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Multivariate Cuscore control charts for monitoring themean vector in autocorrelated processesShuohui Chen a & Harriet Black Nembhard aa Laboratory for Quality Engineering and System Transitions, The Harold & Inge MarcusDepartment of Industrial & Manufacturing Engineering , The Pennsylvania State University ,University Park, PA, 16802, USAPublished online: 13 Feb 2011.

To cite this article: Shuohui Chen & Harriet Black Nembhard (2011) Multivariate Cuscore control charts for monitoring themean vector in autocorrelated processes, IIE Transactions, 43:4, 291-307, DOI: 10.1080/0740817X.2010.523767

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IIE Transactions (2011) 43, 291–307Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/0740817X.2010.523767

Multivariate Cuscore control charts for monitoring the meanvector in autocorrelated processes

SHUOHUI CHEN and HARRIET BLACK NEMBHARD∗

Laboratory for Quality Engineering and System Transitions, The Harold & Inge Marcus Department of Industrial & ManufacturingEngineering, The Pennsylvania State University, University Park, PA 16802, USAE-mail: hbnembhard@psu.edu

Received April 2009 and accepted July 2010

In many systems, quantitative observations of process variables can be used to characterize a process for quality control purposes.As the intervals between observations become shorter, autocorrelation may occur and lead to a high false alarm rate in traditionalStatistical Process Control (SPC) charts. In this article, a Multivariate Cuscore (MCuscore) SPC procedure based on the sequentiallikelihood ratio test and fault signature analysis is developed for monitoring the mean vector of an autocorrelated multivariateprocess. The MCuscore charts for the transient, steady and ramp mean shift signal are designed; they do not rely on the assumptionof known signal starting time. An example is presented to demonstrate the application of the MCuscore chart to monitoring threeautocorrelated variables of an online search engine marketing tracking process. Furthermore, the simulation analysis shows that theMCuscore chart outperforms the traditional multivariate cumulative sum control chart in detecting process shifts.

Keywords: Cuscore, vector ARMA model, autocorrelation, fault signature, multivariate process control, statistical process control

1. Introduction

The control chart is one of the primary techniquesof Statistical Process Control (SPC). In particular, theCumulative score (Cuscore) SPC chart can be devised tobe especially sensitive to deviations or signals of an ex-pected type. This idea tends to have intuitive appeal withpractitioners because they often develop a sense about howa process behaves and falters based on their first-hand ex-perience. For example, in health care surveillance, sentinelphysicians report flu-like illness data to a central collec-tion agency that is monitored for signs of an outbreakor epidemic; a moderate spike in symptoms is expectedsometime after the onset of holiday travel. In chemical pro-cessing, a valve is used to maintain pressure in a pipeline;engineers are concerned that it may fatigue more rapidlythan normal in extremely cold climates. In both exam-ples, a completely unexpected special cause may occurthat would result in an out-of-control signal on a tradi-tional SPC chart. However, some special causes can beanticipated and their likelihood designed into the Cuscorechart.

This actually raises another important point. In prac-tice, the Cuscore chart should be considered as an addi-tional chart in the monitoring arsenal—not a replacement

∗Corresponding author

of existing tools. In fact, it may be helpful in some casesto conceive of a set of Cuscore monitoring charts for aprocess for various expected causes and their periods oflikely occurrence, although mathematically this set may becollapsed into one computational pass.

The univariate Cuscore statistic takes the form:

Qt =t∑

i=1

ai0ri ,

where ai0 is the white noise of a process that is assumedto be in control and ri is the detector for the anticipatedsignal. The univariate Cuscore statistic have been used incontrol charts to monitor for anticipated process signals(see, for example, Box and Ramırez (1991), Box and Luceno(1997), Shu et al. (2002), Luceno (2004), Nembhard (2006),and Changpetch and Nembhard (2008)). Although not ex-pressly for the Cuscore, Box and Jenkins (1962) suggestedthe idea of using multiple SPC charts to monitor a process.Box (1980) provided an analogy for the application of Cus-cores drawing on the idea of a possible military air attackfrom an expected direction.

This research extends the univariate Cuscore statisticsto the multivariate environment and provides the the-oretical derivation of Multivariate Cuscore (MCuscore)statistics in monitoring the mean vector of autocorrelatedprocesses.

0740-817X C© 2011 “IIE”

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292 Chen and Nembhard

1.1. Autocorrelation

Traditional univariate statistical process analysis charac-teristically ignores the correlation among the process vari-ables and constructs a separate chart for each of them. Thisapproach often leads to inaccurate control limits and poordetection performance in monitoring mean shift signals forcorrelated variables.

