An Improved PCA Scheme for Sensor FDI- Application to an Air Quality Monitoring Network

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Journal of Process Control 16 (2006) 625–634

An improved PCA scheme for sensor FDI: Application to an airquality monitoring network

Mohamed-Faouzi Harkat a,*, Gilles Mourot b, Jose Ragot b

a Universite Badji Mokhtar, ANNABA, Faculte des Sciences de l’Ingenieur, Departement d’Electronique, BP. 12, Sidi Amar, 23000 Annaba, Algerieb Institut National Polytechnique de Lorraine, Centre de Recherche en Automatique de Nancy, UMR-CNRS INPL/UHP 7039,

2, Avenue de la Foret de Haye 54516 Vandoeuvre-les-Nancy, France

Received 9 January 2005; received in revised form 13 July 2005; accepted 26 September 2005

Abstract

In this paper a sensor fault detection and isolation procedure based on principal component analysis (PCA) is proposed to monitor anair quality monitoring network. The PCA model of the network is optimal with respect to a reconstruction error criterion. The sensorfault detection is carried out in various residual subspaces using a new detection index. For our application, this index improves the per-formance compared to classical detection index SPE. The reconstruction approach allows, on one hand, to isolate the faulty sensors and,on the other hand, to estimate the fault amplitudes.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Fault diagnosis; Principal component analysis; Sensor failure detection and isolation; Variable reconstruction; Air quality monitoring network

1. Introduction

Many human activities produce primary pollutants likenitrogen oxides (NO2 and NO) and volatile organic com-pounds (VOC) which formed in the lower atmosphere, bychemical or photochemical reactions, secondary pollutantslike ozone. The acceptable concentrations of these pollu-tants, harmful for human health and the environment,are defined by European standards. Air quality monitoringnetworks have the following main missions: the measure-ment network management (recording of pollutant concen-trations and a range of meteorological parameters relatedto pollution events) and the diffusion of data for permanentinformation of population and public authorities in refer-ence to norms. To ensure these missions, the validity ofthe delivered information is essential before any use of

0959-1524/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2005.09.007

* Corresponding author.E-mail address: [email protected] (M.-F.

Harkat).

measurements, mainly those of pollutants. Moreover, thegeographical area monitored by a network being large,fault diagnosis procedure also enables to optimize themaintenance actions. Therefore sensor validation is anissue of great importance for the development of reliableenvironmental monitoring and management systems. Incollaboration with AIRLOR, an air quality monitoringnetwork (France), the aim of this work is to develop amethod to perform sensor validation, mainly sensors mea-suring the concentrations of ozone and nitrogen oxides.

The task for sensor validation is to detect, isolate, andidentify faulty sensors by examining the sensor measure-ments. The sensor validation is usually performed eitherusing ‘‘outlier’’ detection methods which only enable toidentify extreme values out of measurement range or man-ually by an operator. Unfortunately, this latter approachis too subjective and often impractical in real-time dueto high process dimensionality which produces a largeamount of collected data.

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626 M.-F. Harkat et al. / Journal of Process Control 16 (2006) 625–634

However, the availability of many sensors in the net-work provides valuable redundancy for fault detectionand isolation because some sensor measurements are highlycorrelated under normal conditions. The analytical redun-dancy approach consists of checking the consistencybetween the measurements and the estimates provided bythe relationships existing between the various variablesof the process [3]. This analysis may lead to detect andisolate the faulty sensors. In most practical situations, faultdiagnosis needs to be performed in the presence of distur-bances, noise and modelling errors. These mathematicalrelations between the plant variables generally take twoforms [4]. Analytical redundancy methods used explicitinput–output model usually derived from system identifica-tion. However, in the considered sensor network, this expli-cit formulation of the redundancy relationships may bedifficult to obtain (owing to complexity of the processand high process dimensionality) because it must be guidedmainly by performance criteria of the diagnosis system. Asan alternative, implicit modelling approaches, which aredata-driven techniques (like principal component analysis),are particularly well adapted to reveal linear relationshipsamong the plant variables without formulating themexplicitly. Moreover the number of these relations couldbe determined by minimizing a criterion based on the bestreconstruction of the variables with respect to the numberof principal components in the PCA model [12]. PCA hassome other nice features. It can handle high dimensionaland correlated process variables, provides a natural solu-tion to the errors-in-variables problem and includes distur-bance decoupling [4].

