintained in the NSLOF method. The results of the contribution-based fault identification method in the NSLOF approach are shown in Figure 8. The highest contributing variables during the fault period are shown at the bottom of Figure 8. The variable x5 in fault case 1 and the variable x1 in fault case 2 make the highest contributions at sample 450. It is obvious that the NSLOF contribution plots clearly point out the root cause of the two faults. Particularly, sample 450 is not a special case. The highest contributing variable always points to variable x5 during the fault period, which is the faulty variable of fault case 1. Because the fault magnitudes of the first few fault samples in case 2 are quite small, the variation of variable x1 cannot be detected. Thus, the contribution plots lead to a wrong identification result. However, once the fault is detected, the NSLOF method can effectively diagnose the root cause of the fault. 4.2. Tennessee Eastman Process. The TE process, which is based on a practical industrial process, has been widely applied to test the performance of multivariate process monitoring techniques.7,9,15 It contains five major units: a reactor, a condenser, a recycle compressor, a stripper, and a vapor/liquid separator. According to the G/H mass ratios, the process exhibits six operating modes.45 In this work, the process data under modes 1 and 3 were generated using the Simulink programs, which are based on the decentralized control strategy designed by Ricker.46 The Simulink programs can be downloaded from http://depts.washington.edu/control/LARRY/TE/download. html.

Figure 4. LOF stem plot in fault case 1 in the numerical example.

Figure 5. NSLOF stem plot in fault case 1 in the numerical example.

the local outlier factor of each normal training sample and test sample is added as the z axis. The value of the local outlier factor is represented as the height along the z axis. Figure 4 shows that the local outlier factor used in LOF cannot separate the fault samples from normal samples. Because the local neighborhood standardization strategy is employed during the calculation of the local outlier factor, the NSLOF method can detect the fault samples more effectively than the LOF method. The stem plot in Figure 5 reconfirms this statement. The local outlier factor values of fault samples are significantly higher than the LOF values of the normal samples in the NSLOF method. The false detection rates and missed detection rates of the PCA, LOF, and NSLOF methods are presented in Table 1. For a good visual presentation, the monitoring plots of the three methods in the two fault cases are shown in Figures 6 and 7. It is easy to find that the false detection rates of all the three methods in the two fault cases are acceptable. Because the T2 and SPE

Table 1. False Detection Rates and Missed Detection Rates of Different Methods in the Two Fault Cases false detection rate (%) 2

missed detection rate (%) 2

fault

PCA (T )

PCA (SPE)

LOF

NSLOF

PCA (T )

PCA (SPE)

LOF

NSLOF

fault case 1 fault case 2

2.8 0

3 0

0.5 1.3

0.3 1.3

98.5 100

97 90.5

98.8 24

0 11.5

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Figure 6. Monitoring results obtained using different methods in fault case 1: (a) PCA, (b) LOF, (c) NSLOF.

Figure 7. Monitoring results obtained using different methods in fault case 2: (a) PCA, (b) LOF, (c) NSLOF.

little effect on the process, were not used in this work.47 For the test purposes, the remaining 12 faults, which are listed in Table S2 (Supporting Information), were utilized for the comparisons. Five hundred samples were collected under each of the two modes, so 1000 samples in all constituted the training data set. The test data set contained 1000 samples. Process faults happened at sample 201 and lasted until the end. All of the statistical limits were set as 99% control limits. For PCA, the number of principal components was also determined by the cumulative percentage of explained variance (≥85%). Fifty neighbors were selected in the LOF and NSLOF methods.

There are 22 continuous process measurements, 12 manipulated variables, and 19 composition measurements in the process.46−48 During the simulation, the steady-state values of the recycle valve and steam valve in mode 1 always equal 1, and the agitator rates in the two modes are both 100. Therefore, these 3 manipulated variables are not selected as monitored variables in this work. The remaining 9 manipulated variables and 22 continuous process measurements, listed in Table S1 in the Supporting Information, are used as the monitored variables. A total of 20 faults can be added to the simulation. Five unknown faults and faults 3, 9, and 15, which are difficult to detect and have 2396



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Figure 8. Contribution plots of the NSLOF method in the two fault cases: (a) fault case 1, (b) fault case 2.