The alternative is to use time series modeling techniquesto remove any autocorrelation structure in the observa-tions. In most cases it is assumed that the Box–Jenkinsmodel can be used to describe the behavior of observations.Consider a p-dimensional stationary multivariate time se-ries x1, x2, . . . , xt, which is represented by the vector autoregressive moving average VARMA(p, q) model:

xt = �q (B)�p(B)

εt,

where the white noise εt, t = 1, 2, . . ., is assumed to follow amultivariate normal distribution with p-dimensional meanvector 0 and variance-covariance matrix, MVN(0, �), and�q (B) and �p(B) are polynomials in B with �0(B) =�0(B) = I chosen to meet the criteria of stationary timeseries models (Montgomery et al. 1991; Hamilton, 1994).

To remedy the problem of high false alarms (Type I er-rors), some multivariate SPC approaches have been pro-posed for monitoring the mean vector of the autocorrelatedmultivariate process, such as Bakshi (1998), Noorossanaand Vaghefi (2005), and Jarrett and Pan (2007). However,the issue of analysis of the fault signature, the change pat-tern of the fault signal imposed on the output residual inremoving the process autocorrelation by filtering, and theimpact of the autocorrelation for different type of signalshave not been directly and thoroughly addressed.

1.2. Multivariate SPC

By assuming an independent and identically distributedmultivariate Gaussian process with constant covariancematrix among sequential observations, Shewhart, Cumu-lative sum (Cusum) and Exponentially Weighted MovingAverage (EWMA) charts can be extended to the multivari-ate environment to monitor the process mean. Well-knownapproaches are Hotelling’s T2 chart (the multivariate coun-terpart to the Shewhart chart; Hotelling, 1947), the Multi-variate Cusum (MCusum) control chart (Healy, 1987), andthe Multivariate EWMA (MEWMA) control chart (Lowryet al., 1992).

Because the MCuscore is characteristically similar to theMCusum in that they are both accumulating statistics, wewill make some comparisons between the two procedures.Healy (1987) derived an MCusum procedure based on thesequential likelihood ratio test of multivariate variables.The MCusum statistic for detecting a specific shift in a

p-variate process mean vector γ is

St ={

max(St−1 + aTxt − K, 0

)> H,

min(St−1 + aTxt + K, 0

)< −H,

(1)

where S0 = 0, xt is the sample mean vector at time t, H is afixed threshold, a is a p × 1 vector of constants defined as

aT = γT�−1√γT�−1γ

,

and

K = γT�−1γ

2√

γT�−1γ.

Specifically, aTxt ∼ N(0, 1) when the process operates incontrol with mean vector 0, and aTxt ∼ N(0, aTγ) whenthe process has shifted to an out-of-control state. Bydefining the non-centrality parameter D =

√γT�−1γ, the

MCusum statistic can be written in the form of Equation (1)using K = D/2.

Noorossana and Vaghefi (2005) applied the MCusumcontrol chart to monitor the residuals from a vector AR(1)time series model for a mean vector shift by assuming thatthe parameters of the model were estimated accurately byusing the historical data. They showed by simulation thatthe Average Run Length (ARL) properties of MCusumcontrol charts can be improved considerably if the resid-uals from a time series model were used instead of theoriginal data. Jarrett and Pan (2007) developed a VectorAuto Regressive (VAR) chart for monitoring an autocorre-lated multivariate process with transient mean shift basedon the T2 chart. They discussed the distribution of theadditional effects of autocorrelation to the non-shifted T2

statistic.Various techniques, including theoretical derivation,

Markov chain approximation, Monte Carlo simulation,or their combinations, have been used to investigate thein-control and out-of-control run length properties ofunivariate monitoring schemes (see, for example, Brooksand Evans (1972), Apley and Shi (1999), Luceno (2004),Han and Tsung (2005, 2006), and Shu et al. (2008)) andMCusum and MEWMA schemes (see, for example, Crosier(1988) and Runger and Prabhu (1996)). We note that Hanand Tsung (2005) proved that there is a lower (or upper)bound in the definition of classical Cuscore charts for uni-variate processes as in the Cusum chart. Compared withunivariate control charts, the run length properties of multi-variate control charts have been less well studied, especiallywhen autocorrelation exists in the process. The particulartime-varying form of the fault signature depends heavilyon the signal type and the VARMA model that describesthe original process and therefore affects the performanceof the applied control chart.

There have been many other studies on the MCusum,including Crosier (1988) and Hawkins and Olwell (1998).Indeed, the body of literature on multivariate SPC charts

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MCuscore chart for autocorrelated vectors 293

in general is quite large. A recent review of multivariateSPC by Bersimis et al. (2007) contains 195 references. Ofcourse, this area of research continues to develop; rele-vant articles that have appeared since the Bersimis reviewinclude Aparisi et al. (2006), Reynolds and Cho (2006),Testik and Runger (2006), Hawkins and Maboudou-Tchao(2007), Camci et al. (2008), Joner et al. (2008), Chenouriet al. (2009), Jobe and Pokojovy (2009), and Zamba andHawkins (2009).