The widely used detection index SPE (squared predic-tion error) indicates how much each sample deviates fromthe PCA model. Indeed SPE performs fault detection in thethe residual space. However, Harkat et al. [6] have shownexperimentally on real data that the SPE is sensitive tothe modelling errors. Indeed modelling errors could be pro-jected into the residual space which results in residuals withhigher variance than the others. Then the SPE will be heav-ily biased in favour of those equations with the largestresidual variance whereas the residuals with the smallestresidual variances are most useful for sensor fault diagnosisbecause they are associated with linear relationships.

After a fault has been detected, it is important to isolatefaulty sensor. In the PCA framework, the well known isola-tion approaches are residual enhancement, contributionplots and variable reconstruction methods [13]. The residualenhancement technique generates structured residuals thatselectively respond to subsets of faults [5]. Such residualsmay be obtained by algebraic transformation, or by a directtechnique that involves a bank of partial PCA models [4].However, for high dimensionality process, it is not alwayspossible to find the algebraic transformation that enablesto obtain the desired isolation properties because theseproperties are only defined according to the occurrence ofthe faults in the residuals without taking into account thesensitivities of the residuals to the faults. These comments

are also true for the direct method. For fault isolation con-tribution plots show the contribution of each process vari-able to the detection statistic [8]. It is assumed that theprocess variable with the highest contribution is likely thefaulty one. However, the contribution plots may not explic-itly identify the cause of an abnormal condition [9], andsometimes lead to incorrect conclusions [13]. An alternativeapproach for fault isolation is the variable reconstructionmethod proposed by Dunia et al. [2]. It is based on the ideathat the influence of fault on the detection index is elimi-nated when the faulty variable is reconstructed using thePCA model from the variables without defect.

In this paper, a sensor fault detection and isolation pro-cedure based on principal component analysis is proposedto monitor an air quality monitoring network. Taking intoaccount the high redundancy among the process variables,this procedure is based on the variable reconstructionapproach in order to design the PCA model of the network,to isolate the faulty sensors and to estimate the fault ampli-tudes. To improve fault detection, a new fault detectionindex is proposed which monitors the last principal compo-nents (which have smallest variances) in different residualsubspaces, starting with a single direction (the last principaldirection) and gradually increasing the dimension of theresidual to the full residual space. Mathematically, it is dif-ficult to prove that the new index has better fault sensitivitythan the SPE because they perform fault detection in differ-ent residual spaces where the sensitivities of the residuals tothe faults may be different. Therefore experimental data setwas used to evaluate the performance of the proposedindex. For our application, this index improves the faultdetection compared to classical detection index SPE.

First, we present the principle of PCA modelling basedon the variable reconstruction approach. Then, after hav-ing summarized the principle of sensor fault detection inthe PCA framework, we propose our new detection index.For isolation of the faulty sensors, we combine the pro-posed index and the variable reconstruction principle. InSection 4, we present the application of the proposedmethod to sensor fault detection and isolation of an airquality monitoring network in Lorraine. Conclusions andfuture works are finally presented in the last section.

2. PCA fault detection and isolation

2.1. PCA modelling

Principal component analysis is one of the most popularstatistical methods, for extracting information from mea-sured data, which finds the directions of significant variabil-ity in the data by forming linear combinations of variables.

Let us consider xðkÞ ¼ ½ x1ðkÞ x2ðkÞ � � � xmðkÞT �the vector formed with m observed plant variables attime instant k. Define the data matrix X ¼½ xð1Þ xð2Þ � � � xðNÞ �T 2 RN�m with N samples x(k)

M.-F. Harkat et al. / Journal of Process Control 16 (2006) 625–634 627

(k = 1, . . . ,N) which is representative of normal processoperation.

PCA determines an optimal linear transformation of thedata matrix X in terms of capturing the variation in thedata:

T ¼ XP and X ¼ TP T ð1Þwith T ¼ ½ t1 t2 � � � tm � 2 RN�m, where the vectors ti arecalled scores or principal components and the matrixP ¼ ½ p1 p2 � � � pm � 2 Rm�m, where the orthogonal vec-tors pi, called loading or principal vectors, are the eigenvec-tors associated to the eigenvalues ki of the covariancematrix (or correlation matrix) R of X:

R ¼ PKP T with PP T ¼ P TP ¼ Im ð2Þwhere K = diag(k1 � � � km) is a diagonal matrix with diago-nal elements in decreasing magnitude order.

The partition of eigenvectors and principal componentsmatrices gives:

P ¼ P ‘j~P m�‘� �

; T ¼ T ‘j~T m�‘� �

ð3Þ

where ‘ will be defined below.Eq. (1) can be rewritten as

X ¼ T ‘PT

‘ þ ~T m�‘~PT

m�‘ ¼ X þ E ð4Þwith

X ¼ X C‘ and E ¼ X ~Cm�‘ ð5Þwhere the matrices C‘ ¼ P ‘P

T

‘ and ~Cm�‘ ¼ Im � C‘ consti-tute the PCA model.

