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Computers and Chemical Engineering 71 (2014) 307–322

Contents lists available at ScienceDirect

Computers and Chemical Engineering

j ourna l ho me pa g e: www.elsev ier .com/ locate /compchemeng

on-causal data-driven monitoring of the process correlationtructure: A comparison study with new methods

iago J. Rato, Marco S. Reis ∗

IEPQPF, Department of Chemical Engineering, University of Coimbra, Rua Sílvio Lima, 3030-790 Coimbra, Portugal

r t i c l e i n f o

rticle history:eceived 12 May 2014eceived in revised form 27 August 2014ccepted 3 September 2014vailable online 15 September 2014

eywords:

a b s t r a c t

Current approaches for monitoring the process correlation structure lag significantly behind the effective-ness already achieved on the detection of changes in the mean levels of process variables. We demonstratethat this is true, even for well-known methodologies such as MSPC-PCA and related approaches. Onthe other hand, data-driven process monitoring approaches are typically non-causal and based on themarginal covariance between process variables. We also show that such global measure of associationis unable, by design, to effectively discern changes in the local correlation structure of the system and

rocess monitoring of the correlationtructureultivariate dynamic processes

ensitivity enhancing data transformationsartial correlationarginal correlation

propose, for the first time, the explicit use of partial correlations in process monitoring. As a second con-tribution, we introduce the use of sensitivity enhancing data transformations (SET) with the ability tomaximize the detection ability of all monitoring procedures based on (partial or marginal) correlation,and show how they can be constructed. Results confirm the added-value of the proposed monitoringscheme.

© 2014 Elsevier Ltd. All rights reserved.

. Introduction

The optimized and safe operation of current industrial processesequires the simultaneous monitoring of a large number of multi-le related variables. A variety of multivariate statistical processontrol (MSPC) methods, namely control charts, have been devel-ped and applied in order to determine whether the process isnly subject to common causes of variability or if a special orssignable cause, related with some abnormality inside or outsidehe process, has occurred. Analysing the literature, one can verifyhat most multivariate process monitoring methodologies devel-ped so far, including the latent variables methodologies (Jackson,959; Jackson and Mudholkar, 1979; Ku et al., 1995; Li et al., 2000;acGregor et al., 1994; Wise and Gallagher, 1996) and state-space

r time-series approaches (Negiz and C inar, 1997a,b), are essen-ially non-causal and focused on detecting changes in the process

ean (Abbasi et al., 2009; Ghute and Shirke, 2008; Yeh et al., 2006;en et al., 2012). The important complementary problem of mon-

toring the process correlation structure has been almost absent

rom the research efforts, creating a significant gap in what regardso the high level of performance achievable today in detectinghanges in the mean levels of the process variables, contrasting

∗ Corresponding author. Tel.: +351 239 798 700; fax: +351 239 798 703.E-mail address: marco@eq.uc.pt (M.S. Reis).

ttp://dx.doi.org/10.1016/j.compchemeng.2014.09.003098-1354/© 2014 Elsevier Ltd. All rights reserved.

with the rather limited ability to effectively signal out perturba-tions in the correlation structure. In fact, even though this situationhas been pointed out several times, including in the recent litera-ture, as for instance by Xiao (2013) and Liu et al. (2013), only a fewcontributions have been put forward for explicitly monitoring theprocess correlation structure (also called multivariate dispersion).

Therefore, this article is devoted to the analysis and develop-ment of new process monitoring methods specifically designedfor detecting changes in the process correlation structure and forfinding out, in a second stage, the associated root causes. Withthis effort, we aim to significantly contribute to shorten the gapbetween the expected performances of the approaches availablefor monitoring the variables mean levels and those for monitor-ing multivariate dispersion, providing a more balanced set of toolsto practitioners that have to deal with both aspects in real worldproblems.

In this regard, some causal methodologies were proposed, suchas those developed by Bauer et al. (2007), that uses transfer entropyin order to identify the directionality of the fault’s propagation path,and the methodology proposed by Yuan and Qin (2012), where acombination of Granger causality and principal component analy-sis is employed to perform feature selection for locating the origin

of faults with oscillatory characteristics. However, these method-ologies are strongly oriented to fault diagnosis rather than faultdetection, which is an obvious pre-requirement before their appli-cation. Chiang and Braatz (2003) also suggested the combined use

3 Chemi

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08 T.J. Rato, M.S. Reis / Computers and

f the Kullback–Leibler information distance and the process causalap to detect and diagnose faults. Yet, this approach requires the

nowledge of the causal map in the form of a digraph. Thus, givenheir pervasive use, shorter development times, good performancend the fact of not relying on a priori information about the sys-em structure, this paper will focus on the class of non-causalpproaches for monitoring the process correlation structure.

Among the non-causal procedures specifically developed foronitoring the process covariance, the most widely adopted ones

re based on the generalized variance, for which several approachesere proposed, namely by Alt (1984), Aparisi et al. (2001) andjauhari (2005). However, the generalized variance is a rathermbiguous measure of multivariate variability, as quite differentovariance matrices can lead to similar values for the determinant.ther approaches are based on the likelihood ratio test (LRT), as

or example those found in the works of Alt and Smith (1988) andevinson et al. (2002). More recently, Yen and Shiau (2010) pre-ented a control chart, based on LRT, specifically designed to detectn increase in process dispersion, which was later on extended toariations in both directions (increase and decrease) (Yen et al.,012).

Analysing all these contributions for non-causal monitoring ofhe correlation structure it is possible to verify that all of themre strictly based on the process general information describedy the covariance matrix, also referred as the marginal covari-nce matrix. Even multivariate statistical process control based onrincipal component analysis (MSPC-PCA) (Jackson, 1959; Jacksonnd Mudholkar, 1979; Kresta et al., 1991), which implicitly hashe potential to detect changes in the correlation structure of datahrough the Q or SPE statistics, is based on the marginal covari-nce matrix and therefore lacks the ability to effectively detect localtructural changes. This situation arises from the fact that MSPC-CA mainly considers changes in the variables’ mean and variance,hich might not occur during a structural change (Wang and He,

010). The belief that the Q/SPE statistics may be enough for detec-ing changes in the process correlation structure will be challengedn Section 5.1, where we will demonstrate that MSPC-PCA is eas-ly outperformed by monitoring statistics devoted to monitor theorrelation structure, when problems are really located at this level.

As process variables may present a significant mutual marginalovariance even though they do not directly interact in a causalay (as long as they are affected by some common causes of varia-

ion), monitoring procedures based on this quantity are unable, byesign, to effectively detect and discern changes in the local causalorrelation structure. They lack the necessary resolution to performuch an analysis, and any change detected in the marginal covari-nce between two variables may be due to changes directly relatedo them, or that happened in any other variables whose variation

ay, directly or indirectly, affect them. Moreover, some structuralhanges might not even have an impact on the marginal covari-nce, especially if several deviations occur at the same time andompensate each other. Therefore, in order to access and use theocal information of the variables correlation structure, alternative

easures of association must be adopted in the process monitor-ng procedures. Partial correlation (PC) is one such quantity (Sokalnd Rohlf, 1995), as it evaluates the correlation between pairs ofariables, after controlling for the effect of others, i.e., after remov-ng their indirect effect in inducing any association between theariables under analysis. This leads us to the following premise: asartial correlation coefficients are able to retain, to a larger extent,

nformation about the local association of variables (even though in non-causal sense, i.e., without the associated causal directional-

ty), they can provide a finer map of the inner connective structuref variables. Thus, statistical process monitoring (SPM) based onartial correlations should be able to detect changes in the localssociation structure of variables (fault detection) and to identify

cal Engineering 71 (2014) 307–322

the root causes of specific process upsets (fault diagnosis), in a moreeffective way. Therefore, the total time invested in fault detectionand diagnosis activities may be improved using such an alternativemeasure of local association, as both the primary detection andespecially the diagnosis process, will be improved.

We would like to point out that, even though partial correla-tions have been proposed a short time after PCA, their potentialto improve process monitoring and fault detection activities havenot yet been explored. This fact is quite surprising, as detectionand, in particular, diagnosis tasks, can potentially benefit signifi-cantly from the use of local measurements of association. This is inmajor contrast with the widely explored use of marginal correla-tion approaches such has PCA and most of the current monitoringmethodologies.

In this context, we propose in this work two new contributionsto the process monitoring field. Firstly, a new set of methodologiesis proposed for monitoring changes in the process correlation struc-ture, that are based on the use of partial correlation information.Secondly, we introduce the use of sensitive enhancing transfor-mations (SET) for maximizing the ability of the methodologiesto detect changes in the process correlation structure, indepen-dently of their marginal or local nature. The proposed methodsbased on the use of partial correlations are applied to multivariatesystems and their performances compared to the marginal-basedapproaches available in the literature. The results obtained indicatethat partial correlation based statistics were indeed able to improvethe detection of changes in the systems structure. However, thegreatest contribution to this outcome is the use of a proper variabletransformation (the SET) that efficiently increases the sensitivity tosmall changes on the correlation structure of processes.

This article is organized as follows. In the next section, we reviewthe current monitoring statistics based on marginal covariance.Then, we describe the new proposed monitoring statistics based onpartial correlations and introduce the set of sensitivity enhancingtransformations, which play a critical role in the methods’ per-formance. Next, we present and discuss the results obtained fromthe application of all methods considered to systems with differ-ent degrees of complexity. Finally, we summarize the contributionsproposed in this paper and present the main conclusions.

2. Statistical process monitoring based on marginalcovariance

As the topic of this article is on the monitoring of the multivari-ate process dispersion, we devote this section to a review of themethodologies that were proposed for specifically addressing thisproblem. These approaches will constitute the benchmarks againstwhich the performance of the new proposed methodologies will beassessed and compared in Section 5.