�t =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

diag{0, . . . , 0} for t �= t0 and diag{1, . . . , 1} for t = t0 (transient shift signal),diag{0, . . . , 0} for t < t0 and diag{1, . . . , 1} for t ≥ t0 (steady shift signal),diag{0, . . . , 0} for t < t0 and diag{t − t0, . . . , t − t0} for t ≥ t0 (ramp signal),

diag{0, . . . , 0} for t < t0 and diag{e−i(t−t0), . . . , e−i(t−t0)} for t ≥ t0 (sinusoidal signal),

1.3. Organization of this article

For comprehensiveness, in Section 2, the MCuscore statis-tic is strictly derived with the aid of fault signatures in aVARMA model. Underlying assumptions for MCuscorecharts are summarized and the MCuscore charts for tran-sient, steady, and ramp mean shift signals are designed.Then, in Section 3, the MCuscore chart approach is ap-plied to an online search engine marketing process in whichexperimental design and MCuscore chart are applied toachive better process control. Section 4 provides a discus-sion of the fault signature of transient and steady meanshift signals in the output of the inverse VARMA filter,especially for the VAR process. In Section 5, the perfor-mance of the MCuscore chart is investigated and com-pared with the residual-based MCusum control chart, andthe robustness of the control chart is briefly discussed. Theapplication of the MCuscore chart to industrial applica-tions is addressed in Section 6. Finally, in Section 7, wepresent our conclusions and a discussion of future researchopportunities.

2. MCuscore statistics for monitoring the process meanvector

In this section, a p-dimensional multivariate autocorrelatedprocess is formulated using the VARMA time series model,and the MCuscore chart is derived to monitor the processmean vector. The distribution of MCuscore statistics forcertain types of mean shift signals is discussed.

2.1. Assumptions for the MCuscore chart

In extending the Cuscore chart from monitoring the uni-variate process mean to monitoring the multivariate processmean vector, we consider a mean vector shift signal γ ap-plied to a sequence of in-control p-dimensional observation

vectors x1, x2, . . . , xt, which is fitted by a VARMA(p,q)model:

xt = �q (B)�p(B)

εt + �tγ, (2)

where �tγ represents the mean shift signal, γ indicates thevector of the shift size, and �t is a diagonal pattern matrixindicating the type of mean shift signal, such as

and t0 is the signal’s starting time or the change-point inthe process.

Equation (2) can be transformed into

εt = �p(B)�q (B)

xt − �p(B)�q (B)

�tγ. (3)

The second term in Equation (3) is defined as theVARMA(p,q) fault signature for mean shift �tγ

δt = �p(B)�q (B)

�tγ,

and it will be discussed in detail in Section 4.We state the underlying assumptions for the MCuscore

statistic as follows.

1. The signal’s size vector γ and pattern matrix �t areknown a priori. For the ramp shift signal, the transitionperiod �t from the current steady-state level to the newsteady-state level is known a priori.

2. The noise vector εt ∼ MVN(µ, �), and µ and � areknown. We suggest µ be normalized to 0 before usingthe MCuscore chart.

3. The variance-covariance matrix � is constant and inde-pendent of signal �tγ.

4. A stationary and invertible VARMA model has beenidentified and fitted. The cross-correlation matrix, �p,and moving average matrix, �q , if they exist, are con-stant and independent of signal �tγ.

Note that the starting time, t0, does not need to be knowna priori for this proposed MCuscore chart. Instead, it isto be detected by correlating the fault signature with thefiltered process residuals, which will be discussed in detailin the following sections.

Similar to the univariate time series, stationarity andinvertibility are two important properties for a VARMAtime series model. Stationarity requires that neither theprocess mean vector nor the covariance matrix depend ontime t, and invertibility requires that a VMA(1) model can

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294 Chen and Nembhard

be inverted to a VAR(∞) model. For the example of aVARMA(1,1) model such as

xt = �xt−1 + εt − �εt−1,

stationarity requires that all elements λt of the root vectorλ for equation:

det(I − λ�) = 0, (4)

are inside the unit circle, or |λt| < 1. All elements λt of theroot vector λ for equation:

det(I − λ�) = 0, (5)

are also inside the unit circle, or |λt| < 1. This article pro-vides a minimum background on the multivariate time se-ries model in order to clarify the derivation and analysisof the MCuscore chart. Hamilton (1994), Tsay (2005), andBox et al. (2008) have provided detailed discussions on sta-tionarity and invertibility of VARMA time series models.Box et al. (2008) discussed the procedure of VARMA modelidentification, estimation, and checking. In the SAS statisti-cal software package, the subroutine PROC STATESPACEcan be used to perform VARMA model identification andfitting.