The matrices X and E represent, respectively, the mod-elled variations and nonmodelled variations of X basedon ‘ components (‘ < m). The first ‘ eigenvectors P ‘ 2Rm�‘ constitute the representation space whereas the last(m � ‘) eigenvectors ~P m�‘ 2 Rm�m�‘ constitute the residualspace.

The identification of the PCA model thus consists inestimating its parameters by an eigenvalue/eigenvectordecomposition of the matrix R and determining the num-ber of principal components ‘ to retain. Here PCA is usedfor extracting redundancy relationships between the vari-ables. In most practical cases (noisy measurements), thesmall eigenvalues indicate the existence of linear or quasi-linear relations among the process variables. However thedistinction between significant or not significant eigen-values may not be obvious due to modelling errors (distur-bances and nonlinearities) and noise. A key issue todevelop a PCA model is to choose the adequate numberof principal components.

2.1.1. Determining the number of principal components

Most methods to determine the number of principalcomponents are rather subjective in the general practiceof PCA [14]. However, the number of principal compo-nents has a significant impact on each step of the sensorfault detection, isolation and reconstruction scheme, such

as the ability to detect small faults, the degree of freedomfor fault identification and the accuracy of reconstruction.

Qin and Dunia [12] proposed to determine ‘ by minimi-zation of the variance of reconstruction error. The variablereconstruction consists in estimating a variable from othersplant variables using the PCA model, i.e. using the redun-dancy relations between this variable and the others. Thereconstruction accuracy is thus related to the capacity ofthe PCA model to reveal the redundancy relations amongthe variables, i.e. to the number of principal components.Indeed, if too many principal components are chosen, eachvariable tends to rely too much on itself, which means itsrelationship to other variables is weakened. If too few prin-cipal components are used, the model is inaccurate to rep-resent the normal variation of the data and thus results inpoor reconstruction.

First, the variable reconstruction approach is presented.Let us denote xj(k) the measurement vector where the jthcomponent is reconstructed. The reconstruction of the jthvariable, while taking into account the reconstruction con-dition cjj 5 1, is

xjðkÞ ¼ GjxðkÞ ð6Þwith

GTj ¼ n1 � � � gj � � � nm

� �gT

j ¼cT�j 0 cT

þj

� �1� cjj

ð7Þ

where nj ¼ ½ 0 � � � 1 � � � 0 �T is the the jth columnof an identity matrix, cT

j ¼ ½ c1j c2j � � � cmj � ¼½ cT�j cjj cT

þj � is the jth column of C‘ and the subscripts�j, +j, respectively, allow to denote a vector formed bythe first (j � 1) and the last (m � j) elements of the consid-ered vector.

The variance of the reconstruction error, in the directionnj, is defined as

qjð‘Þ ¼~n

T

j R~nj

~nT

j~nj

� �2ð8Þ

where ~nj ¼ ðI � C‘Þnj.To find the number of principal components, the nor-

malized VRE (variance of reconstruction error) is mini-mized with respect to the number ‘:

VREð‘Þ ¼Xm

j¼1

qjð‘ÞnT

j Rnj

ð9Þ

An important feature of this approach is that the pro-posed criterion has a minimum corresponding to the bestreconstruction [12]. Moreover these authors proposed todiscard from process monitoring the variables for whichthe individual variances of reconstruction error are greaterthan their variances. Indeed variables which are little corre-lated with others cannot be reliably reconstructed from thePCA model.


2.2. Fault detection

After the PCA model has been built, we now examine itsuse for sensor fault detection.

2.2.1. Residual generationLet us consider a new sample vector, x(k). Following

decomposition (4), this vector can also be represented as

xðkÞ ¼ xðkÞ þ eðkÞ ð10Þwhere xðkÞ ¼ C‘xðkÞ is the estimation vector, eðkÞ ¼ðI � C‘ÞxðkÞ is the vector of estimation errors.

The principal component vector is given by

tðkÞ ¼ P TxðkÞ ¼ ½ t‘ðkÞ ~tm�‘ðkÞ � ð11Þ

where

t‘ðkÞ ¼ PT

‘ xðkÞ; ~tm�‘ðkÞ ¼ ~PT

m�‘xðkÞ ð12ÞThere is an equivalence between the residual vector and

the last principal component vector ~tm�‘:

eðkÞ ¼ ~P m�‘~tm�‘ðkÞ ð13ÞSo, it becomes simpler to work with the residual vector

~tm�‘ with dimension (m � ‘).Let us consider now the fault propagation on the resid-

ual vector ~tm�‘. In the presence of fault with a magnitudef(k), the measurement vector x(k) can be expressed as

xðkÞ ¼ x�ðkÞ þ �ðkÞ þ njfðkÞ ð14Þ

where x�(k) is the fault-free measurement vector, �(k) is thewhite measurement noise vector and nj is the fault direction(fault on the jth component of x(k)).