2.1. W Statistic

Alt and Smith (1988) presented three procedures for monitoringprocess variability by following its marginal covariance. One of theschemes is based on the likelihood ratio test, which is defined as,

W = −p(n − 1) − (n − 1) ln( |S|

|�0|)

+ (n − 1)tr(�−10 S) (1)

where p is the number of variables, n the number of observations,�0 the in-control covariance matrix and S the sample covari-ance matrix. Anderson (2003) showed that W is asymptotically

2

distributed as �p(p+1)/2, and therefore the process dispersion is con-

sidered to be out-of-control if W exceeds the UCL = �2p(p+1)/2,˛

(�2d,˛

is the 100 × (1 − ˛)th percentile of a Chi-squared distribution withd degrees of freedom).

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.2. |S| control charts

The other two approaches presented by Alt and Smith (1988) areased on the sample generalized variance (i.e., the determinant ofhe sample covariance matrix, |S|). One of these approaches makesse of the sample distribution of |S|. For the case of two variables andssuming Gaussian conditions, it can be shown that |S| is distributeds |�0|(�2

2n−4)2/[4(n − 1)2]. Consequently, the control limits for the

S|-chart with two variables are (Alt, 1984; Alt and Smith, 1988),

LCL = |�0|(�22n−4,1−˛/2)

2/[4(n − 1)2]

UCL = |�0|(�22n−4,˛/2)

2/[4(n − 1)2]

(2)

here LCL and UCL stand for the lower control limit and upper con-rol limit, respectively (symbols have the same meaning as in Eq.1)). When more than two variables are available, one can use anpproximation of the distribution presented by Anderson (2003) toompute the control limits.

Finally, the third |S|-chart presented by Alt and Smith (1988)ses only the first two moments of |S| and the property that most ofhe distribution of |S| is confined in the interval E(|S|) ± 3

√var(|S|),

here E(|S|) = b1|�0| and var(|S|) = b2|�0|2, with,

b1 = (n − 1)−pp∏

i=1

(n − i)

b2 = (n − 1)−2pp∏

i=1

(n − i)

⎡⎣ p∏

j=1

(n − j + 2) −p∏

j=1

(n − i)

⎤⎦

(3)

From this result, it follows that the control limits for this controlhart are given by (Alt, 1984),

LCL = |�0|(b1 − 3b1/22 )

UCL = |�0|(b1 + 3b1/22 )

(4)

However, since |S| is positive definite, it is not meaningful toave a negative LCL, and therefore the LCL is usually set to zero.

The only difference between the last two |S|-charts proceduresies in the control limits adopted: in the former case they are prob-bility limits, Eq. (2), while in the last case, they are 3-sigma limits,q. (4).

.3. G Statistic

Levinson et al. (2002) proposed a monitoring scheme based onhe idea that, in a stable process, the covariance matrix estimatedith the complete set of collected data should be approximately

qual to the one obtained from the mean square of successive dif-erences. To obtain the monitoring statistics, G, one has to calculaterst the in-control sample covariance matrix, S0, from a referenceata set with n0 observations. After that, the ith test sample covari-nce matrix, S1,i, determined from a subgroup with n1 observationss combined with S0 in the pooled estimator of the covariance

Levinson et al., 2002),

pool,i = (n0 − 1)S0 + (n1 − 1)S1,i

n0 + n1 − 2(5)

LCL = kp[

˚′(

n − 12

)− ln

(n − 1

2

)]− z1−˛/2k

√

UCL = kp[

˚′(

n − 12

)− ln

(n − 1

2

)]+ z1−˛/2k

cal Engineering 71 (2014) 307–322 309

The monitoring statistic is then computed as Gi = mMi, where,

Mi = (n0 + n1 − 2) ln |Spool,i| − (n0 − 1) ln |S0| − (n1 − 1) ln |S1,i|

m = 1 −(

1n0 − 1

+ 1n1 − 1

− 1n0 + n1 − 2

)(2p2 + 3p − 1

6(p + 1)

)

(6)

When the process is under statistical process control, the Gstatistic is distributed as �2

p(p+1)/2. Therefore, the control limits canbe determined as (Levinson et al., 2002),

LCL = �2p(p+1)/2,˛/2

UCL = �2p(p+1)/2,1−˛/2

(7)

2.4. E Statistic

All the previous monitoring statistics are mainly based on someform of the generalized variance, which is somewhat insensi-tive to the correlation structure of the variables and thereforesome structural changes may pass undetected. To address thisissue, Guerrero-Cusumano (1995) proposed the use of a conditionalentropy measure. The entropy of a vector is a measure of the dis-persion of its values and, for a continuous p-multivariate randomvariable x, it is defined as (Guerrero-Cusumano, 1995),

H(x) = −∫

f (x) ln f (x)dx = E[− ln f (x)] (8)

where f(x) is the probability density of x.If x follows a normal distribution with mean � and covariance

�, then the entropy is given by (Guerrero-Cusumano, 1995),

H(x) = 12

p ln(2�e) + 12

ln |�| = 12

p ln(2�e)

+ 12

ln |�2d| + 1

2ln |P0| = 1

2p ln(2�e) + 1

2

p∑i=1

ln(�2i ) − T(x) (9)

where P0 is the correlation matrix, �i is the standard deviation ofthe ith variable, �d = diag(�i) is a diagonal matrix with �i in its maindiagonal and T(x) the mutual information. Estimating �i using thesample standard deviation, si, the sample entropy is given by,

H(x) = 12

p ln(2�e) + 12

p∑i=1

ln(s2i ) − T(x) (10)

The difference between the sample and theoretical entropy,ı = H(x) − H(x) = 1/2

∑pi=1 ln(s2

i/�2

i), is then considered to be the

conditional entropy, since it is conditioned on P0. Based on this, themonitoring statistic, E, is defined as,

E = kı =√

n − 12p

p∑i=1

ln

(s2

i

�2i

)(11)

where k = [2(n–1)/p]1/2 is a normalization constant.The control limits of E are calculated by (Guerrero-Cusumano,

1995),

p˚′′(

n − 12

)+ 2

n − 1tr (P0 − I)2

√p˚′′

(n − 1

2

)+ 2

n − 1tr (P0 − I)2

(12)

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here ˚′(·) and ˚′′(·) are the first and second derivative of theatural logarithm of the gamma function and z1−˛/2 is the upper

− ˛/2 percentile of the standard normal variable.Although the E statistic is much simpler than the statistics based

n the generalized variance, it requires P0 to be constant. Thereforene must verify if P0 indeed remains under statistical control beforeesting E.

.5. VMAX statistic

A simpler and more efficient control chart than the |S|-chartas proposed by Costa and Machado (2009). They proposed these of the VMAX statistic, which is the maximum value of theample variances of the data after normalization, i.e., VMAX =ax{s2

1, s22, . . ., s2

p}, where s2i

= zTizi/n − 1 and zi = (xi − �i)/�i. The

ormalization of the data is an important step on this monitoringtatistic since it guarantees that all the sample variances have theame probability to exceed a certain UCL. The upper control limitor a two dimensional process is obtained through,

= 1 −∫ n·UCL

0

Pr

[�2

n,(t�2/1−�2)<

n · UCL

1 − �2

]1

2n/2� (n/2)e−t/2t(n/2)−1dt (13)

here �2n,(t�2/1−�2)

is the non-central Chi-square distribution with

degrees of freedom and non-centrality parameter given byt�2/1 − �2) and � is the correlation between the variables.

. Statistical process monitoring based on local associationeasures

The current approaches to monitor process correlation structurend dispersion are based on the application of a sequence of sta-istical hypothesis tests in order to determine if some significanthange has occurred in the process covariance matrix. However,he covariance matrix does not convey a detailed information abouthe intrinsic local association structure of the system (Melissa et al.,009). Therefore methodologies based on the marginal covarianceresent intrinsic limitations regarding the detection ability of localbnormalities, as well as to the subsequent diagnosis of the fault’srigin once it is signalled. For instance, two variables, x and y, can beelated in several different ways such as (i) direct relation x → y, (ii)o-regulated by a third variable z (z → x and z → y) or (iii) indirectelation x → z → y (Melissa et al., 2009). In all these examples, theorrelation between x and y may be similar, and a change on theirnderlying relationship may pass undetected by just monitoringhe marginal correlation. One way to capture the real inner associ-tions between variables is through the use of partial correlations.

The basic idea of partial correlations is to remove the effect ofhird-party variables before checking for an association betweenhe two designated variables. Therefore the correlation betweenwo variables is quantified, after conditioning upon (i.e., control-ing for, or holding constant) one or several other variables. In thebove example, the partial correlation between x and y conditionedo z would remove the common effect of z on x and y, providing

clearer picture of the local correlation structure between theseariables. More specifically, this can be achieved by first regressing

on z, and y on z, after which the regression residuals are saved.hese residuals are the parts of x and y that are uncorrelated with z.he correlation between these residuals corresponds to the partialorrelation between x and y conditioned on z (rxy·z).

The order of the partial correlation coefficient is determined byhe number of variables it is conditioned on. For instance, rxy·z is

1st order partial correlation coefficient because it is conditionedolely on one variable (z). Partial correlations can be obtained eithersing the above referred regression based approach, or throughnalytical formulas, in a recursive way. Eqs. (14)–(16) illustrate

cal Engineering 71 (2014) 307–322

the computation of the partial correlation coefficients for orders0, 1 and 2. Similar equations exist for higher order representa-tives, however only partial correlations up to 2nd order are typicallyused to map the correlation structure (Fuente et al., 2004; Rao andLakshminarayanan, 2007; Reverter and Chan, 2008).

0th order partial correlation:

rxy = cov(x, y)√var(x)var(y)

(14)

1st order partial correlation:

rxy.z = rxy − rxzryz√(1 − r2

xz)(1 − r2yz)

(15)

2nd order partial correlation:

rxy.zq = rxy·z − rxq·zryq·z√(1 − r2

xq·z)(1 − r2yq·z)

(16)

In this study we will only consider the 0th order and 1st orderpartial correlations in order to assess if there is any advantage inemploying such association metrics. Again, higher orders could beconsidered when addressing more complex systems, but due tothe sensitivity enhancing transformation proposed in Section 4 theproblem of selecting the more adequate order of partial correla-tions is averted, since it implicitly performs this task. Therefore, forconvenience of notation, we define r0 as the (p(p − 1)/2) × 1 columnvector containing all distinct correlation coefficients (0th order par-tial correlations) and r1 as the corresponding (p(p − 1)(p − 2)/2) × 1column vector of 1st order partial correlation coefficients.