The first and fourth of the previously listed assumptionsrequire the system identification and pre-determination ofsignals that are usually conducted in Phase I. Once thisis done, the control limits for a given in-control ARL canbe determined through simulation. In addition, as will beshown in Section 2.2, the MCuscore chart is equivalentto the common multivariate control charts in many cases,such as T2 chart and MCusum chart, etc., depending onthe signal types, and it can be applied to Phase I and PhaseII monitoring.

2.2. MCuscore statistics for the process mean vector

Similar to the derivation of the univariate Cuscore statistic,we use the sequential probability ratio test to derive theMCuscore statistic. Let the hypothesis test be

H0 : γ = γ0 versus. H1 : γ �= γ0,

where γ 0 is from the in-control process and can be assumedto be 0. Rearrange Equation (2) and let the null model be

at(γ0) = εt = �p(B)�q (B)

(xt − �tγ0), (6)

and the discrepancy model be

at(γ) = �p(B)�q (B)

(xt − �tγ). (7)

at : R → �p is defined on a subset γ of �p, and that at(γ ) isdifferentiable at point γ 0. Based on the second assumptionin Section 2.1, the log sequential probability ratio of the

test takes the form

LRk = −12

k∑t=1

[aT

t (γ)�−1at(γ) − aTt (γ0)�−1at(γ0)

]

= −12

k∑t=1

[( f ◦ at)(γ) − ( f ◦ at)(γ0)], (8)

where ( f ◦ at)(·) denotes the composite function f (at(·));f (x) = xT�−1x and f : x → � is a quadratic function de-fined on a subset x of �p; and f (x) is differentiable at pointx0.

Based on the third and fourth assumptions in Section 2.1,using vector calculus and the chain rule to expand ( f ◦at)(γ) into a second-order Taylor series around γ0 = 0 gives

( f ◦ at)(γ) = ( f ◦ at)(γ0) + 2γT(

∂at(γ)∂γ

)T ∣∣∣∣γ=0

�−1at(0)

+γT(

∂at(γ)∂γ

)T ∣∣∣∣γ=0

�−1 ∂at(γ)∂γ

∣∣∣∣γ=0

γ. (9)

(Note that this procedure is discussed in detail in the Ap-pendix.) Substituting Equation (9) into Equation (8) gives:

LRk = −12

k∑t=1

[2γT

(∂at(γ)

∂γ

)T ∣∣∣∣γ=0

�−1at(0)

+γT(

∂at(γ)∂γ

)T ∣∣∣∣γ=0

�−1 ∂at(γ)∂γ

∣∣∣∣γ=0

γ

], (10)

where ∂at(γ)/∂γ is the p × p Jacobian matrix. Define theMCuscore detector matrix:

d(γ) = −∂at(γ)∂γ

∣∣∣∣γ=0

, (11)

and d : γ → �p is defined on a subset γ of �p. Equa-tion (10) can be written as the MCuscore form of

LRk=k∑

t=1

[(d(γ)γ)T�−1at(γ0)−1

2(d(γ)γ)T�−1(d(γ)γ)

].

(12)

Equation (11) provides a generic form for the detector ma-trix, which is a key concept of MCuscore. Equation (12)provides a generic form for the log sequential probabilityratio. Equations (11) and (12) can be applied to the casesin which the signal is not only with the process mean, butalso with the process autocorrelation or cross-correlation.In this article, we limit our study to monitoring the processmean. Therefore, the detector matrix d(γ) takes the follow-ing form by substituting Equation (7) into Equation (11):

d(γ) = �p(B)�q (B)

�t. (13)

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MCuscore chart for autocorrelated vectors 295

It can be seen that the detector matrix for MCuscore statis-tics is the multiplication of the VARMA filter and the sig-nal’s pattern matrix. The impact of the VARMA filter onsignals will be discussed in Section 4. Substituting the nullmodel (6) and Equation (13) into Equation (12) and lettingγ0 = 0 gives the MCuscore statistics:

MCk =k∑

t=1

[(�p(B)�q (B)

�tγ

)T

�−1εt

− 12

(�p(B)�q (B)

�tγ

)T

�−1(

�p(B)�q (B)

�tγ

)](14)

=k∑

t=1

[δT

t �−1 �p(B)�q (B)

xt − 12δT

t �−1δt

](15)

=k∑

t=1

[(�tγ)T�∗−1xt − 1

2(�tγ)T�∗−1(�tγ)

], (16)

where

�∗−1 =(

�p(B)�q (B)

)T

�−1(

�p(B)�q (B)

). (17)

Remark 1: It can be verified that Equations (14) to Equa-tion (16) reduce to the classic Cuscore charting statistic(for residuals) in Box and Ramırez (1991) for univariateprocesses.

