Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

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Chemometrics and Intelligent Laboratory Systems

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Fault detection in the Tennessee Eastman benchmark process usingdynamic principal components analysis based on decorrelatedresiduals (DPCA-DR)

Tiago J. Rato, Marco S. Reis ⁎CIEPQPF, Department of Chemical Engineering, University of Coimbra, Rua Sílvio Lima, 3030-790 Coimbra, Portugal

⁎ Corresponding author. Tel.: +351 239 798 700; faxE-mail address: marco@eq.uc.pt (M.S. Reis).

0169-7439/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.chemolab.2013.04.002

a b s t r a c t

a r t i c l e i n f o

Article history:Received 5 October 2012Received in revised form 25 March 2013Accepted 2 April 2013Available online 10 April 2013

Keywords:Multivariate statistical process controlPrincipal component analysisDynamic principal component analysisMissing data imputationTennessee Eastman benchmark process

Current multivariate control charts for monitoring large scale industrial processes are typically based on latentvariable models, such as principal component analysis (PCA) or its dynamic counterpart when variables presentauto-correlation (DPCA). In fact, it is usually considered that, under such conditions, DPCA is capable to effectivelydeal with both the cross- and auto-correlated nature of data. However, it can easily be verified that the resultingmonitoring statistics (T2 andQ, also referred by SPE) still present significant auto-correlation. Tohandle this issue,a set of multivariate statistics based on DPCA and on the generation of decorrelated residuals were developed,that present low auto-correlation levels, and therefore are better positioned to implement SPC in a more consis-tent and stable way (DPCA-DR). The monitoring performance of these statistics was compared with that fromother alternative methodologies for the well-known Tennessee Eastman process benchmark. From this study,we conclude that the proposed statistics had the highest detection rates on 19 out of the 21 faults, and are statis-tically superior to their PCA and DPCA counterparts. DPCA-DR statistics also presented lower auto-correlation,which simplifies their implementation and improves their reliability.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Current chemical process industries strive to improve the operationstandards of their processes and quality levels of their products, withthe central goal of reducing the variability of the main quality featuresaround their target value [1]. Statistical Process Control (SPC) providesa toolbox for conducting such activities, where control charts have aparticularly important role (so much so, that quite often the two desig-nations, SPC and control charts, are used interchangeably). The goal ofcontrol charts is to provide a simple and objective way to monitor theprocess variability over time, in order to verify, at each instant, whetherit remains “normal”, i.e., in a state of statistical control, or whether aspecial cause of variation has occurred, driving it to an out of statisticalcontrol state [1,2]. The state of statistical control is essentially charac-terized by process variables remaining close to their desired or averagelevels, affected only by common causes of variation, i.e., variationsources affecting the process all time and that are essentially unavoid-able within the process normal operation conditions [1].

In these context, several SPC charts were developed for univariate pro-cesses, such as the classical Shewhart's control charts [1], CUSUM [3] andEWMA [4], and then, for multivariate (Hotelling's T2 control chart [5],MCUSUM [6], MEWMA [7]) and megavariate (PCA-SPC, [8–10]) systems,as data and computational power become increasingly available. ThePCA-SPC control chart, is based on a latent variables model (Principal

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Component Analysis, PCA), whose ability to deal with a large number ofhighly correlated variables is well known. It uses two complementarymonitoring statistics, one of them for monitoring the variability withinthe PCA subspace (the Hotelling's T2 applied to the first p latent vari-ables, p being the process pseudo-rank)while the other follows the var-iability around such subspace, being a function of the projectionresiduals, usually referred as Q or square predicted error, SPE.