Table 2. False Detection Rates and Missed Detection Rates of 12 Faults in TE Mode 1 false detection rate (%) 2

missed detection rate (%) 2

fault

PCA (T )

PCA (SPE)

LOF

NSLOF

PCA (T )

PCA (SPE)

LOF

NSLOF

1 2 4 5 6 7 8 10 11 12 13 14 average

0 0 0 0.5 0 0 0 0 0 0 0 0.5 0.1

0 0 0.5 0 1 0.5 1 0.5 2 2 0 0.5 0.7

0.5 0 1 0.5 0.5 2.5 0.5 0.5 1.5 3 0.5 1.5 1

2.5 2.5 6 4 1.5 4 6 4 5 4 4 2 3.8

65 87.6 99.5 99.6 6.3 98.9 40.5 98.1 79.8 95.6 14.6 99.6 73.8

0.4 1.3 0 99.3 0 34.6 7.5 97.6 10.1 87 8.1 24.9 30.9

0.4 0.8 0 99.5 0 0 4.6 96.4 6 81.5 7 14.9 25.9

0.1 0.6 0 94.8 0 0 2.3 17.8 1 48 2.6 0 13.9

Table 3. False Detection Rates and Missed Detection Rates of 12 Faults in TE Mode 3 false detection rate (%)

missed detection rate (%)

fault

PCA (T2)

PCA (SPE)

LOF

NSLOF

PCA (T2)

PCA (SPE)

LOF

NSLOF

1 2 4 5 6 7 8 10 11 12 13 14 average

1.5 2 1.5 1.5 1 0.5 0.5 1 2.5 3 0.5 1 1.4

3.5 3 1 2.5 3 2.5 2 2 2 2 2.5 2 2.3

2.5 1.5 1 2 0.5 2 1 1 1.5 4 1.5 0.5 1.6

1.5 3.5 3.5 1 1.5 2.5 1 3 1.5 4 2.5 2.5 2.3

68.8 98.8 99.3 97.8 0.8 95.6 37 94.1 90.1 16.4 48.8 99 70.5

1.1 89.9 0 77.6 0 62.4 6.8 91.4 11.1 4.3 31.5 28.8 33.7

1.3 66.4 0 56.3 0 0 5.3 92.3 8.4 2.9 27 21.1 23.4

0.1 2.1 0 0 0 0 2.4 15.1 2.1 0.9 11.6 0 2.9

To evaluate the feasibility of the proposed method, the data sets of all 12 faults in modes 1 and 3 were tested. The false

detection rates and missed detection rates of the three methods are listed in Tables 2 and 3. The average detection results are 2397



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Figure 9. Monitoring results of different methods for fault 14 in mode 1: (a) PCA, (b) LOF, (c) NSLOF.

Figure 10. Monitoring results of different methods for fault 5 in mode 3: (a) PCA, (b) LOF, (c) NSLOF.

complicated, the global mean and standard deviation of the whole training data set employed in the z-score method could not accurately describe the real situation of the data. Therefore, it was unsuitable to use PCA and not sufficient to use the LOF approach directly to monitor the multimode TE process. In comparison with PCA and the LOF approach, the NSLOF method employs the neighborhood standardization strategy. In the NSLOF method, the means and standard deviations of the local neighborhoods are taken into account, which means that the local information is utilized more efficiently. As shown in Tables 2 and 3, the monitoring results of the NSLOF method were the best among the three methods. An obvious improvement in reducing the missed detection rates can be seen by

presented at the bottom of each table. The false detection rate was calculated from the rate of misclassified normal samples compared to the entire set of normal samples from sample 1 to 200, and the missed detection rate was calculated as the rate of misclassified fault samples relative to the entire set of fault samples from sample 201 to 1000. Compared to PCA, the LOF approach had lower missed detection rates for most faults in modes 1 and 3, whereas the false detection rates of the LOF approach were similar to those of PCA. This means that the utilization of the density-based statistic is reasonable and that it has some superiority for monitoring multimode processes, which reconfirms the previous statements in the numerical case study. Because the data distribution of the multimode TE is 2398



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Figure 11. Variable plots and contribution plots of the NSLOF method for fault 14 in mode 1.

Figure 12. Variable plots and contribution plots of the NSLOF method for fault 5 in mode 3.