Remark 2: Equations (16) and (17) provide an addi-tional way to interpret the effect of � and � on thevariance-covariance matrix �. It is shown that by using theoriginal signal size and an equivalent variance-covariancematrix �∗, the MCuscore chart takes the same form ofthe MCusum statistics as in Equation (1), if the processstandardization is not considered. The multivariate nor-mal distribution assumption in Section 2.1 requires that thevariance-covariance matrix � is positive-definite. It is easyto prove that the transformed matrix �∗ is also positive-definite. The ratio of �p(B)/�q (B) determines the cumu-lated change of the original matrix � at time t. Therefore,it can be seen clearly that applying traditional multivari-ate charts to monitor the mean vector of a VARMA(p,q)model is not accurate. With this advantage, MCuscore chartconsiders the change of � and explains it in its model.

Remark 3: The charting statistic in Equation (16) is di-rectionally variant, which means the chart performance isaffected not only by the magnitude but also by the directionof each mean vector component’s shift. The directionallyvariant T2 control chart was also discussed in Wang andTsung (2008) in monitoring univariate dynamic systems,in which the T2 statistic is computed recursively using aprocess quality characteristics vector vt of L1 successiveoutput observations and L2 input observations and theEWMA forecasting of vt as a reference shift vector. Both theMCuscore statistic and the adaptive T2 statistic in Wangand Tsung (2008) assume a multivariate normal distri-

bution of the monitored vectors and both use MonteCarlo simulation to calculate ARLs and show that theyoutperform their corresponding directionally invariantT2-type chart (this will be shown in Sections 3 and5 for the MCuscore statistic). However, the MCuscorestatistic differs from the adaptive T2 statistic in a fewaspects. First, the MCuscore is for monitoring multi-variate autocorrelated processes and considers the autocorrelation and cross-correlation among the monitoredvariables as well as the change patterns of differenttypes of mean shift signals; conversely, the adaptiveT2 statistics is for monitoring the dynamic univariateprocess using constructed multivariate statistic. Second,the dimension of the observation vector is fixed in theMCuscore statistic by the nature of the multivariate pro-cess; conversely, it is determined by algorithm in the adap-tive T2 statistic depending on the structure and magnitudeof the process autocorrelation, which is shown by simula-tion to be an effective selection method. Third, the referenceshift vector, denoted by �tγ in Equation (16), is assumedto be known from the process in the MCuscore statistic;it is, however, estimated from the EWMA forecasting re-cursively of using the latest observation vector vt in theadaptive T2 statistic.

Remark 4: In the scenario of constant �, �, and � matricesof the system, the derivation of MCk in Equations (14) to(16) is straightforward. Substituting at(γ0) and at(γ) ofEquations (6) and (7) into Equation (8), expanding theproduct of vectors and matrices and rearranging the termsgives Equation (14) directly. However, Equations (8) to (16)illustrate the generic form and the concept of MCuscorestatistics for the process mean shift, which also provides thepotential use for monitoring the shift in the components of� and � matrices in future work.

2.3. MCuscore chart for detecting specific signals

Similar to the Cuscore chart for univariate processes, theMCuscore chart provides a generic approach to monitor-ing a variety of signals on the mean vector of multivariateprocesses, and it can be transformed into various specificmultivariate charts depending on type of signal.

2.3.1. Transient mean shiftFor a transient mean shift signal �tγ, where �t =diag{1, . . . , 1} for t = t0, and �t = diag{0, . . . , 0} for t �= t0,the MCuscore statistics in Equation (16) can be written as

MCt = 2Kt(aTt xt − Kt), (18)

where MC0 = 0, Kt is defined as

Kt = 12

√δT

t �−1δt,

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and at is a vector of constants defined as

aTt = δT

t �−1√δT

t �−1δt

�p(B)�q (B)

.

As has been noted, the starting time t0 is not required to beknown a priori for the MCuscore chart.

The term aTt xt in Equation (18) is a linear combination

of normally distributed variables, which can be seen moreclearly in Equation (14), and the term Kt is independentof the observations xt. Therefore, MCt is normally dis-tributed with mean −2K2

t and variance δTt �−1��−1δt =

δTt �−1δt = 4K2

t , or written as MCt ∼ N(−2K2

t , 4K2t

).

It can be seen from the derivation of Equations (14)to (16) that the MCuscore statistics is the additional partto the Hotelling T2 statistics of the null model εt = at(γ0),at(γ0)T�−1at(γ0) ∼ χ2(p) with mean p and variance of2p, where p is the dimension of the process. Therefore, anMCuscore-χ2 chart based on a combination of Hotelling’sT2 statistics and MCuscore statistics can be constructed formonitoring the transient mean shift as follows:

MCt =(

�p(B)�q (B)

xt

)T

�−1 �p(B)�q (B)

xt − 4KtaTt xt + 4K2

t .