However, as for all previous methodologies, PCA-SPC tacitly as-sumes that the underlying data generation process is i.i.d., meaningin particular that the process mean vector is constant over time,therefore excluding any auto-correlated or non-stationary behavior.This is currently a serious limitation, which strongly hinders the prac-tical application of approaches based upon the i.i.d. assumption, dueto the mass, energy and momentum inertial effects presented inmost industrial systems, coupled with the high sampling rates thatare currently easily achieved with modern process instrumentationand acquisition systems. To address this issue, Ku et al. [11] proposedan SPC procedure that extends PCA-SPC, based on a dynamical ver-sion of principal component analysis, called dynamic principal com-ponent analysis (DPCA). DPCA includes time-shifted versions of theoriginal variables, in order to accommodate and tacitly model the dy-namic behavior of variables within the same PCAmodel. Unfortunate-ly, one can easily verify that this method still leads to auto-correlatedstatistics, meaning that the fundamental problem raised by data auto-correlation still remains to be properly addressed.

In order to handle this issue, Rato and Reis [9] recently studiedseveral combinations of approaches to deal with data correlation

102 T.J. Rato, M.S. Reis / Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

and autocorrelation, including DPCA, PLS, Time Series modelling anddecorrelated residuals based on missing data imputation techniques,in a total of 22 monitoring statistics, most of them being new. Fromthis screening study, a combination of DPCA and decorrelated resid-uals based on missing data imputation (DPCA-DR) stands out byits potential for dealing with data cross- and auto-correlation, andthe lower levels of auto-correlation in the monitoring statistics, im-plying that the dynamical behavior is being properly described andincorporated in the methods' model structure. Furthermore, for thesystems studied, this method has also shown better monitoringperformances when compared to its current counterparts. After suchscreening and characterization work, and given the good performancesachieved as well as the stable monitoring behavior of the DPCA-DR sta-tistics, it is now both important and opportune to test them in alarge-scale benchmark data set, in order to consolidate the preliminaryresults obtained in an independently generated data set. The case studyselected is the Tennessee Eastman process [12]. This case study is awidely adopted and cited benchmark in Multivariate Statistical ProcessControl and Fault Detection and Diagnosis, and therefore is especiallysuitable to test the proposed approach and to make our results compa-rable with those obtained from other methodologies proposed.

The rest of this article is organized as follows. In the next section,we describe the current multivariate statistics and latent variablemodels used in this study (PCA, and DPCA) as well as our proposedmethod (DPCA-DR). Next, we present and discuss the results obtainedfrom their application to the Tennessee Eastman benchmark process,after properly defining the criteria that provide the basis for compari-son. Finally, we summarize the contributions presented in this paperand present the main conclusions.

2. Methods

In this section, we briefly revise the main control chart proceduresused currently for performing SPC on multivariate and megavariateprocesses. These methods will be used as reference against whichour methodology based on DPCA-DR will be compared. We alsopresent the new statistics based on this procedure.

2.1. Multivariate statistical process control

The natural extension of the univariate Shewhart control chart formonitoring multivariate process is the Hotelling's T2 chart [12]. Thischart assumes the process to be i.i.d., following amultivariate normal dis-tribution, and the monitoring statistic, for single observations samples isjust the Mahalanobis distance between each multivariate observationand the overall referencemean. Assuming themean and covariancema-trix to be known, the monitoring statistic has the form [12–14]:

χ20 ¼ x−μð ÞTΣ−1 x−μð Þ ð1Þ

where xm × 1 is a measurement vector, μm × 1 is the population meanvector and Σm × m is the population covariance matrix. Under multivari-ate normal conditions, this statistic follows a centralχ2 distributionwithm degrees of freedom. Therefore, a multivariate χ0

2 control chart canbe constructed by plotting χ0

2 versus time with an upper controllimit (UCL) given by χα,m

2 where α is an appropriate level of significance(e.g. α = 0.01) [2,12].