From (12) and (14), the residual vector is given by

~tm�‘ðkÞ ¼ ~PT

m�‘x�ðkÞ|fflfflfflfflfflffl{zfflfflfflfflfflffl}

¼0

þ~PT

m�‘�ðkÞ þ ~PT

m�‘njfðkÞ ð15Þ

In the fault-free case, the expectation of the residual vec-tor is zero. In the presence of faults, the expectation of theresidual vector is no more zero and the fault affects all thecomponents of residual vector.

2.2.2. Definition of the detection index Di

Sensor fault detection using PCA is performed by mon-itoring the residuals. The SPE (squared prediction error) isa statistic that measures the lack of fit of the PCA model tothe data. At time k, the detection index SPE is given by

SPEðkÞ ¼ ~tm�‘ðkÞT~tm�‘ðkÞ ¼Xm

j¼‘þ1

t2j ðkÞ ð16Þ

This quantity suggests the existence of an abnormal sit-uation in the data when:

SPEðkÞ > d2a ð17Þ

where d2a is a control limit for SPE given by [1] or estimated

using the historical data.

The SPE is formed by summing the squares of residualsobtained from PCA model. However modelling errors couldbe projected into the residual space which results in residualswith higher variance than the others. Then the SPE will beheavily biased in favour of those equations with the largestresidual variance whereas the residuals with the smallestresidual variances are most useful for sensor fault diagnosisbecause they are associated with linear relationships.

To overcome this problem, a new index based on the lastprincipal components is suggested and given, at time k, bythe following expression:

DiðkÞ¼~tiðkÞT~tiðkÞ¼Xm

j¼m�iþ1

t2j ðkÞ with i¼ 1;2; . . . ;ðm� ‘Þ

ð18Þ

This index is calculated by using successive sum ofsquared last principal components in different subspacesof the residual space, starting with a single direction (thelast principal direction associated to the smallest variance)and gradually increasing the dimension of the residual tothe full residual space. Therefore, the index Di may be lesssensitive to the modelling errors than the SPE. However,mathematically, it is difficult to prove that the new indexhas better fault sensitivity than the SPE because theyperform fault detection in different residual spaces wherethe sensitivities of the residuals to the faults may bedifferent.

Considering the definitions (16) and (18), the expressionof the index Di can be written as follows:

DiðkÞ ¼ SPEiðkÞ ð19Þ

where SPEi(k) is a squared prediction error calculated byretaining (m � i) components in the PCA model. The con-trol limits d2

i;a for the detection index Di with a significancelevel a, can be calculated as in the case of SPE [1].

The process is considered out of control limit if:

DiðkÞ > d2i;a i ¼ 1; 2; . . . ; ðm� ‘Þ ð20Þ

To improve quality of detection by reducing the rate offalse alarms (due to noise), filter EWMA (exponentiallyweighted moving average) can be applied to the residuals.The filtered residuals are thus obtained:

�~tiðkÞ ¼ ð1� cÞ�~tiðk � 1Þ þ c~tiðkÞ ð21Þ�~tiðkÞ being the filtered vector at time k and c is a forgettingfactor chosen in the interval [0,1].

The control limits �d2

i;a for Di (filtered Di) are computedas proposed by Qin et al. [11] for the index SPE (filteredSPE).


2.3. Fault isolation

After the presence of a fault has been detected, it isimportant to identify the fault and apply the necessarycorrective actions to eliminate the abnormal data.

The variable reconstruction approach for sensor faultisolation was proposed by Dunia et al. [2]. This approachassumes that each sensor may be faulty (in the case of a sin-gle fault) and suggests to reconstruct the assumed faultysensor using the PCA model from the remaining measure-ments. By examining the residuals given by PCA modelbefore and after reconstruction, we can determine thefaulty sensor.