The MCuscore-χ2 chart can be used in both Phase I andPhase II with control limits determined by the χ2 for pro-cess dimension p. Jarrett and Pan (2007) developed a VARchart for monitoring autocorrelated multivariate processwith a transient mean shift. Their control statistics consistof a non-shift T2

t statistic and two additional terms, whichhave a form and distribution similar to MCuscore statistics.

2.3.2. Steady mean shiftFor a steady mean shift signal �tγ, where �t =diag{0, . . . , 0} for t < t0, and �t = diag{1, . . . , 1} for t ≥ t0,the MCuscore statistic in Equation (16) can be written as atwo-sided chart:

MCt ={

max(MCt−1 + aT

t xt + Kt, 0)

> H,

min(MCt−1 + aT

t xt + Kt, 0)

< −H,(19)

where MC0 = 0; H is the control limit, which can be deter-mined in Phase I by simulation; and at and Kt are defined inEquation (18). The procedure is to simulate a large numberof runs where each run has enough in-control observationvectors. We recommend at least 5000 runs and 1000 obser-vations in each run. The MCt can be computed with thepredetermined signal size γ. Then, various control limitsare tested in the simulation until the ARL is approximatelyequal to the required in-control ARL0, normally 200 formultivariate control.

Similar to the transient mean shift case, the starting timet0 is not required to be known a priori for the MCuscorechart in the steady mean shift case. The chart in Equa-tion (19) is applied to the whole monitored process sequen-tially until an alarm is fired.

In addition, the upper-sided part of Equation (19) is de-rived directly from Equation (16). The lower-sided part isan analog for testing the shift in the negative direction, andit does not play any important role in detecting the positivemean shift. Therefore, the upper-sided part can be used asa stand-alone chart in the case of a positive mean shift.The two-sided MCuscrore statistic for the steady meanshift signal takes a similar form to Equation (1), exceptthat the mean shift term becomes the fault signature inEquation (19) instead of the constant mean shift vector.Depending on the process dimension, the variance-covariance matrix �, the auto-correlation matrix �p, andmoving-average matrix �q , the fault signature can take acomplicated pattern. A brief discussion on fault signaturesis presented in Section 4.

2.3.3. Ramp signalFor a ramp signal �tγ, where �t = diag{0, . . . , 0} for t < t0,and �t = diag{t − t0, . . . , t − t0} for t ∈ [t0, t0 + �t], it hasbeen assumed in Section 2.1 that the transition period �tfrom the old steady-state level to the new steady-state levelis known a priori. Therefore, to remove the assumption ofprior knowledge of signal starting time t0, we suggest thefollowing two-sided MCuscore chart, which uses a movingtime window of length �t:

MCt,k ={

max(MCt,k−1 + aTk xt+k + Kk, 0) > H,

min(MCt,k−1 + aTk xt+k + Kk, 0) < −H,

(20)

where MC0,0 = 0; t is the observation time; k =1, 2, . . . , �t; MCt,k represents the MCuscore statistic atthe kth observation within the moving window, whichstarts from time t; ak and Kk are defined in Equation (18)for each window t + �t; and �k = diag{k, . . . , k} for k =1, 2, . . . , �t.

Similar to the steady mean shift case, the upper-sided partof Equation (20) is derived directly from Equation (16), andthe lower-sided part is an analog for testing the shift in thenegative direction. Therefore, the upper-sided part can beused as a stand-alone chart in this case of positive meanshift.

The control limit H can be determined by simulation. Werecommend large runs of 50 000 or more, �t observationsin each run for the simulation, and H be established for thein-control false alarm rate 1/200 in Phase I. The evaluationcriteria for MCuscore chart performance on ramp signalsare slightly different from those on steady-mean shift sig-nals because of the limited length of the transition period.We recommend the false alarm rate for the Phase I criterionand combining the detection rate and ARL ratio, definedas the ratio of ARL and �t, for the Phase II criterion.Therefore, for two charts with the same false alarm rate,the one having the higher detection rate and small ARLratio wins. An application example that uses an MCuscorechart to monitor the ramp signal is illustrated in the secondpart of the next section.

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Fig. 1. An example of search engine listing with Monthly-Credit.coms ad at rank 3.

The transition period �t is required for this MCuscorechart in Equation (20). For a more general case where �t isunknown, the MCuscore chart in Equation (20) can still betried but the optimal window length needs to be determinedby simulation. Such a study may be appropriate in futureresearch on the MCuscore chart.

3. Application example

MonthlyCredit.com is a credit consulting service that en-ables American consumers to receive a free annual credit re-port for better credit management. They use Search EngineMarketing (SEM) to attract online traffic to their website.In a typical SEM campaign, MonthlyCredit.com placestheir bid price on certain keywords, such as “credit” and“credit report.” The rank at which their advertisement isdisplayed on the search result page depends on the bidprices of their keywords and their landing pages relevanceto the search words. Figure 1 illustrates a listing of searchresults where MonthlyCredit.coms advertisement is at rank3. Once a user clicks its link, he or she is directed to Month-lyCredit.coms website and is requested to register for a freecredit report.