When the in-control mean vector μ and the covariance matrix Σare unknown, they can be estimated from a sample of n past multi-variate observations, using the usual well-known unbiased estima-tors of these population parameters, namely the sample mean andthe sample covariance matrix [2]:

x ¼ 1n

Xni¼1

xi ð2Þ

S ¼ 1n−1

Xni¼1

xi−xð Þ xi−xð ÞT: ð3Þ

In this case, when newmultivariate observations are obtained, theHotelling's T2 statistic is given by [2,12,14],

T2 ¼ x−xð ÞTS−1 x−xð Þ ð4Þ

whose control chart has now the following upper control limit (UCL)[2,14–16]:

UCL ¼ m n−1ð Þ nþ 1ð Þn2−nm

Fα;m;n−m ð5Þ

where, Fα,m,n − m is the upper α percentile of the F distribution withmand n − m degrees of freedom. This chart is just a representative(perhaps the simplest and most well-known) of the charts that canbe applied to multivariate systems of limited size (order of a dozenor less) and without strong problems of collinearity, otherwise the in-version of the covariance matrix would be highly unstable or evenimpossible in case of full redundancy or rank deficiency. For largescale systems, the control charts presented in the next section offera more stable and effective solution.

2.2. Megavariate statistical process control

When the number of measured variables (m) becomes large(order of several dozens or higher), alternative approaches must beused, mostly due to the problems raised by the inversion of the co-variance matrix in the Mahalanobis distance computation. A commonsolution for dealing whit this issue consists of using a latent variablemodelling framework, developed for these types of processes,whose parameters can be estimated with simple and stable methods.Examples of suchmodels are principal component analysis (PCA) [12]and partial least squares (PLS) [16], the former for problems involvinga single block of variables and the latter for those where two blocks ofvariables need to be explicitly and simultaneously handled.

Regarding the analysis of problems with a single block of variables(the situation covered in this article), the use of PCA allows a reduc-tion of the dimensionality of the space under monitoring, but thatstill preserves the essential features of the original data variability.This is achieved by transforming the original variables into a newset of uncorrelated variables, called the principal components (PCs).The first principal component (PC) of x is defined as the linear combi-nation t1 = p1

Tx with maximum variance subject to ‖p1‖2 = 1. Thesecond PC, is the linear combination defined by t2 = p2

Tx with maxi-mum variance subject to ‖p2‖2 = 1, and to the condition that it mustbe uncorrelated with (orthogonal to) the first PC (t1). Additional PCsare similarly defined, decomposing the entire observation matrix,Xn × m, as [12,14]:

X ¼ TPT þ E ð6Þ

where, Tn × p is the matrix of scores, Pm × p is the loading matrix, andEn × m is the residual matrix that contains the accumulated contribu-tion of the last principal components, with small (or residual) contri-butions for explaining the variability exhibited by the X matrix.

After applying the PCA decomposition to a reference data set, aHotelling's T2 statistic for future observations can be obtained fromthe first p PCs by,

T2PCA ¼

Xpi¼1

t2iλi

¼ xTPΛ−1p PTx ð7Þ

where Λp is a diagonal matrix with the first p eigenvalues in the maindiagonal and ti is the new score for the ith PC.

103T.J. Rato, M.S. Reis / Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

The upper control limit (UCL) of TPCA2 statistic is given by [11,17]:

UCL ¼ p n−1ð Þ nþ 1ð Þn2−np

Fα;p;n−p ð8Þ

where Fα,p,n − p is the upper α percentile of the F distribution with pand n − p degrees of freedom. Since TPCA

2 only monitors the variabil-ity within the PCA subspace, spanned by the first p PCs, it must becomplemented by a residual or lack of fit statistic, that accounts forthe variation not captured by the PCA model, and monitored byTPCA2 . This is achieved by computing the squared prediction error

(SPE) of the residuals of a new observation (em × 1), also known asthe Q statistic, which is defined as [11]

Q ¼ eTe ¼ x−x̂ð ÞT x−x̂ð Þ ¼ xT I−PPT� �

x ð9Þ

where x̂ is the projection of a given observation onto the PCA sub-space. This statistic is usually quite useful as it is sensitive to specialevents that cause the data to move away from the PCA subspacewhere normal operation data is mostly concentrated. The UCL forthis statistic is given by [9],

UCL ¼ θ1cα

ffiffiffiffiffiffiffiffiffiffiffiffiffi2θ2h

20

qθ1

þ 1þ θ2h0 h0−1ð Þθ21

0@

1A

1=h0

ð10Þ

where

θi ¼Xnj¼pþ1

λij; i ¼ 1;2;3 ð11Þ

h0 ¼ 1−2θ1θ33θ22

ð12Þ

and p is the number of retained principal components; cα is the stan-dard normal variable corresponding to the upper 1 − α percentile.