detection performance of the LOF method was still not satisfactory. It can readily be seen that about half of the faulty samples were below the confidence limit of the LOF approach in Figure 10b. In contrast to PCA and the LOF approach, the monitoring performance of the NSLOF method was the best. Fault 14 in mode 1 and fault 5 in mode 3 were both easily detected by the NSLOF method with acceptable false alarms. Furthermore, the NSLOF method has the ability to identify the root causes of a fault. The corresponding variable plots and contribution plots of the NSLOF method for the two faults are shown in Figures 11 and 12, respectively. It can be seen that the variables that changed significantly after the faults had the highest contributions. For example, fault 5 involved a step change in the

comparing the average detection rates of the NSLOF method with those of the other two methods. In particular, the average missed detection rate of the NSLOF method in mode 3 is 2.9%, which is much lower than the average missed detection rate of either PCA or the LOF method. To visually illustrate the superiority of the NSLOF method, the monitoring results for fault 14 in mode 1 and fault 5 in mode 3 are presented in Figures 9 and 10, respectively. It can be seen that the T2 statistic of PCA barely detected the two faults. A significant number of faulty samples were wrongly detected as normal ones. The missed alarms of SPE were less than those of the T2 statistic. Compared to PCA, the monitoring results of the LOF approach showed some improvements. However, the fault 2399



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Figure 13. Average detection results of three methods with different k values in the TE process: (a) mode 1, (b) mode 3.

distances between the samples and their neighbors. The contribution-based fault identification method of NSLOF was developed to help operators assess the real state of the process. Finally, the proposed NSLOF method was applied in a numerical example and the multimode TE process. The monitoring results of the NSLOF method, PCA, and the conventional LOF approach were compared and analyzed in detail. In contrast to PCA and the LOF approach, the missed detection rates of the NSLOF method were significantly decreased with acceptable false detection rates. Compared to the multiple-model strategy, the NSLOF method builds only one off-line model for multimode process monitoring without needing a priori process knowledge. Furthermore, neither additional pattern-matching rules nor integrating steps are needed for the further analysis of the monitoring results of the local models. Therefore, the NSLOF method can potentially be applied for the monitoring of complex industrial processes. However, some more efforts are still needed to amend the contribution metric method of the NSLOF approach. In addition, between-mode transitions in multimode processes are not included in the present NSLOF method. Because the training data set used for modeling was assumed to be uncontaminated in this work, the robust extension of the NSLOF method can be meaningful for practical process monitoring. Moreover, it would be worth some research efforts to extend the new method to batch processes or some other outlier-detection applications.

condenser cooling water inlet temperature. When fault 5 occurred, the flow rate of the outlet stream from the condenser to the separator increased, which resulted in an increase in both the separator temperature (variable 11) and the separator cooling water outlet temperature (variable 22). At the same time, the stripper temperature (variable 18) also displayed a significantly rise.47 For variables 18 and 22, the faulty data showed a large difference from the normal data, and these variables were correctly identified in the contribution plot in Figure 12. Similar fault identification results were obtained for fault 14 in mode 1. As shown in Figure 11, the contribution plot of the NSLOF method clearly pointed out the three major variables with large contributions. To explore the effects of parameters in the NSLOF method, the TE multimode process was monitored by the three methods with different k values. The average false detection rates and missed detection rates of 12 faults in modes 1 and 3 are summarized in Figure 13. Because the number of neighbors is not a parameter of PCA, the average detection results of PCA are straight lines. It can be seen that the average false detection rates of the LOF and NSLOF methods are both smaller than 5%, which is similar to that of PCA. When k is extremely small, such as 2, the NSLOF method has a high missed detection rate. The monitoring performance of the NSLOF method is even worse than PCA in this situation. However, the superiority of the NSLOF method is revealed when k increases. The average missed detection rate of the NSLOF method is the smallest of the three methods for a wide range of k values. Furthermore, the influence of k is very small once k is larger than 10, which could corroborate the previous statements in section 3.1. From Figure 13, it can be seen that the outstanding performance of the NSLOF method is insensitive to parameter changes over a wide range. Thus, it can be concluded that the parameter k in the NSLOF method is easy to determine.

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ASSOCIATED CONTENT

S Supporting Information *

Details of the monitored variables and process faults used in the Tennessee Eastman process. This material is available free of charge via the Internet at http://pubs.acs.org.

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5. CONCLUSIONS In this article, a novel neighborhood standardized local outlier factor (NSLOF) method is proposed for fault detection and identification of complex chemical processes. A neighborhood standardization strategy was employed to normalize the

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-021-64252189. Fax: +86-021-64252349. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 2400