DðjÞi ðkÞ denotes the index DiðkÞ calculated after recon-

struction of the jth variable. To show the propagation ofa fault f(k) in the direction nf on the index D

ðjÞi , let us con-

sider the expression of the measurement vector xj(k) (6):

xjðkÞ ¼ Gjðx�ðkÞ þ �ðkÞÞ þ Gjnf fðkÞ ð22Þ

If we consider the case when the faulty variable is thereconstructed one (j = f), then:

Gjnf ¼ 0 ð23Þ

Therefore, if the faulty variable is reconstructed (j = f),the index D

ðjÞi is in the control limit because the fault f is

eliminated (23). If the reconstructed variable is not faulty(j 5 f), the index D

ðjÞi being always affected by the fault f,

DðjÞi is outside its control limit. In summary, when a fault

has been detected, all the indices DðjÞi are computed, and

if DðjÞi 6

�d2

i;a, the jth sensor is considered as a faulty one.From the above analysis, we can define an identification

index based on DðjÞi , (j = 1, . . . ,m) as follows:

ADðjÞiðkÞ ¼ D

ðjÞi ðkÞ�d

2

i;a

ð24Þ

Thus at time k, if the faulty variable is the reconstructedvariable, then A

DðjÞiðkÞ is lower than one. If the recon-

structed variable is not faulty ADðjÞiðkÞ will be larger than

one. So, the jth variable for which ADðjÞiðkÞ is lower than

one, is the faulty variable.

3. Proposed sensor FDI scheme

The aim of the proposed FDI (Fault Detection and Iso-lation) scheme is to detect the faulty sensors using the indexDi and to identify the faulty sensor using the variablereconstruction approach. The measurement delivered bythe faulty sensor is then reconstructed using Eq. (6).

The algorithm for implementing the proposed faultdetection and isolation scheme is as follows:

(i) Perform a standard PCA on the data matrix X; deter-mine the model by a proper selection of the numberof PC.

(ii) Calculate Diði ¼ 1; . . . ;m� ‘Þ and determine the con-trol limit �d

2

i;a for each index under normal conditions.(iii) For each observation x(k):

(a) Select the residual subspace, i = 1.(b) Calculate DiðkÞ.

if DiðkÞ > �d2

i;a, a fault is detected, go to (c),else i = i + 1, go to (b),repeat until i = m � ‘.

(c) Identify the faulty variable:• The index A

DðjÞiðkÞ, calculated after the recon-

struction of the jth variable, which is lower than1 indicates that this variable is the faulty one.

• Suggest a replacement value for the faulty mea-surement (estimation of the fault amplitude) byreconstructing the faulty variable.

4. Application to an air quality monitoring network

In this section the proposed FDI scheme is applied tosensor fault detection, isolation and reconstruction of airquality monitoring network.

4.1. Problem settings

The air quality monitoring network AIRLOR workingin Lorraine, France, consists of more than twenty stationsplaced in rural, peri-urban and urban sites. Each stationconsists of a set of sensors, dedicated to the acquisitionof the following pollutants: carbon monoxide CO, nitrogenoxides NO and NO2, sulfur dioxide SO2 and ozone O3.Moreover, seven stations are dedicated to the acquisitionof additional meteorological parameters. The measuresare averages calculated over fifteen minutes in order tolimit spatial and temporal sampling problems.

The purpose is to detect functioning abnormalities ofthe sensors principally those of the ozone concentration(O3) and nitrogen oxides (NO and NO2). First, let usdescribe in some words the phenomenon. Ozone is a sec-ondary pollutant produced by complex photochemicalreactions between primary pollutants (mainly NO, NO2

and VOC) emitted into the atmosphere. Their concentra-tions also depend highly on the vertical and horizontalmovements of the atmosphere that are linked to the mete-orological conditions. On one hand, ozone is a secondarypollutant whose spatial distribution of the maximum val-ues is rather homogeneous at our local scale. On the otherhand, the nitrogen oxides are primary pollutants which aremore localized because their concentrations dependdirectly on the sources of emissions. PCA can handle thehigh dimension of the measurement network and the highdegree of correlation among some variables. However, wecan wonder about the performances of PCA since the phe-


nomenon of photochemical pollution exhibits a nonlinearand time-varying behaviour.

In this paper, only six neighbour measurement stationsare considered. The matrix X contains 18 variables, v1 tov18, corresponding, respectively, to ozone O3 and nitrogenoxides (NO2 and NO) collected on these six stations. Itshould be noted here that the paper presents a feasibilitystudy. Thus short period of measurements are only consid-ered here and we simulated sensor faults whose magnitudeis chosen according to measurement uncertainties providedby an expert of AIRLOR (approximately 15% of measure-ment for ozone). However, in the considered period, wehave weak, average and high levels of pollutants (Figs. 2–4). Here, only 1080 samples are used: the first 800 samplesto elaborate the PCA model and the last samples 801–1080to test the proposed fault detection and isolation scheme.