2.3. Megavariate statistical process control of dynamic processes

Ku et al. [11] showed that a linear time series relationship can bedescribed by a conventional PCAmodel, through an implicit multivar-iate autoregressive model (VAR; processes containing moving aver-age terms can also be approximated by finite VAR models). This isachieved by the additional inclusion of time-shifted versions of theoriginal variables, as follows:

x lð Þi ¼ xT

i xTi−1 ⋯ xT

i−l

� �T ð13Þ

where i represents an arbitrary sampling instant, l is the number oflags (or time-shifts) to be considered and xi(l) is the resulting aug-mented vector of variables for the instant i. The augmented matrixis obtained by the straightforward superposition of these lines ofaugmented observations, xi(l). Then, PCA is applied over such aug-mented matrix, providing a description of not only the (static)cross-correlations among variables but their auto-correlations andlagged cross-correlations, due to the additional incorporation oftime-shifted variables. Furthermore, by properly choosing the num-ber of lags to include, l, both the static and dynamic relations shouldappear in the noise subspace corresponding to the PCs with smallvariance [11].

2.4. Megavariate statistical process control of dynamic processes: theDPCA-DR approach

The introduction of time-shifted variables on DPCA has the pur-pose of describing the autocorrelation and lagged cross-correlations

present in data, besides the cross-correlation features. However,looking to the behavior of the resulting DPCA T2 and Q statistics,one can verify that they, in fact, still present autocorrelation, andthe underlying problem of removing it from the monitoring statistics,so that they can be handled with simple charts based on the i.i.d. as-sumption, was not properly solved yet. In order to mitigate thisissue, Rato and Reis [9] have recently proposed a new methodologythat combines a DPCA model and decorrelated residuals obtainedfrom a conditional data imputation technique, in order to obtainbetter time-decorrelated statistics, in a simple way, within the samemodelling approach, without the need to resource on further time-series modelling in order to compensate for the remaining dynamicpatterns of the statistics. The underlying reasoning is the following.In DPCA, a matrix with current and past measurements is build. Ateach new incoming observation, i, a new observed score and projec-tion can be computed. If one assumes that the current multivariateobservation vector is missing, the associated values for the currentscores and projections can still be estimated from past data using aconditional missing data imputation technique for PCA (in this case,DPCA). This essentially amounts to perform a one-step-ahead predic-tion of the scores and observations, obtained with an implicit latentvariable VAR model estimated by DPCA. We have verified that theresiduals obtained from the differences between the observed andestimated scores and projections are almost serially decorrelated,meaning that such residuals are ready to be monitored by simplecontrol charting procedures. The conditional data imputation methodchosen was the conditional mean replacement [18,19]. In this meth-od, a measurement vector with missing data is rearranged, withoutloss of generality, as follows,

xT ¼ x#T x�T� � ð14Þ

where x# denotes themissingmeasurements and x∗ the observed ones.Correspondingly, the P matrix is also partitioned as PT ¼ P#T P�T� �

.The missing measurements (x#) can be estimated by applicationof the Expectation–Maximization (EM) algorithm, where at each it-eration, their values are replaced by the expected ones from theconditional normal distribution given the known data and thecurrent estimate of the mean and covariance matrix (expectationstage), that is [18],

x̂# ¼ E x#jx�; x ; S

� �ð15Þ

which will then be used to update the model (maximization stage),and so forth, until a convergence criteria regarding the change onthe successive solutions, is met. In our case, we assume that aPCA model is already available (the DPCA model built from refer-ence data), and therefore, only the expectation step of the EM algo-rithm is required, in order to calculate the estimates for the missingmeasurements. Substituting P into the expression for S results in[18],

S ¼ S11 S12S21 S22

� �¼ P#ΛP#T P#ΛP�T

P�ΛP#T P�ΛP�T

� �: ð16Þ

Using this expression for S, the conditional expectation of themissing measurements is simply given by [18]

x̂# ¼ S12S−122 x� ¼ P#ΛP�T P�ΛP�T� �−1

x�: ð17Þ

Table 1Process faults for the Tennessee Eastman process simulator.