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(21) Yu, J. A particle filter driven dynamic Gaussian mixture model approach for complex process monitoring and fault diagnosis. J. Process Control 2012, 22, 778−788. (22) Xie, X.; Shi, H. B. Dynamic Multimode Process Modeling and Monitoring Using Adaptive Gaussian Mixture Models. Ind. Eng. Chem. Res. 2012, 51, 5497−5505. (23) Yu, J.; Qin, S. J. Multiway Gaussian mixture model based multiphase batch process monitoring. Ind. Eng. Chem. Res. 2009, 48, 8585−8594. (24) Yu, J. B. Hidden Markov models combining local and global information for nonlinear and multimodal process monitoring. J. Process Control 2010, 20, 344−359. (25) Rashid, M. M.; Yu, J. Hidden Markov Model Based Adaptive Independent Component Analysis Approach for Complex Chemical Process Monitoring and Fault Detection. Ind. Eng. Chem. Res. 2012, 51, 5506−5514. (26) Mehranbod, N.; Soroush, M.; Piovoso, M.; Ogunnaike, B. A. Probabilistic model for sensor fault detection and identification. AIChE J. 2004, 49, 1787−1802. (27) Mehranbod, N.; Soroush, M.; Panjapornpon, C. A method of sensor fault detection and identification. J. Process Control 2005, 15, 321−339. (28) Ge, Z. Q.; Song, Z. H. A distribution-free method for process monitoring. Expert Syst. Appl. 2011, 38, 9821−9829. (29) Breunig, M. M.; Kriegel, H.; Ng, R. T.; Sander, J. LOF: Identifying density-based local outliers. In Proceedings of the 2000 ACM SIGMOD International Conference on Mangement of Data; Chen, W., Naughton, J. F., Bernstein, P. A., Eds.; ACM: New York, 2000; pp 93−104. (30) Chen, S. Y.; Wang, W.; van Zuylen, H. A comparison of outlier detection algorithms for ITS data. Expert Syst. Appl. 2010, 37, 1169− 1178. (31) Duan, L.; Xu, L.; Guo, F.; Lee, J.; Yan, B. A local-density based spatial clustering algorithm with noise. Inf. Syst. 2007, 32, 978−986. (32) Gao, J.; Hu, W.; Zhang, Z.; Zhang, X.; Wu, O. RKOF: Robust kernel-based local outlier detection. In Advances in Knowledge Discovery and Data Mining; Springer: Heidelberg, Germany, 2011; Vol. 6635, pp 270−283. (33) Silverman, B. W. Density Estimation for Statistics and Data Analysis; Chapman and Hall: New York, 1986. (34) Lee, J. M.; Yoo, C. K.; Lee, I. B. Statistical process monitoring with independent component analysis. J. Process Control 2004, 14, 467−485. (35) Hsu, C.; Chen, L.; Liu, C. A process monitoring scheme based on independent component analysis and adjusted outliers. Int. J. Prod. Res. 2010, 48, 1727−1743. (36) Lee, J.; Kang, B.; Kang, S. Integrating independent component analysis and local outlier factor for plant-wide process monitoring. J. Process Control 2011, 21, 1011−1021. (37) Chiang, L. H.; Pell, R. J.; Seasholtz, M. B. Exploring process data with the use of robust outlier detection algorithms. J. Process Control 2003, 13, 437−449. (38) Daszykowski, M.; Kaczmarek, K.; Heyden, Y. V.; Walczak, B. Robust statistics in data analysisA review: Basic concepts. Chemom. Intell. Lab. Syst. 2007, 85, 203−219. (39) Wang, D. Robust data-driven modeling approach for real-time final product quality prediction in batch process operation. IEEE Trans. Ind. Inf. 2011, 7, 371−377. (40) Yu, J. A Bayesian Inference Based Two-Stage Support Vector Regression Framework for Soft Sensor Development in Batch Bioprocesses. Comput. Chem. Eng. 2012, 41, 134−144. (41) Yu, J. Localized fisher discriminant analysis based complex chemical process monitoring. AIChE J. 2011, 57, 1817−1828. (42) Yu, J. Nonlinear Bioprocess Monitoring Using Multiway Kernel Localized Fisher Discriminant Analysis. Ind. Eng. Chem. Res. 2011, 50, 3390−3402. (43) Nomikos, P.; MacGregor, J. F. Monitoring batch processes using multiway principal component analysis. AIChE J. 1994, 40, 1361−1375. (44) Wold, S.; Esbensen, K.; Geladi, P. Principal component analysis. Chemom. Intell. Lab. Syst. 1987, 2, 37−52.

ACKNOWLEDGMENTS This research was supported by the National Natural Science Foundation of China (No. 61074079) and Shanghai Leading Academic Discipline Project (No. B504).