0 100 200 300 400 500

50

100

150

0 100 200 300 400 5000

50

100

0 100 200 300 400 5000

50

100

Fig. 2. Measurements and estimations o

Table 1Variances of the reconstruction error for the variables

i 1 2 3 4 5 6 7 8 9

‘ = 7 0.1 0.1 0.2 0.1 0.4 0.4 0.0 0.3 0.6

2 4 6 8 10

5

10

15

20

Fig. 1. Variances of the reconstruction error for the number of compo-nents in the PCA model.

4.2. PCA modelling

First, the correlation matrix of data matrix X was calcu-lated and an analysis of theses correlations was carried out.Only measurements of the ozone concentrations from thevarious sites have significant coefficients of correlation(higher than 85%) whereas measurements of NOx levelsfrom the various sites are poorly correlated. These firstresults are in agreement with our remarks on the phenom-enology of these pollutants.

A preliminary analysis of the data enabled us to notethat the relations between the variables could be approxi-mated by static linear relations. The analysis of eigenvaluesof the data correlation matrix and the cumulative percent-ages of correlation explained by each eigenvalue did notmake it possible to determine the number of componentto be retained because the eigenvalues have an exponentialdecrease. Fig. 1 shows the evolution of the variance of thereconstruction error according to the number of principalcomponents. This curve having a minimum for ‘ = 7, aPCA model with seven components was retained with apercentage of cumulative correlation equal to 91%.

In Table 1, the variances of the reconstruction errorusing seven principal components for each variable areindicated. It should be noted that all variables can bereconstructed because their respective variance is lowerthan 1. However, variables 9, 11, 12 and 15 having coeffi-cients higher than the other variables (in bold in the table),some detectability difficulties may appear for these vari-ables. These variables are the concentrations of NO andNO2 on urban sites or peri-urban. Broadly, the ozone con-

600 700 800 900 1000

600 700 800 900 1000

600 700 800 900 1000

f O3 level for the first three stations.

10 11 12 13 14 15 16 17 18

0.0 0.5 0.6 0.0 0.4 0.6 0.1 0.4 0.4

0 100 200 300 400 500 600 700 800 900 1000

100

200

300

0 100 200 300 400 500 600 700 800 900 1000

50100150200250

0 100 200 300 400 500 600 700 800 900 10000

50

100

Fig. 4. Measurements and estimations of NO level for the first three stations.

0 100 200 300 400 500 600 700 800 900 1000

100200300400500

0 100 200 300 400 500 600 700 800 900 100020406080

100120

0 100 200 300 400 500 600 700 800 900 1000

20406080

100

Fig. 3. Measurements and estimations of NO2 level for the first three stations.


centrations have variances of reconstruction error muchweaker than nitrogen oxide levels.

Figs. 2–4, respectively, present measurements and esti-mations of O3, NO2 and NO levels for the first three sta-tions, the estimations being given by the PCA model.These three stations are representative of the whole mea-surement network. Station 1 is a peri-urban station whichhas the highest ozone levels. Station 2 is an urban stationwhich has the lowest ozone levels and the highest nitrogenoxide levels. Station 3 behaves like the others peri-urbanstations.

By taking into account the nature of the considered pro-cess, the results are very satisfactory. With this PCA model,the concentrations of NO, O3 and NO2 are generally cor-rectly estimated for weak, average and high levels. How-ever, for some variables we can have modelling errors asshown in Figs. 2 (station 1) and 4 (station 3).

In conclusion, the linear PCA was able to model therelations between the various variables. However, as wecould note it, certain variables being less better estimatedthan others, we now will examine the effect of these model-ling errors on the detection and isolation of faulty sensors.

4.3. Sensor fault detection and isolation

Firstly, we will present some simple examples of sensorfault detection in order to show the results that can beobtained using the proposed fault detection, isolation andreconstruction scheme. The simulated fault is an additivestep whose amplitude is rather higher than the measure-ment uncertainties. Then we will present a systematic studyto compare the performances of the proposed index Di withthose of the SPE.

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Fig. 9. D1 with a fault on the variable v2.

0 5 10 150

0.5

1

1.5

2

Fig. 10. Localization of the faulty variable v2 using the index ADðjÞ1

.

800 850 900 950 1000 10500

100

200

300

400

500

Fig. 11. Reconstruction of the faulty variable v2 (NO2).

500 600 700 800 900 10000

0.1

0.2

0.3

0.4

500 600 700 800 900 10000

0.2

0.4

0.6

Fig. 7. Localization using the reconstruction approach: two indices DðjÞ2 ,

(j = 1,7).