Variable Description Type

IDV(1) A/C feed ratio, B composition constant (Stream 4) StepIDV(2) B composition, A/C ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss — reduced availability

(Stream 4)Step

IDV(8) A, B, C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) UnknownIDV(17) UnknownIDV(18) UnknownIDV(19) UnknownIDV(20) UnknownIDV(21) The valve for Stream 4 was fixed at the

steady state positionConstant position

104 T.J. Rato, M.S. Reis / Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

The estimated measurements can then be used in the score calcu-lations along with the observed data, as if no measurements weremissing. For PCA this leads to [18]

t̂ ¼ PT1:p

x̂#

x�

� �¼ PT

1:p

P#ΛP�T P�ΛP�T� �−1

x�

P�ΛP�T� �

P�ΛP�T� �−1

x�

264

375

¼ PT1:p

P#

P�

� �ΛP�T P�ΛP�T

� �−1x�

¼ PT1:pPΛP

�T P�ΛP�T� �−1

x�

¼ I 0½ �ΛP�T P�ΛP�T� �−1x�

ð18Þ

where P1 : p is the matrix of the first p eigenvectors, I is an (p × p)identity matrix and 0 is an (p × (m − p)) matrix of zeros.

The same approach can be applied to DPCA for generatingdecorrelated residuals. In this case, we consider that the current vari-ables, xiT in Eq. (13) are unknown. Therefore, the application of thismethodology will give us an estimate of the scores that best agreewith the last l known measurements. Given such estimated scores,we have defined the following Hotelling's T2 statistic:

T2PREV ¼ t� t̂

TS−1

t�t̂t� t̂ ð19Þ

where St�t̂

is the sample covariance matrix of the difference betweenthe observed and estimated scores, t−t̂

, that monitors the DPCA

reference subspace. Likewise, a monitoring statistic for the residualsubspace can be defined as:

T2RES ¼ rTS−1

r r ¼ x−Pt̂ TS−1

r x−Pt̂ ð20Þ

Condenser

1

2

3

A feed

D feed

E feed

A/B/C feed

6

7

CWS

CWR

13

CWS

CWR12

Reactor

4

Fig. 1. The Tennessee Eastm

where Sr is the sample covariance matrix of the residuals in thereconstructed data, obtained with the estimated scores r ¼ x−Pt̂

.

These two statistics present low levels of autocorrelation and very in-teresting detection performances, as will be illustrated in the follow-ing section for the Tennessee Eastman case study.

Vap/Liquid Separator

Stripper

Compressor

5

10

P-14

11

Purge

Steam

Cond

Product

8

9

an process flow sheet.

Table 2Fault detection rates for the various methods under study, regarding each faultyscenario (a description of each fault can be found in Table 1). The best results achievedfor each fault are signalled in boldface format.

Fault PCA DPCA DPCA-DR

T2 Q T2 Q TPREV2 TRES

2

1 0.991 0.995 0.990 0.994 0.996 0.9982 0.985 0.984 0.984 0.981 0.985 0.9833 0.036 0.006 0.035 0.010 0.021 0.0164 0.218 0.980 0.165 0.999 0.998 0.9995 0.257 0.217 0.293 0.228 0.999 0.9996 0.989 0.999 0.989 0.999 0.999 0.9997 0.999 0.999 0.986 0.999 0.999 0.9998 0.974 0.968 0.973 0.974 0.985 0.9819 0.034 0.010 0.030 0.002 0.020 0.01010 0.367 0.154 0.439 0.172 0.956 0.93311 0.414 0.638 0.340 0.829 0.965 0.86512 0.985 0.925 0.990 0.964 0.998 0.99813 0.943 0.950 0.943 0.950 0.958 0.95614 0.988 0.999 0.990 0.999 0.998 0.99915 0.035 0.007 0.059 0.009 0.385 0.04716 0.174 0.137 0.217 0.145 0.976 0.94517 0.787 0.905 0.790 0.953 0.976 0.97518 0.893 0.901 0.890 0.898 0.905 0.90019 0.115 0.059 0.046 0.298 0.971 0.84320 0.340 0.423 0.408 0.493 0.908 0.91621 0.362 0.414 0.429 0.409 0.539 0.577