As partial correlation coefficients bring out differences betweenthe direct and indirect relationships between variables, we havestudied for the first time in the broad field of chemical engineering(and possibly beyond it) several methods to monitor changes onthe partial correlation coefficients, namely in the 1st order partialcorrelations, with the potential to detect finer local changes in theprocess structure, and to identify their source in a more effectiveway. The proposed procedures will be presented in detail in thenext subsections.

3.1. R0MAX and R1MAX

One way to detect changes in the process structure is by usingsequential hypothesis tests to verify if the partial correlationsremain close to their respective NOC values (i.e., the populationcorrelation, �). To formalize the hypotheses tests to be applied inthis situation, one will first define the probability distribution of thecorrelation coefficients and their extension to partial correlations.

When the number of samples is large, the distribution of thecorrelation coefficients (0th order partial correlation), transformedaccording to Eq. (17), tends to be normally distributed with zeromean and variance one (page 133 of Ref. (Anderson, 2003)).

w1 =√

n − 1(r − �)1 − �2

(17)

This tendency to normality can be strengthen by the use of theFisher’s z transformation (pages 133–134 of Ref. (Anderson, 2003)),resulting in,

w2 =√

n − 1(

ln(

1 + r)

− ln(

1 + �))

(18)

2 1 − r 1 − �

In both cases, the underlying distribution for hypothesis test of� = �0 against the alternative of � /= �0, corresponds to the stan-dardized normal distribution.

Chemical Engineering 71 (2014) 307–322 311

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Table 1Algorithm to determine the PCA model for the qth order partial correlation.

1. Obtain a reference data set:a. For i = 1 to k,

i. Collect n observations (X);ii. Determine the sample covariance matrix:

Si = 1n − 1

n∑j=1

(xj − x)(xj − x)T

iii. Compute rq,i from Si , using Eqs. (14)–(16);b. Determine the sample mean (rq) and sample covariance (Srq ) of rq:

rq = 1k

k∑i=1

rq,i

Srq = 1k − 1

k∑i=1

(rq,i − rq)(rq,i − rq)T

2. Construct PCA model for the partial correlations (rq,i):a. Perform the spectral decomposition: Sr = ���T;

T.J. Rato, M.S. Reis / Computers and

Since the distribution of the qth order partial correlationoefficients based on n observations is the same as the correla-ion coefficients based on (n − q) observations, its hypothesis tests exactly the same, except that n is replaced by (n − q) (Anderson,003).

If the transformation w1 or w2 is applied to all the partial corre-ations, then they will present the same distribution and probabilityf exceeding a certain limit. As only one partial correlation coeffi-ient needs to exceed the control limit to consider that a changen the process structure has occurred, we propose to monitor the

aximum of the normalized partial correlations, in absolute value,efined as,

0MAX = max{|w(r0)|} (19)

or the correlation coefficients (0th order partial correlations, i.e.,he marginal correlations) and as,

1MAX = max{|w(r1)|} (20)

or the 1st order partial correlations. In these Equations, w(·) standsor either one of the transformations presented in Eqs. (17) and (18),r any other transformation that guarantees that the transformedartial correlations follow approximately a standard normal distri-ution.

.2. Monitoring process correlation through MSPC-PCA

As referred before, MSPC-PCA is a non-causal approach with anmplicit capability to monitor the variables correlation structure,amely through the residual statistic (Q or SPE) (Jackson, 1959;

ackson and Mudholkar, 1979; Kresta et al., 1991) This approachses two monitoring statistics. One of them follows the variability

n the PCA subspace estimated with NOC data, corresponding to aotelling’s T2 statistic applied to the retained principal components

PCs). The other statistic monitors the complementary variability,round the PCA subspace, through a lack of fit or residual statisticsually called Q or SPE (squared prediction error). More specifically,he MSPC-PCA monitoring statistics are defined as follows:

2PCA =

∑a

i=1

t2i

�i= xT P�−1

a PT x (21)

= eT e = (x − x)T (x − x) = xT (I − PPT )x (22)

here P is a matrix containing the first a eigenvectors, �a =iag (�1, . . ., �a) is a diagonal matrix with the first a eigenvalues, �i,

n the main diagonal, x is the projection of x onto the PCA subspace, is the number of retained PCs (pseudorank) and I is an identityatrix. Control limits for these monitoring statistics can be found

lsewhere (Jackson and Mudholkar, 1979; MacGregor and Kourti,995; Tracy et al., 1992).

This approach will be further analyzed in the preliminary studyresented in Section 5.1, where it will be demonstrated thatSPC-PCA has a relatively low ability to detect structural changeshen compared with other methodologies designed to monitor

he covariance matrix. Moreover, MSPA-PCA is even prone to missuch changes since it is primarily focused in detecting deviations onhe variables’ means and variances. For instance, a decrease in theigenvalues of the covariance matrix is hardly detected throughSPC-PCA. To mitigate this issue Wang and He (2010) proposed

o monitor an extended vector composed by the variables’ mean,ovariance, skewness and kurtosis, via MSPC-PCA, in order to cap-ure dissimilarities on the operation conditions. Even though thispproach leads to higher detection capabilities than MSPC-PCA, it

till does not consider the variables inner relationships in any ofhe monitored terms. To do so, it is here explored the application of

SPC-PCA to monitor the marginal correlations (r0) and 1st orderartial correlations (r1). This is done by replacing x in the above

q

b. Determine the number of principal components to retain (a);c. Define the loading matrix, P, as the first a columns of �.

MSPC-PCA procedure by r0 or r1 (henceforth generically defined asrq, q = 0, 1).

To apply MSPC-PCA in the proposed context, it is required tocompute the respective covariance of rq, in order to construct theNOC PCA model. The covariance matrix of rq can be obtained fromk subgroups with n observations, leading to k observations of rq

(see Table 1). The number of subsets (k) required to construct thePCA model is one of the disadvantages of this method, since it maytranslate into an overall number of observations that can be large,especially when the number of process variables is substantial (theestimation of a full rank covariance matrices require k > p(p − 1)/2for r0 and k > p(p − 1)(p − 2)/2 for r1). In the current study we alwaysconsidered that a sufficient amount of data is availed for estimatingthe PCA model.

In practice, the pair of T2PCA and Q statistics can be merged into a

single combined index (Yue and Qin, 2001) or combined through alogical gate “or”. In the current work, we chosen to use a combinedstatistic, C, that signals an alarm when either T2

PCA or Q, or both, fallbeyond their control limits. In the comparison study, the controllimits of T2

PCA and Q where adjusted to the same false alarm rate andalso in such a way that C presents the desired global false alarm rate.

4. Sensitivity enhancing transformations

The performance of the methodologies presented in the pre-vious section can be significantly enhanced by application of a“sensitivity enhancing transformation”. This can be understoodusing the following simplified rational. If a change on the rela-tionship between two process variables occurs, the correspondingpartial correlation coefficients should also change. However thechange on the partial correlation can be quite subtle and difficult todetect. For instance, consider that x = kz + wε, where both z and ε fol-lows a N(0, 1) and k and w are constants. In this case, the correlationbetween x and z (rxz) can be represented as a function of the ratiok/w according to Fig. 1(a). It is apparent that there can be ratherdifferent sensitivities to changes on rxz, depending on the partic-ular position in the curve. The 1st derivative of rxz (see Fig. 1(b))is a proper measure of such sensitivity for detecting changes onrxz, and one can verify that changes are more difficult to detect as|rxz| gets closer to 1 and easier when |rxz| is close to 0. The maxi-mum sensitivity is reached at rxz = 0, which corresponds to the case

where variables are uncorrelated (either k = 0 or w → ∞, i.e., w � k).Therefore, changes in the correlation coefficients will be easier todetect when a relationship emerges from a previously non-existentone. Furthermore, we have also found out that the same behaviour

312 T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322

-2 -1 0 1 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

k/w

r xz

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

k/w

1st d

eriv

ativ

e of

rxz

k/w o

ofs

m(a

�

w

u

Tmaxatup

sslvdms(bpRbtoaXb

X

(a)

Fig. 1. Graphical representation of (a) the effect of

ccurs with partial correlations. Thus, in order to exploit thiseature, the original variables are preliminarily rotated to a regres-ion subspace of uncorrelated variables.

One of the transformations usually applied in the treatment ofultivariate Gaussian data relies on the Cholesky decomposition

Press et al., 2007), which factorizes the covariance matrix, �, into lower triangular matrix L, such that,

= LLT (23)

This matrix, can then be used to obtain uncorrelated variablesith unit variance through the transformation (Press et al., 2007):

= L−1(x − �) (24)

This transformation was adopted by Hawkins and Maboudou-chao (2008) on their extension of the W statistic to on-lineonitoring and corresponds to a succession of regression oper-

tions, where ui is the residual of the regression of xi on x1,. . .,i−1, rescaled to unit variance. Yet, the W statistic is invariant tony linear transformation and therefore does not benefit from suchransformation. On the rest of this article the use of the original,ntransformed data will be referred by TX, whereas the use of thisarticular data transformation will be denoted as TCh.