MonthlyCredit.com tracks the SEM listing Click-Through Rate (CTR) to its website, which measures theprobability of users’ clicks on its delivered advertisement tothe SEM listing; as well as Registration Rate (RR), whichis defined as a complete registration process through theregistration page. Usually, an advertisement’s SEM listingrank is correlated to CTR and RR.

The keywords’ bid prices and the landing page’s rele-vance are two major factors that an advertiser can controlto improve their SEM traffic. MonthlyCredit.com uses ex-perimental designs on these two factors to maximize theCTR and RR. After establishing the factors’ settings, they

use the MCuscore chart to monitor the multivariate re-sponses of rank, CTR and RR. If they find out-of-controlsignals, they adjust the experiment settings and the controlscheme to achieve a new state of control.

The MCuscore chart is applied to this process for thefollowing reasons.

1. The mean shift size is predicted from the experiments,which can be used for setting up the MCuscore chartscheme.

2. The process has been identified as a VAR(1) station-ary process with constant non-zero cross-correlationsamong neighboring observation vectors.

3. After each change of the experiment settings, the meanshift needs to be captured as quick as possible at a lowfalse alarm rate.

To model this problem, let xt = (x1t, x2t, x3t)′ denote theprocess response vector (rank, CTR, RR). Its mean vector,standard deviations, variance-covariance coefficient ma-trix, and cross-correlation coefficient matrix are estimatedfrom the historical data as follows:

µ =⎛⎝ 3

0.0030.05

⎞⎠ , σ =

⎛⎝ 1.5

0.0030.02

⎞⎠ ,

ρ0 =⎛⎝ 1.00 −0.70 −0.30

−0.70 1.00 0.60−0.30 0.60 1.00

⎞⎠

and

ρ1 =⎛⎝ 0.3 −0.20 −0.09

−0.20 0.3 0.22−0.15 0.23 0.5

⎞⎠ .

For ease of comparing the levels among each componentof the responses, the process is standardized by its meanµ and standard deviations σ. The shift size vector γ inthe following description represents γ units of standarddeviations.

3.1. Sudden steady mean shift

In one of recent SEM campaigns, MonthCredit.com ad-justed its bid price and land page designs at day t0 = 10.A sudden steady mean shift with γ = (1 − 0.1 − 0.1)′ unitsof standard deviations is expected to occur. The time seriesof process observations and fault signatures are illustratedin Fig. 2 and three multivariate control charts are shownin Fig. 3. The mean shift signal is detected in 8 days by theone-sided MCuscore chart as shown in Fig. 3(a), and 9 daysby Healy’s MCusum chart as shown in Fig. 3(b), and barely13 days by Crosier’s MCusum chart as shown in Fig. 3(c).The MCuscore chart control limit is 3.84, Healy’s MCusumchart limit is ±3.3, and Crosier’s MCusum chart’s limit is±2.84 with all values being determined through 10 000 runsin a Monte Carlo simulation at an in-control ARL0 = 200.

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Fig. 2. (a) Time series plot of steady mean shifted process and (b) fault signature.

As a general comparison of the three charts for this pro-cess, 10 000 runs of a Monte Carlo simulation are usedto estimate the out-of-control ARLs by fixing the con-trol limit at 3.84 for the MCuscore chart, ±3.3 for Healy’sMCusum chart, and ±2.84 for Crosier’s MCusum chart.For the stated shift γ, the MCuscore chart’s ARL1 is7.3, Healy’s MCusum chart’s ARL1 is 7.7, and Crosier’sMCusum chart’s ARL1 is 12.5.

3.2. Ramp mean shift

In another campaign, MonthCredit.com changes the bidprice and system design at day t0 = 10. A mean shiftwith γ = (1 − 0.1 − 0.1)′ units of standard deviations isexpected to occur. However, the process mean is shiftedas a ramp this time and it takes 30 days to fully trans-fer to the new level. The time series plot of process ob-servations, the original fault signal, and its fault signa-tures are illustrated in Fig. 4 and three multivariate con-trol charts are plotted in Fig. 5. The ramp shift signal isdetected in 18 days by the one-sided MCuscore chart (20)as shown in Fig. 5(a), 28 days by Healy’s MCusum chart in

Fig. 5(b), and it is not caught by Crosier’s MCusum chartas shown in Fig. 5(c). The MCuscore chart’s control limit is9.85, Healy’s MCusum chart’s limit is ±5.70, and Crosier’sMCusum chart’s limit is ±4.33 and they were determinedby a 50 000 run Monte Carlo simulation at the in-controlfalse alarm rate 1/200.