Figures in bold represent the best scores achieved in the analysis of each faultcondition. Therefore, they indicate the methods that performed better in each case.

105T.J. Rato, M.S. Reis / Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

3. A comparison study based on the Tennessee Eastmanbenchmark process

In order to test and compare the monitoring features and perfor-mance of the proposed methodology, we have selected an application

10 20 30 400

50

100

150

200

Time (hr)

T2 P

CA

T2 D

PC

AT

2 PR

EV

10 20 30 400

100

200

300

Time (hr)

10 20 30 400

1

2

3

4x 10

6

Time (hr)

Fig. 2. The multivariate statistics under test for Fault 5: PCA statistics (first or top row), D

scenario which has been widely used in process monitoring and faultdetection studies: the Tennessee Eastman benchmark process. Thiscase study not only provides a challenging testing environment forthe specific comparison study carried out in this work, but also en-ables and simplifies the extension of the comparison scope to othermethods tested in the same system, such as [11,20–23]. A model ofthis process was developed by Downs and Vogel [10], consisting offive major transformation units, which are a reactor, a condenser, acompressor, a separator, and a stripper, as shown in Fig. 1. Fromthis model, 41 measurements (XMEAS) are generated along with 12manipulated (XMV) variables. A total of 21 different process upsetsare simulated for testing the detection ability of the monitoringmethods, as presented in Table 1 [20,24]. In the current study wehave conducted our analysis with the data set used by Russell et al.[25] (available at http://web.mit.edu/braatzgroup), where the Ten-nessee Eastman process is controlled with the approach suggestedby Lyman and Georgakis [24]. Each data set contains 960 observationscollected at a sample interval of 3 min and the faults were introduced8 h after the simulations start. All the manipulated and measurementvariables, except the agitation speed of the reactor's stirrer (which isalways constant), were collected, giving a total of 52 variables.

A data set with no faults, representing normal operation condi-tions was used to estimate the reference PCA, DPCA and DPCA-DRmodels. The number of principal components for PCA and DPCA wasdetermined by parallel analysis and the number of lags was selectedby the approach proposed by Ku et al. [11]. Using these methods weconstructed a PCA model with 17 PCs and a DPCA model with 3 lagsand 29 PCs. These results are in good accordance with those obtainedby Russell et al. [20]. For selecting the number of lags for the DPCA-DRmodel we have used the algorithm proposed by Rato and Reis [26].This algorithm is based on a succession of singular value decomposi-tion problems, and subsequent analyses of an auxiliary function from

T2 R

ES

10 20 30 400

50

100

Time (hr)

QP

CA

QD

PC

A

10 20 30 400

100

200

300

400

Time (hr)

10 20 30 400

2

4

6

8x 10

4

Time (hr)

PCA statistics (second or middle row) and DPCA-DR statistics (third or bottom row).

10 20 30 400

20

40

60

80

Time (hr)

T2 P

CA

T2 D

PC

AT

2 PR

EV

T2 R

ES

QP

CA

QD

PC

A

10 20 30 400

20

40

60

Time (hr)

10 20 30 400

50

100

Time (hr)10 20 30 40

50

100

150

200

Time (hr)

10 20 30 400

2

4

6

8x 10

5

Time (hr)10 20 30 40

0

2000

4000

6000

Time (hr)

Fig. 3. The multivariate statistics under test for Fault 10: PCA statistics (first or top row), DPCA statistics (second or middle row) and DPCA-DR statistics (third or bottom row).