However, the transformation based on the Cholesky decompo-ition described on Eq. (24) is only suitable for stationary linearystems. In order to extend its application to dynamic and non-inear systems, we propose in this work the addition of time-shiftedariables or polynomial terms (depending of the situation) to theata matrix. This type of approach is in accordance with otherethodologies, such as Dynamic PCA which incorporates time-

hifted variables to implicitly model a vector autoregressive modelVAR) process (Ku et al., 1995). As in the case of DPCA, the num-er of lags can be selected based on algorithms that estimate therocess lagged structure, such as the ones described in (Rato andeis, 2013). Likewise, the approximation of non-linear functionsy polynomial terms is justifiable by the Taylor series expansionheorem, since under normal conditions the process experimentsnly mild fluctuations around a target value. These additional vari-bles should be placed at the beginning of the extended data matrix,

˜ , which for the case of the inclusion of time-shifted variablesecomes:

˜ =

[X(l) · · · X(1) X(0)

](25)

(b)

n the correlation (rxz) and (b) first derivative of rxz .

where X(j) is an n × p matrix of variables shifted j times into thepast (i.e., with j lags). When the inclusion of polynomial terms isrequired, X(j) are replaced by powers of the type x(j + 1).

After this step, the regular Cholesky decomposition can be per-formed, Eq. (26), from which a new set of uncorrelated variablesare obtained by Eq. (27).

� = LLT

(26)

u = L−1

(x − �) (27)

In the case where X is obtained by the addition of time-shiftedvariables, only the regression variables related to the present stateare of interest (i.e., the last p variables in u), since they correspond tothe residuals of the linear regressions of the variables in the present,onto those from the past. Using this procedure, both cross- andauto-correlations are handled simultaneously for whitening dataat the current time. The same procedure can also be applied to non-linear systems through the use of polynomial terms instead of time-shifted variables. This type of transformation will be referred asTChExt. A summary of the studied transformations is given in Table 2.

In order to better illustrate the advantage of employing the sen-sitivity enhancing transformations, let us consider Table 3, wherefour systems are presented. In each system the variables are con-nected in different ways, according to the networks represented(for instance, in system (a) variable 1 has a direct influence on vari-able 2, which in turn affects variable 3). Under normal operationconditions the connected variables are related by a linear equa-tion and, during a fault, the slop of one of these equations wasincreased in 5%, which consequently changes the systems correla-tion. In this study, 200 sample covariance matrices were collectedfor NOC and Fault conditions. Each sample covariance matrix wasdetermined based on 3000 observations (n). The obtained par-tial correlation coefficients, normalized according to Eq. (17), aredepicted in Table 3 for both original (TX) and transformed (TCh) vari-ables. The results presented clearly show the importance of usinga sensitivity enhancing transformation (SET), since when a changeoccurs, a deviation of more than 24 standard deviations is observ-able when the transformation is used, while for raw data, withoutany transformation, only a maximum change of 2.60 standard devi-ations is observed. This general result has a great impact on the

monitoring statistics performance, especially when the originalpartial correlations remain mostly unchanged and the statisticsbased on the untransformed data are unable to detect any change.Additionally, as fewer observations are used to estimate the partial

T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322 313

Table 2Nomenclature associated with the sensitivity enhancing transformations covered in this study.

Transformation Description Application

TX Original variablesTCh Variables transformed using the Cholesky decomposition, Eq. (24) Stationary linear systemsTChExt Variables transformed using the Cholesky decomposition after incorporation of

time-shifted variables and/or polynomial terms, Eq. (27)Dynamic non-linear systems

Table 3Effect of the sensitivity enhancing transformation on the partial correlation coefficients. The results presented are the average value of 200 samples (standard deviations areall close to 1).

Network

31 2

(a)

2

1

3

(b)

TX TCh TX TCh

NOC Fault NOC Fault NOC Fault NOC Fault

r1,2.3 0.11 0.01 −0.03 24.11 −0.03 1.74 0.08 24.39r1,3.2 −0.20 0.08 0.002 −0.03 0.11 −2.06 −0.03 −0.09r2,3.1 0.17 −0.09 0.20 −0.07 −0.06 0.17 −0.05 0.17

Network

3

21(c)

2

31(d)

TX TCh TX TCh

NOC Fault NOC Fault NOC Fault NOC Fault

−0.2324.79

0.02

cttvatwt

nsipd

TD

r1,2.3 −0.06 −1.35 −0.04

r1,3.2 0.09 2.55 −0.01

r2,3.1 0.01 −0.003 0.001

orrelation coefficients, their uncertainty increases. This implieshat, for untransformed variables, process deviations can easily leado statistics still falling under the NOC region, while for transformedariables the partial correlation coefficients are significant even for

small number of observations. Therefore, under the same detec-ion power, monitoring statistics based on transformed variablesill require fewer observations for detecting the same fault and

herefore they will also present shorter detection times.By applying the proposed monitoring statistics to both origi-

al and transformed variables and also to r0 and r1, the complete

et of new statistics under analysis in this work is obtained, whichs summarized in Table 4. The general workflow of the proposedrocedure is also represented in Fig. 2. The main stages includeata pre-processing, where original variables are transformed

able 4efinition of the new monitoring statistics proposed in this work.

Sensitivity enhancing transformation Statistic

RMAX

r0 r1

TX R0MAXX R1

TCh R0MAXCh R1

TChExt R0MAXChExt R1

−0.07 2.60 0.06 24.32 0.03 0.02 0.02 0.13

−0.02 −0.02 −0.02 −0.02

according to one of the sensitivity enhancing transformationsdescribed earlier, and process monitoring, performed by eitherMSPC-PCA or RMAX on the vectors of marginal or partial correla-tion coefficients. As an example, R1MAXChExt results from applyingR1MAX to data transformed according to the transformation, TChExt.The performance of all these monitoring statistics will be assessedand compared in the next section, together with the monitor-ing statistics already proposed in the literature, namely the onesdescribed in Section 2.

5. Results

The analysis of the process monitoring statistics and sensitiv-ity enhancing transformations proposed in this work is divided in

MSPC-PCA

r0 r1

MAXX Cr0,X

{T2

r0,X

Qr0,XCr1,X

{T2

r1,X

Qr1,X

MAXCh Cr0,Ch

{T2

r0,Ch

Qr0,Ch

Cr1,Ch

{T2

r1,Ch

Qr1,Ch

MAXChExt Cr0,ChExt

{T2

r0,ChExt

Qr0,ChExt

Cr1,ChExt

{T2

r1,ChExt

Qr1,ChExt

314 T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322

TX TCh TChExt

Data

Pre-Processing

R1MAXChExt

Marginal Correla�ons

1st order par�al

corr ela�on s

Computa�on of correla�ons

MSPC- PCAC = { T2

PCA, Q } RMAX

Process Monitoring

F itorina

tMo5sl(sbcspd5sit

ig. 2. Schematic representation of the main blocks that compose the proposed monre highlighted as an example.

hree parts. In the first part (Section 5.1) a preliminary analysis ofSPC-PCA is performed in order to justify and explain its absence

n the remaining parts of the study. The second part (Section.2), is focused on the performance assessment of the monitoringtatistics considering different degrees of complexity in the under-ying dynamics of the causal network model: (i) linear stationarywithout dynamics); (ii) linear dynamic system; (iii) non-lineartationary system. As the purpose of this work is to assess the capa-ility of methods to detect and diagnose changes in the variablesorrelation structure, the use of a network system of reasonableize provides a suitable testing scenario, offering the flexibility toerform and analyze a variety of local and global perturbations, ofifferent types and magnitudes. Finally, in the third part (Section

.3), the study is orientated towards the analysis of a more realisticystem with dynamic non-linear features. In this case, our purposes to move the test scenario to a typical industrial system, oncehe methods were characterized in detail under well controlled

g procedures. The modules involved in the construction of the R1MAXChExt statistic

and easily interpretable conditions. For those reasons the samplecovariance matrices used to compare the monitoring schemes arecomputed based on 3000 observations. Even though this value rep-resents some memory overhead, it is easily managed by currentcomputers and corresponds to less than 1 h of process operation ata sampling rate of 1 Hz. However, more important than this, it willallow us to focus our study on the relative impact of using partialcorrelations to detect structural changes in the process, relativelyto the application of current approaches based on the marginalcovariance, under the same circumstances. This approach will alsoidentify the most suitable methodologies worthy to consider forfurther development in the context of on-line monitoring, a taskthat will be addressed subsequently and based on the fundamental

results of this study. Additionally, we would like to point outthat the effect of the number of observations was also carefullystudied. However, the only hard constrain to the number of obser-vations arises from the prerequisite of positive definitiveness of the

T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322 315

(b) (a)

0.5 1 1.50

0.2

0.4

0.6

0.8

1

δ

Det

ectio

n R

ate

WMSPC-PCAMSPC-PCACh

0

0.2

0.4

0.6

0.8

1

W MS

PC

-PC

AC

h

MS

PC

-PC

A

N

F lt det( rrespo

cmtlewo

sttoeUtfittsptt

poi1c

5

ttoaHisb

other studied monitoring statistics dedicated to detect correlationchanges are able to attain a detection rate of 1 long before MSPC-PCA is able to achieve the same detection rate, which implies thatits performance is considerable lower.

Table 5Definition of the faults and respective variables involved for the network systemwithout dynamics or non-linearity. ı is a multiplicative factor that changes themodel parameters (under NOC, ı = 1).

Fault Variables relation changed

ig. 3. Performance comparison of MSPC-PCA and the W statistic in fault D. (a) Fauunder NOC, ı = 1); (b) box-plots of the area under the fault detection curve. Bars co

ovariance matrix, which implies that the number of observationsust be greater than the number of variables. This requirement is

ransversal to all the monitoring schemes considered. In fact, thisimitation can be overcome with resource to modern regularizedstimation approaches of the covariance matrix, but as all methodsere tested under similar circumstances, this was not necessary or

pportune for this work.Throughout this study, the performance of the monitoring

tatistics is summarized by an index (N) based on the area underhe fault detection rate curve (i.e., the integral of the curve detec-ion rate versus fault’s magnitude). This index condenses the resultsbtained by a given method for different fault magnitudes, whichnables the comparison of a large number of monitoring statistics.nder normal operation conditions, the fault detection rate is close

o the pre-established false detection rate. As the magnitude of theault increases, the detection rates tend to 1. Therefore, a monitor-ng statistic with a larger area under the detection rate curve tendso 1 more rapidly, meaning that it will also present higher detec-ions rates. This index was computed for each fault and normalizedo that its values fall in the range [0, 1], where 1 represents the besterformance observed (corresponding to the greatest area underhe fault detection rate curve). More details on the computation ofhis performance index will be provided in Section 5.1.