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REFERENCES

(1) Qin, S. J. Statistical process monitoring: basics and beyond. J. Chemom. 2003, 17, 480−502. (2) Lee, J. M.; Yoo, C. K.; Choi, S. W.; Vanrolleghem, P. A.; Lee, I. B. Nonlinear process monitoring using kernel principal component analysis. Chem. Eng. Sci. 2004, 59, 223−234. (3) Dong, D.; McAvoy, T. J. Nonlinear principal component analysisbased on principal curves and neural networks. Comput. Chem. Eng. 1996, 20, 65−78. (4) Lee, J. M.; Qin, S. J.; Lee, I. B. Fault detection of non-linear processes using kernel independent component analysis. Can. J. Chem. Eng. 2007, 85, 526−536. (5) Ku, W.; Storer, R. H.; Georgakis, C. Disturbance detection and isolation by dynamic principal component analysis. Chemom. Intell. Lab. Syst. 1995, 30, 179−196. (6) Zhao, C. H.; Gao, F. R.; Wang, F. L. Nonlinear Batch Process Monitoring Using Phase-Based Kernel-Independent Component Analysis−Principal Component Analysis (KICA−PCA). Ind. Eng. Chem. Res. 2009, 48, 9163−9174. (7) Ge, Z. Q.; Song, Z. H. Process monitoring based on independent component analysis−principal component analysis (ICA−PCA) and similarity factors. Ind. Eng. Chem. Res. 2007, 46, 2054−2063. (8) Kano, M.; Tanaka, S.; Hasebe, S.; Hashimoto, I.; Ohno, H. Combination of independent component analysis and principal component analysis for multivariate statistical process control. In Proceedings of International Symposium on Design, Operation and Control of Chemical Plants (PSE Asia 2002); Taipei, Taiwan, 2002; pp 319−324. (9) Yu, J.; Qin, S. J. Multimode process monitoring with Bayesian inference-based finite Gaussian mixture models. AIChE J. 2008, 54, 1811−1829. (10) Ge, Z. Q.; Song, Z. H. Mixture Bayesian Regularization method of PPCA for multimode process monitoring. AIChE J. 2010, 56, 2838− 2849. (11) Tan, S.; Wang, F. L.; Peng, J.; Chang, Y. Q.; Wang, S. Multimode Process Monitoring Based on Mode Identification. Ind. Eng. Chem. Res. 2012, 51, 374−388. (12) Ma, H. H.; Hu, Y.; Shi, H. B. A novel local neighborhood standardization strategy and its application in fault detection of multimode processes. Chemom. Intell. Lab. Syst. 2012, 118, 287−300. (13) Ge, Z. Q.; Song, Z. H. Multimode process monitoring based on Bayesian method. J. Chemom. 2009, 23, 636−650. (14) Zhao, S. J.; Zhang, J.; Xu, Y. M. Performance monitoring of processes with multiple operating modes through multiple PLS models. J. Process Control 2006, 16, 763−772. (15) Zhao, S. J.; Zhang, J.; Xu, Y. M. Monitoring of processes with multiple operation modes through multiple principle component analysis models. Ind. Eng. Chem. Res. 2004, 43, 7025−7035. (16) Natarajan, S.; Srinivasan, R. Multi-model based process condition monitoring of offshore oil and gas production process. Chem. Eng. Res. Des. 2010, 88, 572−591. (17) Ge, Z. Q.; Gao, F. R.; Song, Z. H. Two-dimensional Bayesian monitoring method for nonlinear multimode processes. Chem. Eng. Sci. 2011, 66, 5173−5183. (18) Lane, S.; Martin, E. B.; Kooijmans, R.; Morris, A. J. Performance monitoring of a multi-product semi-batch process. J. Process Control 2001, 11, 1−11. (19) Garcia-Alvarez, D.; Fuente, M. J.; Sainz, G. I. Fault detection and isolation in transient states using principal component analysis. J. Process Control 2012, 22, 551−563. (20) Yu, J. A nonlinear kernel Gaussian mixture model based inferential monitoring approach for fault detection and diagnosis of chemical processes. Chem. Eng. Sci. 2011, 68, 506−519. 2401



Article

(45) Ricker, N. L. Optimal steady-state operation of the Tennessee Eastman challenge process. Comput. Chem. Eng. 1995, 19, 949−959. (46) Ricker, N. L. Decentralized control of the Tennessee Eastman Challenge Process. J. Process Control 1996, 6, 205−221. (47) Chiang, L. H.; Russell, E.; Braatz, R. D. Fault Detection and Diagnosis in Industrial Systems; Springer: London, 2001. (48) Downs, J. J.; Vogel, E. F. A plant-wide industrial process control problem. Comput. Chem. Eng. 1993, 17, 245−255.

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Fault Detection and Identification Based on the Neighborhood

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