106 T.J. Rato, M.S. Reis / Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

which the lags for each variable can be set. Table A.1 summarizes thenumber of lags considered for each variable obtained with thismethodology, which led to a model with 69 PCs.

We would like to point out that the direct use of the theoretical sig-nificance levels for establishing the statistical control limits for the

Table 3p-Values for the paired t-test involving the detection rates obtained with method A(see first column) and method B (see first line), on all simulated faults, along with thesignal of the test statistic, i.e. sign(t0). For instance, a plus (+) signal, indicates that methodA leads to higher detections rates, on average, when compared tomethod B. Values in boldindicate p-values lower than 0.05 (i.e., statistically significant differences at this level).

A B PCA DPCA DPCA-DR

T2 Q T2 Q TPREV2 TRES

2

PCAT2 0.388 0.414 0.152 0.002 0.003

(−) (−) (−) (−) (−)Q 0.388 0.540 0.046 0.004 0.008

(+) (+) (−) (−) (−)

DPCAT2 0.414 0.540 0.257 0.002 0.004

(+) (−) (−) (−) (−)Q 0.152 0.046 0.257 0.007 0.013

(+) (+) (+) (−) (−)

DPCA-DRTPREV2 0.002 0.004 0.002 0.007 0.115

(+) (+) (+) (+) (+)TRES2 0.003 0.008 0.004 0.013 0.115

(+) (+) (+) (+) (−)

Another advantage of the DPCA-DR method is the lower autocorrelation levels of its statis-tics, where much of its success may lie, as this characteristic makes the DPCA-DR statisticsmore reliable and consistent with the type of control charts used to monitor them (Fig. 4).

various methods may lead to widely different observed false alarmrates, which distorts any comparison study on the method detectionperformances. This undesirable effect can be removed by manipulatingthe theoretical significance level of the control limits in such a way thatthe effectively observed performance for all methods under normal op-eration conditions (i.e., their false alarm rate), becomes equal. Only insuch condition can all the methods be properly compared with futuretest data. Therefore, the UCL for the various methods was set to a falsealarm rate of 1% under normal operation conditions, by trial and error,on a second data set with no faults.With this preliminary but importantprocedure concluded, the fault detection rates for all the methods re-garding each fault were finally determined. A summary of the resultsobtained is presented in Table 2.

From the analysis of Table 2, it is possible to verify that the DPCA-DRmonitoring statistics tend to present highest fault detection rates. Infact, TPREV2 was the best statistic in 14 out of 21 faults, and TRES

2 in 9 ofthem. Globally, they were capable to detect 19 of 21 faults, failing onlyin the detection of faults number 3 and 9, where all methods alsopresent problems. Fault number 15 is another example of a fault diffi-cult to detect, but where the statistic TPREV

2 achieved the best score.The lower capability for detecting these three specific faults wasexpected, as other methods reported in the literature (e.g. PCA, DPCAand CVA) also fail to detect them [20].

In order to better illustrate the monitoring behavior of the methodsunder analysis, we present in Figs. 2 and 3 the control charts for someof the process faults. From these representations it is possible to clearlyobserve that only the DPCA-DR statistics present a consistent out of con-trol state in both statistics, simultaneously (TPREV2 and TRES

2 , see Fig. 2). Thisis a relevant issue, since the PCA and DPCA statistics may lead to the erro-neous conclusion that the process has returned to their normal operationconditions and it is no longer under the effect of a fault. In the case of Fault10 (Fig. 3) only the DPCA-DR statistics signals an out of control state

0 5 10 15 20-0.5

0

0.5

1

Lag

r k r kr k

r k

r kr k

T2PCA

T2DPCA

T2PREV

0 5 10 15 20-0.5

0

0.5

1

Lag

QPCA

QDPCA

QRES

0 5 10 15 20-0.5

0

0.5

1

Lag0 5 10 15 20

-0.5

0

0.5

1

Lag

0 5 10 15 20-0.5

0

0.5

1

Lag0 5 10 15 20

-0.5

0

0.5

1

Lag

Fig. 4. Auto-correlation plots for the monitoring statistics when the process is under normal operation conditions (data set with no faults). The proposed DPCA-DR statistics presentthe lowest levels of correlation among all the studied ones. PCA statistics — first or top row, DPCA statistics — second or middle row , DPCA-DR statistics — third or bottom row.