The detections rates of each monitoring statistic were also com-ared using a permutation test (Pesarin and Salmaso, 2010) inrder to verify if the differences obtained were statistically mean-ngful. This was done for each pair of monitoring statistics using0,000 permutations, after which the corresponding p-value wasomputed.

.1. Preliminary assessment of MSPC-PCA

MSPC-PCA is a well-established and extensively used procedureo monitor the process operation status, not only due to its abilityo explain most of the process variability with a reduced numberf variables but also for its performance on the detection of devi-tions from the mean levels and even some structural changes.

owever, this procedure presents limitations in detecting changes

n the local correlation structure when compared to monitoringtatistics specifically designed for that purpose. This situation wille exemplified in this section, where the performance of MSPC-PCA

ection curve, where ı is a multiplicative factor that changes the model parametersnd to the associated mean values.

is compared against the current W statistic for the stationary linearsystem described in Section 5.2.1.

The faults simulated in this system are localized and are exclu-sively related with changes in the correlation between variables.The sample correlations to be monitored by the W statistics werecomputed from 3000 observations. Likewise, the average of theseobservations was also used to determine the T2

PCA and Q statis-tics, which in this work were combined through a logical gate “or”(MSPC-PCA). The same procedure was conducted after transfor-mation TCh, resulting in the monitoring statistic MSPC-PCACh. Thecontrol limits were established for a false alarm rate of 1% for allmonitoring procedures and the detection rates were determinedfor the same set of faults used in Section 5.2.1, with magnitudesin the range of ±50%. The simulations were repeated 100 times inorder to assess the consistency of the results.

For faults A, B and C (see Table 5), the detection rates of bothMSPC-PCA procedures are always lower than 0.10 under the rangeof simulated faults, while the W statistic presents detection ratesof 1 for deviations even smaller than ±10%. For fault D, the resultsobtained are represented in Fig. 3. Analysing the detection curves(Fig. 3(a)), it is observed that both MSPC-PCA procedures are ableto detect the simulated faults, but a more effective detection isobtained when the proposed transformation TCh is used. Still, the Wstatistics performs much better, correctly detecting all the pointsas faulty even for small magnitude perturbations. Furthermore, thesmallest fault’s magnitude needed to obtain a full detection ratewith MSPC-PCA (±40%) is considerable larger than the one used inthe main comparison study, which is ±20%. Therefore, most of the

A g8 → g1 (g1 = 1.2ıg8 + 0.80g9 + ε1)B g1 → g3 (g3 = 0.05 + 0.22ıg1 + ε3)C g8 → g10 (g10 = 1 + 0.40ıg8 + ε10)D g3 → g14 (g14 = 1 + 0.40ıg3 + ε14)

3 Chemical Engineering 71 (2014) 307–322

itvaftttfvWoTtfcndftiotcc

rocspftMicatoMsMqNicbt

5s

unitewlucrmr

11

10

9

2

45

6

7

8

12

13 14

15

3

1

16

16 T.J. Rato, M.S. Reis / Computers and

The same result is corroborated by the performance index. Thisndex aims to facilitate the analysis of these results by condensinghe outcomes for the different magnitudes of faults into a singlealue of the index, thus providing a suitable mean for comparing

large number of monitoring statistics. This index is computedrom the area under the detection rate curve (i.e., the integral ofhe curve detection rate versus fault’s magnitude) of each moni-oring statistic as described earlier. For this case, the areas underhe curve, for one of the replications, were 0.9017 for W, 0.5492or MSPC-PCA and 0.6448 for MSPC-PCACh. By normalizing thesealues, the performance index, in this replication, becomes 1 for, 0 for MSPC-PCA and 0.2712 for MSPC-PCACh. The distribution

f the obtained indices is depicted of Fig. 3(b) for 100 replications.his representation clearly shows that the W statistics is consis-ently ranked as the best monitoring statistic and tends towards aull detection rate of 1 more rapidly (note, that the fault detectionurve increases monotonically with the increase of the fault’s mag-itude). It also shows by how much such monitoring proceduresiffer. For instance, the performance of MSPC-PCACh (with trans-ormed variables) is closer to MSPC-PCA (with original variables)han to W. This relative position is easily observable by a simplenspection of the detection rate curve. However, when the numberf monitoring statistics becomes large, it is necessary to summarizehe results in a meaningful way, since the plot of the detection rateurve for all monitoring statistics lead to very dense graphs andonfusing representations.

From these results it can be concluded that even though the cur-ent MSPC-PCA is able to detect changes in the correlation structuref process data, the magnitude after which it becomes signifi-ant is relatively larger than for most of the current monitoringtatistics dedicated to such task (in most simulations, MSPC-PCAresented a detection rate lower than 0.10, regardless of the trans-ormation used) and therefore it has a weak performance underhe range of fault’s magnitude studied. The only situation where

SPC-PCA gave better results happened in the case study referredn Section 5.2.3. Yet, the faults simulated in such system were notompletely constrained to structural changes, and often led to devi-tions in the mean value of the measured variables, which explainshe improved performance obtained by MSPC-PCA. Therefore, inrder to reduce the amount of monitoring statistics to be compared,SPCA-PCA will not be included in the comparison studies pre-

ented in the following sections. It is also worth noticing that whenSPC-PCA is applied to single observations, which is the most fre-

uent situation, the observed detection rates become even lower.evertheless, MSPC-PCA was implemented according to the mon-

toring scheme presented in Section 3.2, which shares a similaronstruction to the statistics pattern analysis procedure proposedy Wang and He (2010), that was shown to be superior to conven-ional MSPC-PCA.

.2. Extended comparative assessment of process monitoringtatistics

In order to assess the performance of the various methodologiesnder study, they were applied to a modified version of the artificialetwork originally presented by Tamada et al. (2003). This network

s composed by 16 nodes (or variables) causally related accordingo the representation provided in Fig. 4. In each scenario consid-red in the following examples, 1000 sample covariance matricesere computed based on 3000 observations each, taken at regu-

ar intervals of time. The sample covariance matrices were thensed to determine the monitoring statistics and to compute the

orresponding fault detection rates (true detection and false alarmates). The same procedure was repeated 10 times in order to esti-ate the confidence levels of the performance indicators (detection

ates). The control limits for all the monitoring statistics were

Fig. 4. Graphical representation of the network structure used in this work.

preliminarily adjusted, by trial and error, so that all monitoringstatistics present the same false detection rate of 1% under nor-mal operation conditions (NOC). This approach to determine thecontrol limits was taken since the direct use of their theoreticalexpressions often fails to produce suitable control limits. This situa-tion happens because the underlying hypothesis for the monitoringstatistics, such as i.i.d. conditions and normality assumptions, arenot found in all data sets. Therefore, by taking such an approach, weare guaranteeing a fair and sound comparative assessment of thedetection performances for the various methods under analysis.

5.2.1. Stationary linear systemIn this case study, the original variable relationships are lin-

earized according to Eq. (28), where εi is a white noise sequencewith a signal-to-noise ratio of 10 dB. This system was then sub-jected to a set of perturbations, as described in Table 5, whereı represents a multiplicative factor that causes a change on themodel’s parameter on the range of ±20%.

g8 = ε8, g9 = ε9, g16 = ε16

g10 = 1 + 0.40g8 + ε10

g11 = 0.56 + 0.15g8 + ε11

g1 = 1.2g8 + 0.80g9 + ε1

g2 = 0.60g1 + ε2

g3 = 0.05 + 0.22g1 + ε3

g4 = 1 + 0.4g1 + ε4

g5 = 0.062 + 0.16g1 + ε5

g6 = 0.60g1 + ε6

g7 = 0.70g1 + ε7

g12 = 0.80g16 + 0.51g3 + ε12

g13 = 1.30g3 + ε13

(28)

g14 = 1 + 0.40g3 + ε14

g15 = 0.028 + 1.30g3 + ε15

T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322 317

0

0.2

0.4

0.6

0.8

1

R0M

AX C

h

R1M

AX C

h

Cr1

,Ch

W Cr0

,Ch

Cr0

,X

R1M

AX X

R0M

AX X

Cr1

,X

E VM

AX

N

Fig. 5. Comparison of the statistics performance for the network system with no dynamics or non-linearity: box-plots of the area under the fault detection curve obtainedfor all perturbations. Bars correspond to the associated mean values.

fppmGtdwwvstsmlciaotnno

tettitbM

tetvac(ao

vpa

ables’ time dependency, failing to detect changes on the variablesdynamics.

Table 6Definition of the faults and respective variables involved for the network systemwith linear dynamics. ı is a multiplicative factor that changes the model parameters(under NOC, ı = 1).

Fault Variables relation changed

The distribution of the comparison index (N) over each type ofault is represented in Fig. 5 and the permutation tests for eachair of monitoring statistics are presented in Table S1 of the Sup-lementary Material (available on the journal website). Among theonitoring statistics already proposed in the literature, the W and

statistics are the ones presenting the best performances, whilehe determinant of the sample covariance matrix, |S|, is unable toetect any change. The E statistic only performs well on fault A,here the perturbation occurs close to the root node of the net-ork and is propagated to almost all variables. As the number of

ariables affected by a fault decreases, the performance of the Etatistic also degrades. This is an expected result, since the E statis-ic is focused in detecting changes in variance for a fixed correlationtructure. From these results, we choose the W statistic as a bench-ark to compare the new proposed statistics. However, we would

ike to point out that the W statistic requires the inversion of theovariance matrix (see Eq. (1)), which in some situations may bell-conditioned or even singular. In such case, a pseudo-inverse or

linear transformation of the variables was performed. The onlybservable change of this modification was on the asymptotic dis-ributional behaviour of the statistic, which seems to follow now aon-central Chi-square distribution. This situation underlines theeed to set the control limits by trial and error adjustment, insteadf using the theoretical control limits.