800 850 900 950 1000 10500

50

100

150

Fig. 8. Reconstruction of the faulty variable v7 (O3).

0 200 400 600 800 10000

1

2

3

4

Fig. 5. SPE with a fault on the variable v7.

0 200 400 600 800 10000

0.2

0.4

Fig. 6. D2 with a fault on the variable v7.


4.3.1. Some examples

For the first example, a fault was simulated on the var-iable v7 from sample 801 to the end of the data file (Fig. 8).The magnitude of the fault amounts to 20% of the range ofvariation of variable v7. Fig. 5 shows the detection indexSPE for this fault. False alarms occur due to the modellingerrors. If the threshold of the SPE is increased in order tobe insensitive to these errors, it will impossible to detect thisfault. Using the proposed fault detection index, the simu-lated fault is detected without ambiguity on the D2 indexusing the two last principal components (Fig. 6).

To identify the fault, the reconstruction approach com-bined with the D2 index, clearly indicates that the variableresponsible for this out-of-control situation is v7. Fig. 7only indicates the time evolution of two indices D

ðjÞ2 ,

(j = 1,7), the others being similar to Dð1Þ2 . The index

Dð7Þ2 clearly shows that variable v7 was contaminated by a

fault because this index is the only one in its control limit.Variable v7 being identified as the faulty variable, then

we can reconstruct this variable in order to give a replace-ment value for the faulty measurement. Fig. 8 shows,for variable v7, the fault-free measurements, the faultymeasurements and the replacement values obtained by

Table 3Good detection ratio (GDR) for Di and SPE, good reconstruction ratio(GRR) for different variables

Variables GDR GRR (%)

Di (%) SPE (%)

v1 (O3) 75 4 94v2 (NO2) 92 4 78v3 (NO) 99 7 92v4(O3) 84 21 85v7 (O3) 89 11 90v8 (NO2) 88 45 86v10 (O3) 89 8 88v11 (NO2) 80 59 85v13 (O3) 97 6 88v14 (NO2) 97 89 85v15 (NO) 100 97 84v16 (O3) 82 15 90v17 (NO2) 70 30 86

20

40

60

80

100


reconstruction. It is clear that the reconstructed measure-ments are good estimations of the fault-free measurements.

For the second example, we will consider nitrogen diox-ide variables (NO2). Now, a fault was simulated on the var-iable v2 (NO2). The simulated fault is detected on the indexD1 using the last principal component (Fig. 9). The isola-tion of this faulty variable is depicted in Fig. 10, wherethe index A

Dð2Þ1

is lower than one which indicates that the

variable v2 is the faulty variable. Taking into account thenature of this variable, its reconstruction gives an accept-able estimation of the fault-free measurements as it isdepicted in Fig. 11.

We carried out the same simulations for the other sen-sors of O3, NO2 and NO. The detection/isolation perfor-mances obtained are completely identical for these sensors.

4.4. Performances of the sensor FDI procedure

The performances of the proposed sensor fault detectionand isolation procedure are mainly based on the effective-ness of the detection index Di and on the capacity of variablereconstruction approach. To evaluate its performances,we have presented here the results of three tests:

• comparison of the detectability conditions of the indicesDi and SPE,

• comparison of the good detection ratio of both indicesDi and SPE,

• test of the quality of the variable reconstructionapproach.

0 10 20 30 40 50 600

0 10 20 30 40 50 600

20

40

60

80

100

4.4.1. Fault detectability conditions

We extended the fault detectability conditions, intro-duced by Qin et al. [11] for SPE, to the index Di. The sen-sitivity to the fault #i,j of Di compared to SPE is defined asthe ratio of the fault amplitudes checking these detectabil-ity conditions of the two indices for the same variable j:

#i;j ¼�da

�di;a

~nðiÞj

��

~nj

�� ð25Þ

where ~nðiÞj ¼ ðI � Cm�iÞnj.

Table 2Sensitivity of the index Di compared to SPE

f1 f2 f3 f4 f5 f6 f7 f8 f9

#1 3.2 11.7 10.6 1.1 0.9 0.3 1.2 0.6 0.1#2 1.7 6.1 5.5 1.2 1.0 0.2 4.4 2.0 1.7#3 1.5 4.3 3.9 1.1 0.8 0.1 3.2 1.4 0.1#4 1.3 3.6 3.3 1.9 1.3 0.2 3.8 1.3 0.2#5 2.4 2.7 2.6 2.3 1.3 0.5 2.8 1.2 0.4#6 2.4 2.2 2.2 2.2 1.4 0.4 2.4 1.0 0.4#7 1.8 1.7 1.7 1.8 1.5 1.1 1.8 1.4 0.5#8 1.6 1.5 1.5 1.6 1.3 1.2 1.6 1.3 0.5#9 1.4 1.3 1.3 1.4 1.2 1.3 1.4 1.2 0.5#10 1.2 1.1 1.1 1.2 1.2 1.2 1.2 1.1 0.5