Table A.1Number of lags for each variable obtained with Rato and Reis [26] lag selection method.

XMEAS(1) 17 XMEAS(14) 4 XMEAS(27) 17 XMEAS(40) 12XMEAS(2) 17 XMEAS(15) 17 XMEAS(28) 13 XMEAS(41) 17XMEAS(3) 8 XMEAS(16) 12 XMEAS(29) 3 XMV(1) 17XMEAS(4) 17 XMEAS(17) 17 XMEAS(30) 17 XMV(2) 17XMEAS(5) 17 XMEAS(18) 17 XMEAS(31) 17 XMV(3) 17XMEAS(6) 16 XMEAS(19) 17 XMEAS(32) 8 XMV(4) 17XMEAS(7) 17 XMEAS(20) 17 XMEAS(33) 8 XMV(5) 15XMEAS(8) 15 XMEAS(21) 17 XMEAS(34) 17 XMV(6) 16XMEAS(9) 17 XMEAS(22) 17 XMEAS(35) 17 XMV(7) 17XMEAS(10) 17 XMEAS(23) 17 XMEAS(36) 17 XMV(8) 17XMEAS(11) 16 XMEAS(24) 17 XMEAS(37) 17 XMV(9) 16XMEAS(12) 17 XMEAS(25) 17 XMEAS(38) 17 XMV(10) 17XMEAS(13) 17 XMEAS(26) 17 XMEAS(39) 4 XMV(11) 17

107T.J. Rato, M.S. Reis / Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

during the total duration of the fault, while the PCA and DPCA statisticsonly became out of control when the data exceeds their normal values.

In order to confirm the overall superiority of the DPCA-DR statisticsin this case study, we have conducted paired t-tests between all the sta-tistics presented in Table 3 (as they were implemented over the samedata sets, they are paired by design in the comparison study). The teststatistic is given by t0 ¼ D= sD=

ffiffiffin

p , where D is the sample average of

the differences between two methods under analysis in the n differenttesting conditions,D1, D2,…, Dn, and sD is the sample standard deviationof these differences [27]. From this analysis it can be concluded that,with a 5% significance level, the DPCA-DR statistics are indeed signifi-cantly better than all the PCA and DPCA statistics.

4. Conclusions

In this paper we have presented a methodology for conductinglarge-scale process monitoring of dynamical systems, called DPCA-DR,and compared its performance against other well-known methodolo-gies used in the same application context, namely PCA and DPCA. Thecomparison study was conducted using the Tennessee Eastman bench-mark process and all faults considered in its design. From the analysis ofthe results obtained, we can conclude that the DPCA-DR statistics weresuperior, in a statistically significant sense, to the other ones, achievingthe highest detection scores in 19 out of the 21 faults.

On the other hand, the DPCA-DR statistics also presented the low-est auto-correlation levels and were able to sustain the out-of-controlsignals during the whole faults duration, while PCA and DPCA statis-tics often return to their in-control regions leading to a false senseof normality. Consequently, the DPCA-DR statistics seems to bemore effective, reliable and consistent regarding their counterparts

tested in this study, features that make them a viable alternative tocurrent monitoring statistics.

Acknowledgments

Tiago J. Rato acknowledges the Portuguese Foundation for Scienceand Technology for his PhD grant (grant SFRH/BD/65794/2009).Marco S. Reis also acknowledges financial support through projectPTDC/EQU-ESI/108374/2008 co-financed by the Portuguese FCT andEuropean Union's FEDER through “Eixo I do Programa OperacionalFactores de Competitividade (POFC)” of QREN (with ref. FCOMP-01-0124-FEDER-010397).

Appendix A

108 T.J. Rato, M.S. Reis / Chemometrics and Intelligent Laboratory Systems 125 (2013) 101–108

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