Regarding the results for the proposed statistics, one can noticehat the sensitivity enhancing transformation proved to be a rel-vant factor in the monitoring task, as it significantly increaseshe statistics performance. In fact, without such transformation,he statistics performance is worse than for the W statistic. Its also noticeable that without the transformation the use ofhe partial correlations tends to decrease the detection capa-ility of the proposed statistics, especially the ones based onSPC-PCA.After applying the sensitivity enhancing transformation, most of

he proposed statistics outperform the current W statistic, with thexception of Cro,Ch (MSPC-PCA applied to the marginal correlation ofhe transformed variables TCh) which has a similar performance (p-alue of 0.048). This shows that the use of partial correlations has

relevant effect on the MSPC-PCA based statistics, especially forhanges of small magnitude and when few variables are involvedfaults C and D). However, these statistics require a significantmount of data in order to construct the PCA model for normalperation conditions.

Regarding the RMAX statistics, it can be concluded that theariables’ transformation also improves their performance. Theerformance of the RMAX statistics with transformed variables waslways found to be better than the W statistic and also presented a

better detection capability than the MSPC-PCA statistics. Curiously,the use of partial correlations seems to have no effect on the per-formance of these statistics after variables transformation, as bothR0MAXCh and R1MAXCh (RMAX applied to marginal and partial cor-relations of the transformed variables TCh, respectively), presenta performance index, N, consistently close to one, meaning thathigher detection rates are being achieved by both monitoring statis-tics (see Fig. 5). The performance of R0MAXChExt and R1MAXChExt(RMAX applied to marginal and partial correlations of the trans-formed variables TChExt, respectively) was not assessed, since thisis a stationary linear system for which there is no need to use morecomplex transformations, which would not bring any added valueto the analysis.

5.2.2. Dynamic linear systemA similar analysis was performed with a dynamic version of

the network system with the addition of a multivariate time seriesdependency between variables according to Eq. (29), where εi is awhite noise sequence with a signal-to-noise ratio of 10 dB. This pro-cess was then subjected to the perturbations presented in Table 6,where ı was set to cause perturbations in the range of ±20%. Theresults of the permutation tests are given in Table S2 of the Supple-mentary Material, while the performance index is depicted in Fig. 6.The results involving the original variables and transformation TChare consistent with the ones obtained for the stationary case, i.e.,the use of uncorrelated variables improves the detection capabil-ity of the monitoring methods. The main difference is the betterperformance of R1MAXCh relatively to R0MAXCh, even though theyshow a similar performance for the W statistic (see Fig. 6 and TableS2 of the Supplementary Material). In general, transformation TChdoes not contribute to a significant increase on the performanceof the monitoring statistics, as it does not handles the vari-

A g8 → g1 (g1,t = 1.2ı (g8,t + 0.60g8,t−1 + 0.30g8,t−2) + 0.80g9,t + ε1,t )B g1 → g3 (g3,t = 0.05 + 0.22ı (g1,t − 0.40g1,t−1 − 0.20g1,t−2) + ε3,t )C g8 → g10 (g10,t = 1 + 0.40ı (g8,t + 0.60g8,t−1 − 0.30g8,t−2) + ε10,t)D g3 → g14 (g14,t = 1 + 0.40ı (g3,t + 0.40g3,t−1 + 0.60g3,t−2) + ε14,t)

318 T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322

0

0.2

0.4

0.6

0.8

1

1MA

X ChE

xt

0MA

X ChE

xt

r1,C

h

1MA

X Ch

0MA

X Ch

r1,C

hExt

W

r0,C

hExt

r0,C

h

r0,X

r1,X 1M

AX X

0MA

X X

MA

X

N

F near dp

)

cTcsstFe

5

aw

TDwp

R R C R R C

ig. 6. Comparison of the statistics performance for the network system with lierturbations. Bars correspond to the associated mean values.

g8,t = ε8,t, g9,t = ε9,t, g16,t = ε16,t

g10,t = 1 + 0.40 (g8,t + 0.60g8,t−1 − 0.30g8,t−2) + ε10,t

g11,t = 0.56 + 0.15 (g8,t + 0.40g8,t−1 + 0.60g8,t−2) + ε11,t

g1,t = 1.2 (g8,t + 0.60g8,t−1 + 0.30g8,t−2) + 0.80g9,t + ε1,t

g2,t = 0.60 (g1,t + 0.50g1,t−1 + 0.20g1,t−2) + ε2,t

g3,t = 0.05 + 0.22 (g1,t − 0.40g1,t−1 − 0.20g1,t−2) + ε3,t

g4,t = 1 + 0.4 (g1,t − 0.20g1,t−1 − 0.10g1,t−2) + ε4,t

g5,t = 0.062 + 0.16 (g1,t + 0.40g1,t−1 + 0.60g1,t−2) + ε5,t

g6,t = 0.60 (g1,t + 0.80g1,t−1 + 0.10g1,t−2) + ε6,t

g7,t = 0.70 (g1,t + 0.40g1,t−1 + 0.20g1,t−2) + ε7,t

g12,t = 0.80 (g16,t + 0.60g16,t−1 + 0.30g16,t−2) + 0.51g3,t + ε12,t

g13,t = 1.30 (g3,t + 0.50g3,t−1 + 0.50g3,t−2) + ε13,t

g14,t = 1 + 0.40 (g3,t + 0.40g3,t−1 + 0.60g3,t−2) + ε14,t

g15,t = 0.028 + 1.30 (g3,t + 0.60g3,t−1 − 0.30g3,t−2) + ε15,t

(29

On other hand, the use of a transformation with dynamicomponents (TChExt) overcomes the deficiencies of transformationCh, by the simultaneous consideration of both cross- and auto-orrelation. This feature, leads to an increase of the monitoringtatistics detection capabilities, especially on the case of the RMAXtatistics. Again, R0MAXChExt and R1MAXChExt presented a consis-ently better performance than the other monitoring statistics (seeig. 6), showing the importance of applying a transformation thatffectively breaks the variables’ correlation.

.2.3. Stationary non-linear systemThe original non-linear structure of the network system was

pproximated by polynomial relationships according to Eq. (30),here εi is a white noise sequence with a signal-to-noise ratio

able 7efinition of the faults and respective variables involved for the network systemith a non-linear model structure. ı is a multiplicative factor that changes the modelarameters (under NOC, ı = 1).

Fault Variables relation changed

A g8 → g1 (g1 = 1.20ıg8 + 0.80g9 + ε1)B g1 → g3 (g3 = (ıg1 − 4)(g1 + 4) + ε3)C g8 → g10 (g10 = 0.02g2

8 + 0.44ıg8 + 0.82 + ε10)D g3 → g14 (g14 = 0.020ıg2

3 + 0.44g3 − 0.82 + ε14)E g8 → g11 (g11 = −0.053ıg3

8 − 0.00068g28 + 0.52g8 + 0.50 + ε11)

F g3 → g15 (g15 = 1.40ıg3 + ε15)

C C C C R R E V

ynamics: box-plots of the area under the fault detection curve obtained for all

of 10 dB. The system was then subject to the faults presented inTable 7, where ı was set to cause changes on the model’s param-eters in the range of ±10%. A summary of the results obtained ispresented in Fig. 7 for the performance index (N) and in Table S3 ofthe Supplementary Material for the permutation tests.

g8 = ε8, g9 = ε9, g16 = ε16

g10 = 0.020g28 + 0.44g8 + 0.82 + ε10

g11 = −0.053g38 − 0.00068g2

8 + 0.52g8 + 0.50 + ε11

g1 = 1.20g8 + 0.80g9 + ε1

g2 = 0.60g1 + ε2

g3 = (ıg1 − 4)(g1 + 4) + ε3

g4 = 0.020g21 + 0.44g1 + 0.82 + ε4

g5 = −0.057g31 − 0.077g2

1 + 0.52g1 + 0.22 + ε5

g6 = 0.60g1 + ε6

g7 = 0.70g1 + ε7

g12 = 0.80g16 + 0.60g3 + ε12

g13 = 1.30g3 + ε13

g14 = 0.020g23 + 0.44g3 − 0.82 + ε14

g15 = 1.40g3 + ε15

(30)

Among the current monitoring statistics, only the E statisticspresent a good performance in detecting fault A, while for theremaining faults, all current statistics (including E) present detec-tion rates of less than 0.05. These poor detection capabilities maybe partially associated with the kind of perturbations simulated,which were made on non-linear terms or in network boundarynodes. This also explains why the E statistic was capable to detectfault A, which occurs close to the root node of the network and ispropagated to the rest of the network nodes. Given these results,the E statistics was chosen as the benchmark for this case study.

On the other hand, the proposed statistics were generally able todetect the simulated faults. Again, the sensitivity enhancing trans-formations and the use of partial correlations increase the detectioncapability of both MSPC-PCA and RMAX based monitoring statistics.However, due to the system’s non-linear nature, transformationsTCh and TChExt are not suitable for the RMAX statistics, as the par-tial correlations do not follow approximately a normal distribution

under these circumstances. For this case, a more complex trans-formation is required or, alternatively, an estimation of the partialcorrelations distribution for describing their variability. The MSPC-PCA based statistics are less sensitive to this problem, as most of the

T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322 319

0

0.2

0.4

0.6

0.8

1

1.2

Cr1

,Ch

R0M

AX C

h

R1M

AX C

h

Cr0

,Ch

Cr1

,X

Cr0

,X

Cr1

,ChE

xt

R1M

AX X

Cr0

,ChE

xt

R1M

AX C

hExt

VM

AX

R0M

AX X

R0M

AX C

hExt

E W

N

F n-linef

sttto

5

sssccsfotTc

mirtT

ig. 7. Comparison of the statistics performance for the network system with a noor all perturbations. Bars correspond to the associated mean values.

cores would still approximately follow a normal distribution dueo the central limit theorem effect induced in the computation ofhe scores. Nevertheless, the RMAX statistics proved to have poten-ial to be applied on non-linear systems and, when properly scaled,utperform all the studied statistics.