Table 2 presents the results of sensitivity calculation (25)for different variables. Thus, for example, with the samelevel of confidence (a = 95%), the D1 index can detect faultson the first variable 3.2 times smaller than those detectableby the SPE. From this table, it is clear that there is, for

f10 f11 f12 f13 f14 f15 f16 f17 f18

0.9 0.1 0.4 1.9 0.7 0.2 0.2 0.1 0.21.3 0.4 0.4 5.4 1.1 0.1 0.7 0.7 0.24.4 1.1 0.3 3.8 0.9 0.5 2.3 1.0 0.13.8 0.9 0.3 3.8 0.9 0.4 2.6 1.2 0.12.8 0.7 0.2 2.9 0.6 0.5 1.9 0.9 0.22.4 0.6 0.4 2.4 0.7 0.4 2.3 1.0 0.11.8 1.1 0.8 1.8 0.6 0.9 1.8 0.7 0.31.6 0.9 0.8 1.6 0.6 0.8 1.5 1.5 1.471.4 1.0 1.0 1.4 1.1 1.0 1.4 1.3 1.31.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

Fig. 12. Di and SPE good detection ratios for two variables.


every variable, at least one index Di that is more sensitive tothe fault than the SPE.

4.4.2. Detection ratio of both indices

The second test compares the good detection ratio forthe two indices SPE and Di. The good detection ratio isdefined as the percentage of good detection on a certainnumber of realizations. Therefore, the best detection indexis the one with the highest fault detection ratio.

In order to discriminate between the two indices, the testconditions are very difficult because the fault magnitudesare nearly equal to the estimation errors obtained withthe PCA model. The additive fault value is randomly gen-erated within the interval [10, 30] lg/m3. Table 3 gives thedetection ratios for different variables using the two indicesDi and SPE. It is very clear that the detection ratios for theindex Di are greater than those of the index SPE which con-firms the better fault sensitivity of the index Di by compar-ison with the SPE.

In order to plot the curves of good detection ratio foreach variable, fault magnitudes are randomly generatedwithin the interval [1, 60] lg/m3 for each realization. Twocharacteristic curves of the results obtained are depicted inFig. 12. All the curves obtained indicate that the good detec-tion ratios of the index Di are always greater than those ofthe index SPE whatever the fault magnitudes may be.

4.4.3. Quality of variable reconstruction

Table 3 presents the rate of good reconstruction for eachsensor (the value is well reconstructed when the recon-struction error is lower than the standard deviation ofthe estimation error of the considered variable). Thoseresults show that the variable reconstruction approachgives generally good replacement values for the faultymeasurements.

5. Conclusion

In this paper, a sensor fault detection and isolation pro-cedure based on principal component analysis is proposedto monitor an air quality monitoring network. Taking intoaccount the high redundancy among the process variables,this procedure is based on the variable reconstructionapproach in order to design the PCA model of the network,to isolate the faulty sensors and to estimate the fault ampli-tudes. To improve fault detection, a new detection index isproposed which monitors the last principal components(which have smallest variances) in different residual sub-spaces, starting with a single direction (the last principaldirection) and gradually increasing the dimension of theresidual to the full residual space. Therefore, the index Di

may be less sensitive to the modelling errors than theSPE. However, mathematically, it is difficult to prove thatthe new index has better fault sensitivity than the SPEbecause they perform fault detection in different residual

spaces where the sensitivities of the residuals to the faultsmay be different.

The proposed sensor fault detection, isolation andreconstruction scheme is successfully applied to data col-lected on an air quality monitoring network. For this appli-cation, the index Di improves the fault detection comparedto classical detection index SPE.

The results of this feasibility study, carried out overshort periods of measurements, must be confirmed, onone hand, over longer periods of measurements and, onthe other hand, by quantifying the performances of themethod by a Monte-Carlo simulation. For longer periodsof measurements it should be important to take intoaccount the nature of phenomena involved (nonlinearand time-varying behaviour). We plan to use extensionsof PCA such as nonlinear PCA [7] or recursive PCA [10].

Acknowledgement

The authors would like to thank AIRLOR which madeavailable the atmospheric pollution data used in thisarticle.

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An Improved PCA Scheme for Sensor FDI- Application to an Air Quality Monitoring Network

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