.3. Analysis of a continuous stirred-tank reactor

In this section, the comparative assessment of the monitoringtatistics performance is conducted with resource to a system pre-enting a dynamic non-linear model structure. This class of modeltructure can be found in real processes and present additionalomplexity features regarding the ones studied before, such aslose-loops and bidirectional dependencies. In this case study, 500ample covariance matrices were computed and collected for eachault. Each sample covariance matrix was calculated from 3000bservations. The fault detection rates were subsequently adjustedo a fault detection rate of 1% under normal operation conditions.his procedure was repeated 5 times in order to obtain approximateonfidence intervals for the fault detection rates.

As an example of this type of complex systems, a dynamicodel of a continuous stirred-tank reactor (CSTR) with a heat-

ng jacket and under feedback control was adopted, schematically

epresented in Fig. 8. In this system, an endothermic reaction ofhe type A → B takes place in a CSTR with free discharge flow.his system is under a PI control system in order to maintain the

Fig. 8. Process flow diagram for the CSTR.

ar model structure: box-plots of the area under the fault detection curve obtained

temperature and fluid level close to their set-points. The inputs ofthe system are the feed stream concentration (CA0) and tempera-ture (T0) and the heating fluid inlet temperature (Tj0). The systemoutputs are the CSTR level (h), concentration (CA), temperature (T)and the heating fluid outlet temperature (Tj).

The transformations were determined based on a reference dataset where the process was operating under normal conditions.Transformation TChExt with 2 lags for all variables was employed inorder to model the dynamic features of the data. In order to com-pare the performance of the monitoring statistics, the system wassubject to several perturbations, namely on the heat transfer coeffi-cient, discharge coefficient, pre-exponential factor and valve’s timeconstant (assuming a 1st order dynamic). Although these devia-tions were introduced in a single parameter at a time, they producemultiple changes in the correlation structure since they affect therelationships between several variables. The results are presentedin Fig. 9 for the performance index. The results obtained for thepermutation tests regarding each pair of monitoring statistics arepresented in Table S4 of the Supplementary Material.

The current monitoring statistics were only capable to detectfaults on the discharge coefficient and decreases on the heat trans-fer coefficient while the proposed monitoring statistics based ontransformation TChExt detected most of the faults, even for changesof small magnitude. This transformation only fails to detect faultson the valve’s time constant. These perturbations were not detectedby any of the studied monitoring statistics. Since this fault is relatedto the time that the control valve takes to respond to a controlaction, changes on the valve’s time constant are translated intohard to detect response time delays. Therefore, the detection ofthis fault relies in the SET ability to model the system dynamics.Thus, a more complex transformation would be required to cap-ture this effect. Nevertheless, the monitoring statistics based ontransformation TChExt were consistently better than all the others(see Fig. 9), confirming the potential of partial correlations to detectstructural changes, especially when coupled with suitable variabletransformations.

6. Discussion

The current statistics for monitoring the process multivariatedispersion are mainly based on the marginal covariance matrixand therefore have poor detection capabilities for changes in theprocess’ local correlation structure. This situation is even more

noticeable when the complexity of the system increases, whichis when these methods perform quite poorly (e.g., for non-lineardynamic systems). In order to improve the ability to detect struc-tural faults in such complex systems, we have explored the use

320 T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322

0

0.2

0.4

0.6

0.8

1

Cr1

,ChE

xt

R1M

AXC

hExt

Cr0

,ChE

xt

R0M

AXC

hExt

Cr0

,X

W VM

AX

R1M

AXX

Cr1

,X

Cr1

,Ch

E Cr0

,Ch

R0M

AXX

R1M

AXC

h

R0M

AXC

h

N

F he aret

ocpfs

tietwnttMbcvditevvomtu

PvTMi

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i

ig. 9. Comparison of the statistics performance on the CSTR system: box-plots of the associated mean values.

f partial correlations as a measure of the variables local asso-iations. This study showed that monitoring schemes based onartial correlation information obtained from process data can inact improve the ability to detect changes in the process correlationtructure.

However, the performance of the resulting monitoring statis-ics is dependent upon several factors. The most important ones the type of transformation implemented. The use of sensitivitynhancing transformations for monitoring the variables’ correla-ion structure is therefore another relevant contribution of thisork. These transformations rotate the original variables to aew set of uncorrelated variables, where changes in the correla-ion and partial correlation coefficients are easier to detect dueo the increase in sensitivity around the zero correlation state.

oreover, they can also be used to break the autocorrelationy including lagged variables in the data matrix. Additionally, asorrelations only capture the linear dependencies between theariables, non-linear data transformations must be adopted forealing with non-linear dependencies. This can be achieved, for

nstance, by extending the original data matrix with polynomialerms and then applying the Cholesky decomposition. However,ven though the Cholesky decomposition produces uncorrelatedariables, the resulting statistics performance is dependent on theariables ordering. To circumvent this problem, we suggest the usef some knowledge of the network structure in order to constructore localized linear/polynomial regression models involving only

he related variables (i.e., only the parents nodes or variables aresed as regressors).

Both approaches for monitoring the process structure (MSPC-CA-based and RMAX) have their performance improved when theariables are transformed and when partial correlations are used.he effect of the partial correlations was more noticeable on theSPC-PCA based statistics, while the RMAX statistics present sim-

lar performances under both circumstances.In the event of simultaneous changes in the correlation structure

as happens in the CSTR case study), the proposed methodolo-ies remain competitive. In general the MSPC-PCA-based approachhould have the same detection properties of the conventionalSPC-PCA for the mean, and react to several deviations at once.n the other hand, for the RMAX statistics the expectation is that

hey behave as if a single fault (with magnitude equal to the great-st deviation) is present. That is, the RMAX performance is driveny the greatest deviation in the correlation. Therefore, under simul-aneous deviations in the correlation structure the performance of

he proposed monitoring statistics is (at worse) the same as if onlyne fault occurred.

The major disadvantage of the proposed monitoring statisticss the amount of data required to construct the NOC PCA model

a under the fault detection curve obtained for all perturbations. Bars correspond to

and the requirement that all partial correlations follow the samedistribution (RMAX). The latter issue can be easily solved throughthe use of a proper data transformation or by adopting an esti-mation procedure for defining the empirical distribution of thecoefficients.

Both correlation and partial correlations monitoring statisticswere found to be rather insensitive to changes in the process vari-ance. Therefore, the use of a complementary monitoring statisticsfor detecting changes on the variance is advised.

The number of observations used to estimate the sample covari-ance matrix is another relevant factor. Even though we only presenthere the results obtained for the case of correlation coefficientsestimated based on 3000 observations, smaller window sizes werealso considered. For all the monitoring statistics, the performancedecreases as the number of observations decrease, but their relativeperformance is maintained in general. The only exception regardsthe MSPC-PCA-based statistics which no longer outperform the Wstatistic when a small number of observations is used. The RMAXstatistics showed to be less sensitive to this factor. This featuremakes the RMAX family of statistics suitable candidates to on-linemonitoring when coupled with an adequate sensitivity enhancingtransformation.

Regarding fault diagnosis based on marginal and partial correla-tions, a simple analysis of the marginal correlations falling outsidesome pre-established threshold can give an indication of the vari-ables involved and fault location. However, since the marginalcorrelation does not distinguish between directly and indirectlyrelated variables, the variables involved may not be correctly iden-tified in this way. This situation is exemplified in Fig. 10(a) for thestationary linear system of Section 5.2.1, regarding fault A (changein the relationship between variables 1 and 8) with a multiplica-tive factor ı of 1.20. 1st order partial correlations can be used toimprove the diagnostic properties. To do so, the number of partialcorrelations with values above the threshold is counted, with theith variable controlled. The controlled variable that gives the lowestcount can then be considered as being related with the fault, basedon the rational that, if the faulty variable is controlled, the corre-sponding partial correlations will remain under control, as a resultof removing the faults origin. For the same fault described ear-lier the distribution of the number of partial correlations obtainedby this procedure unequivocally selects the correct cause in about90% of the cases, as can be seen in Fig. 10(b). Similar results areobtained in other faults, such fault B represented in Fig. 11. Thisfault is a result of a relationship change between variables 1 and 3,

yet, by analysis of the marginal correlation, the correlations involv-ing variables 13–15 are above the threshold more frequently (seeFig. 11(a)). Even though these variables are directly dependent onvariable 3, they do not identify the correct root case. On the other

T.J. Rato, M.S. Reis / Computers and Chemical Engineering 71 (2014) 307–322 321

(b) (a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

80

90

100Marginal correlation

variable

% o

f ide

ntifi

catio

n

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

80

90

1001st order partial correlation

variable

% o

f ide

ntifi

catio

n

Fig. 10. Percentage of instances that each variable was identified as the faults’ root case on fault A (i.e., on the change of the relationship between variables 1 and 8), withı = 1.20, for the stationary linear system, in a total of 1000 cases. The threshold used by both methods was set for the same statistical significance of 0.01. The plot on the left(a) presents the identification results for the diagnosis based on marginal correlation, whereas the plot on the right (b) regards the use of the proposed procedure based onpartial correlations.

(b) (a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60Marg inal correlation

variable

% o

f ide

ntifi

catio

n

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

80

901st order partial correlation

variable

% o

f ide

ntifi

catio

n

Fig. 11. Percentage of instances that each variable was identified as the faults’ root case on fault B (i.e., on the change of the relationship between variables 1 and 3), withı by bo( n, whp

hltmtdstcsudm

= 1.10, for the stationary linear system, in a total of 1000 cases. The threshold useda) presents the identification results for the diagnosis based on marginal correlatioartial correlations.

and, variable 1 is identified as the root cause when partial corre-ations are employed (see Fig. 11(b)). These results, clearly suggesthat partial correlations can help in the isolation of the root cause

ore effectively, as a result of conveying more localized struc-ural information. Note that the presented diagnosis is conductedirectly with the original variables. Therefore, in order to applyuch procedure, it is recommended to use R1MAXX to monitorhe process, since it guarantees that at least one partial correlationoefficient is out-of-control, if a threshold equal to the monitoring

tatistic’s UCL is used. Other variable transformations can also besed for diagnosis purposes, with the advantage of improving theetection speed. Yet, the diagnosis based on transformed variablesight not be as straightforward as above.

th methods was set for the same statistical significance of 0.01. The plot on the leftereas the plot on the right (b) regards the use of the proposed procedure based on

7. Conclusions

In the present study we have proposed several monitoringstatistics based on the use of partial correlations in order to detectchanges in the process’ structure. These statistics were appliedto systems with different degrees of complexity, including linear,dynamic and non-linear systems and compared with the currentstatistics for monitoring process dispersion. Furthermore, severalsensitivity enhancing transformations were considered with the

goal of improving the methods performance regarding their abilityto detect several types of structural changes in the process.

In general, the proposed statistics present higher detection sen-sitivities for the same false alarm rate, especially when only few

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Yuan T, Qin SJ. Root cause diagnosis of plant-wide oscillations using Granger causal-

22 T.J. Rato, M.S. Reis / Computers and

ariables are affected by the faults, making more difficult theiretection. We have also showed that the use of variable trans-ormations that break variables’ correlation is a key element formproving the fault detection capability. We strongly believe, andur accumulated experience has been consistently confirming it,hat these SET carry a significant value to practitioners and toddress real world problems.

The characteristics of the RMAX family of statistics make themuitable to detect changes in the process’ structure even whenew observations are collected. Thus, they are suitable candidateso perform on-line monitoring. In this case the covariance matrixan be either estimated by adoption of moving windows, by usef updating schemes as the one presented by Wang et al. (2005),r through an EWMA recursion as proposed by Yeh et al. (2005).owever, RMAX does not account for changes in the variance and

herefore should be complemented with a monitoring statistic thatollows this particular feature. These issues will be addressed inuture work, along with other sensitivity enhancing transforma-ions that take explicitly into consideration a priori knowledgeegarding the variables underlying causal structure, available fornstance through process flow diagrams or estimated through

ethodologies based on transfer entropy (Bauer et al., 2007) orranger causality (Yuan and Qin, 2012). Part of this work is alreadyeing carried out (see for instance Rato and Reis, 2014).

cknowledgements

Tiago J. Rato acknowledges the Portuguese Founda-ion for Science and Technology for his PhD grant (grantFRH/BD/65794/2009). Marco S. Reis also acknowledges financialupport through project PTDC/EQU-ESI/108374/2008 co-financedy the Portuguese FCT and European Union’s FEDER through “Eixo

do Programa Operacional Factores de Competitividade (POFC)” ofREN (with ref. FCOMP-01-0124-FEDER-010397).

eferences

bbasi B, Niaki S, Abdollahian M, Hosseinifard S. A transformation-based mul-tivariate chart to monitor process dispersion. Int J Adv Manuf Technol2009;44:748–56.

lt FB. Multivariate quality control. In: Johnson NL, Kotz S, Read CR, editors. Theencyclopedia of statistical sciences, vol. 6. New York: Wiley; 1984. p. 110–22.

lt FB, Smith ND. Multivariate process control. In: Krishnaiah PR, Rao CR, editors.Handbook of statistics, vol. 7. Amsterdam: Elsevier; 1988. p. 333–51.

nderson TW. An introduction to multivariate statistical analysis. 3rd ed. NewJersey: Wiley; 2003.

parisi F, Jabaloyes J, Carrión A. Generalized variance chart design with adaptivesample sizes. The bivariate case. Commun Stat – Simul Comput 2001;30:931–48.

auer M, Cox JW, Caveness MH, Downs JJ, Thornhill NF. Finding the direction ofdisturbance propagation in a chemical process using transfer entropy. IEEE TransControl Syst Technol 2007;15:12–21.

hiang LH, Braatz RD. Process monitoring using causal map and multivariate statis-tics: fault detection and identification. Chemom Intell Lab Syst 2003;65:159–78.

osta A, Machado M. A new chart based on sample variances for monitoringthe covariance matrix of multivariate processes. Int J Adv Manuf Technol2009;41:770–9.

jauhari MA. Improved monitoring of multivariate process variability. J Qual Tech-nol 2005;37:32–9.

uente ADL, Bing N, Hoeschele I, Mendes P. Discovery of meaningful associa-tions in genomic data using partial correlation coefficients. Bioinformatics2004;20:3565–74.

hute VB, Shirke DT. A multivariate synthetic control chart for process dispersion.Qual Technol Quant Manage 2008;5:271–88.

cal Engineering 71 (2014) 307–322

Guerrero-Cusumano J-L. Testing variability in multivariate quality control: a condi-tional entropy measure approach. Inf Sci 1995;86:179–202.

Hawkins DM, Maboudou-Tchao EM. Multivariate exponentially weighted movingcovariance matrix. Technometrics 2008;50:155–66.

Jackson JE. Quality control methods for several related variables. Technometrics1959;1:359–77.

Jackson JE, Mudholkar GS. Control procedures for residuals associated with principalcomponent analysis. Technometrics 1979;21:341–9.

Kresta JV, MacGregor JF, Marlin TE. Multivariate statistical monitoring of processoperating performance. Can J Chem Eng 1991;69:35–47.

Ku W, Storer RH, Georgakis C. Disturbance detection and isolation by dynamic prin-cipal component analysis. Chemom Intell Lab Syst 1995;30:179–96.

Levinson WA, Holmes DS, Mergen AE. Variation charts for multivariate processes.Qual Eng 2002;14:539–45.

Li W, Yue HH, Valle-Cervantes S, Qin SJ. Recursive PCA for adaptive process moni-toring. J Process Control 2000;10:471–86.

Liu L-p, Zhong J-l, Ma Y-Z. A Multivariate synthetic control chart for monitoringcovariance matrix based on conditional entropy. In: Qi E, Shen J, Dou R, editors.The 19th International Conference on Industrial Engineering and EngineeringManagement. Berlin, Heidelberg: Springer; 2013. p. 99–107.

MacGregor JF, Jaechle C, Kiparissides C, Koutoudi M. Process monitoring and diag-nosis by multiblock PLS methods. AIChE J 1994;40:826–38.

MacGregor JF, Kourti T. Statistical process control of multivariate processes. ControlEng Pract 1995;3:403–14.

Melissa AS, Raghuraj KR, Lakshminarayanan S. Partial correlation metric based clas-sifier for food product characterization. J Food Eng 2009;90:146–52.

Negiz A, C inar A. PLS, balanced, and canonical variate realization techniques for iden-tifying VARMA models in state space. Chemom Intell Lab Syst 1997a;38:209–21.

Negiz A, C inar A. Statistical monitoring of multivariable dynamic processes withstate-space models. AIChE J 1997b;43:2002–20.

Pesarin F, Salmaso L. Permutation tests for complex data: theory, applications andsoftware. Chichester: John Wiley & Sons Ltd; 2010.

Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes: the art ofscientific computing. 3rd ed. New York: Cambridge University Press; 2007.

Rao KR, Lakshminarayanan S. Partial correlation based variable selectionapproach for multivariate data classification methods. Chemom Intell Lab Syst2007;86:68–81.

Rato TJ, Reis MS. Defining the structure of DPCA models and its impact on processmonitoring and prediction activities. Chemom Intell Lab Syst 2013;125:74–86.

Rato TJ, Reis MS. Sensitivity enhancing transformations for monitoring the processcorrelation structure. J Process Control 2014;24:905–15.

Reverter A, Chan EKF. Combining partial correlation and an information theoryapproach to the reversed engineering of gene co-expression networks. Bioin-formatics 2008;24:2491–7.

Sokal R, Rohlf FJ. Biometry: the principles and practice of statistics in biologicalresearch. 3rd ed. New York: W.H. Freeman & Co.; 1995.

Tamada Y, Kim S, Bannai H, Imoto S, Tashiro K, Kuhara S, et al. Estimating genenetworks from gene expression data by combining Bayesian network modelwith promoter element detection. Bioinformatics 2003;19:ii227–36.

Tracy ND, Young JC, Mason RL. Multivariate control charts for individual observa-tions. J Qual Technol 1992;24:88–95.

Wang J, He QP. Multivariate statistical process monitoring based on statistics patternanalysis. Ind Eng Chem Res 2010;49:7858–69.

Wang X, Kruger U, Irwin GW. Process monitoring approach using fast moving win-dow PCA. Ind Eng Chem Res 2005;44:5691–702.

Wise BM, Gallagher NB. The process chemometrics approach to process monitoringand fault detection. J Process Control 1996;6:329–48.

Xiao P. (Unpublished PhD Thesis) Robust MEWMA-type control charts for monitor-ing the covariance matrix of multivariate processes (Unpublished PhD Thesis).Virginia Tech; 2013.

Yeh AB, Huwang L, Wu C-W. A multivariate EWMA control chart for monitoringprocess variability with individual observations. IIE Trans 2005;37:1023–35.

Yeh AB, Lin DKJ, McGrath RN. Multivariate control charts for monitoring covariancematrix: a review. Qual Technol Quant Manage 2006;3:415–36.

Yen C-L, Shiau J-JH. A multivariate control chart for detecting increases in processdispersion. Stat Sin 2010;20:1683–707.

Yen C-L, Shiau J-JH, Yeh AB. Effective control charts for monitoring multivariateprocess dispersion. Qual Reliab Eng Int 2012;28:409–26.

ity. In: 8th IFAC International Symposium on Advanced Control of ChemicalProcesses, vol. 8; 2012. p. 160–5.

Yue HH, Qin SJ. Reconstruction-based fault identification using a combined index.Ind Eng Chem Res 2001;40:4403–14.