As a general comparison of the three charts for this pro-cess, 50 000 runs of a Monte Carlo simulation are used toestimate the out-of-control ARL1 values by fixing the con-trol limit at 9.85 for the MCuscore chart, ±5.70 for Healy’sMCusum chart, and ±4.33 for Crosier’s MCusum chart.For the stated ramp shift, the MCuscore chart’s detectionrate is 60.4% and its ARL1 is 23.2, Healy’s MCusum chart’sdetection rate is 46.7% and its ARL1 is 24.6, and Crosier’sMCusum chart’s detection rate is 16.3% and its ARL1 is25.0.

4. Fault signature of a process mean shift

For a multivariate correlated process, a properly fitted mul-tivariate time series model can serve as an inverse filter to

Fig. 3. Control charts on steady mean shifted process: (a) the MCuscore chart; (b) Healy’s MCusum chart; and (c) Crosier’s MCusumchart.

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Fig. 4. (a) Time series plot of ramp shifted process: (b) original signal and (c) fault signature.

remove the autocorrelation in the data. The fault signalof a mean vector shift in the original data takes the formof a transient response and is imposed on the multivariatewhite noise residuals. In some process control literature, thechange pattern of a fault signal is called a fault signature be-cause the controller or filter produces a signature or patternof the fault signal in the output series (see, for example, Ap-ley and Shi (1999); Yoon and MacGregor (2001)). The faultsignature depends on the structure of the VARMA modeland the model parameters. Hu and Roan (1996) discussedthe use of fault signatures in the case of the univariateautocorrelated process. We explore the details of fault sig-natures in order to understand the underlying mechanismin detection using the MCuscore approach.

Note that the size and pattern of the fault signature canbe affected by many factors, such as the dimension of theprocess, the size and pattern of the mean shift signal, thestructure of the VARMA model, and the parameter matri-ces of �, �, and �; thus it can be very complicated. Anextensive study on the pattern of the fault signatures formultivariate time series models is beyond the scope of thisarticle. In the next two sections, we will limit our discussionto the effects of the � and � matrices on fault signaturesof the mean shift vector with unit elements in a bivariate

process with fixed variance-covariance matrix � for themultivariate normal distribution of the white noise.

For an autocorrelated process with a mean shift signal,the output from an inverse filter is composed of two parts:a white noise series and a fault signature of the mean shiftsignal (Equation (3)). Figure 6 graphically illustrates thecomposition of the mean shift signal and the vector timeseries and the filtering of an inverse VARMA model.

4.1. Fault Signature in the VAR(1) Process

As in the univariate process analysis, the structure of theVARMA model and its parameters can affect the magni-tude and the pattern of the fault signature. We use a bivari-ate VAR(1) model to illustrate the impact of the � matrixon the fault signature. In the simulation, we assume thewhite noise term εt in the VAR(1) model has a multivariatenormal distribution with µ = (0, 0)T and

� =(

1 0.50.5 1

).

The length of process is 20 and the mean shift vector γ =(1, 1)T starts at the tenth observation.

Fig. 5. Control charts on ramp shifted process: (a) the MCuscore chart; (b) Healy’s MCusum chart; and (c) Crosier’s MCusum chart.

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Fig. 6. Composition and filtering of a bivariate VARMA(1,1)process.

We first study the effect of φ1 on the fault signature bydecreasing the value of φ1 from 0.7 to −0.7 while keepingthe other three parameters fixed (Table 1 and Fig. 7). Then,we study the effect of φ12 on the fault signature by increasingφ12 from −1.2 to 1.6 while keeping others fixed (Table 2and Fig. 8). The values are selected to meet the stationaritycondition of a VAR(1) model (Equation (4)).

4.2. Fault signature in the VARMA(1,1) process

To further study the impact of the parameters of theVARMA model on the fault signature of the mean shiftvector (1, 1)T, we illustrate the fault signatures using three� matrices by varying only the φ1 value, and each � ma-trix corresponds to six � matrices (see Tables 3 and 4).The � and � matrices are selected to meet the stationar-ity and invertibility condition of the VARMA(1,1) model(Equations (4) and (5)).

For comparison, we first plot the fault signatures of themean shift vector (1, 1)T for a VMA(1) filter in Fig. 9. Eachpanel corresponds to the respective value in Table 4 withthe � matrix from rows 1–6, respectively. It can be seen that

Table 1. � matrix obtained by varying φ1

aφ1 φ12 φ21 φ2

a 0.7 0.1 0.2 0.5b 0.4 0.1 0.2 0.5c 0.1 0.1 0.2 0.5d −0.1 0.1 0.2 0.5e −0.4 0.1 0.2 0.5f −0.7 0.1 0.2 0.5

Table 2. � matrix obtained by varying φ12

φ1 φ12 φ21 φ2

a −0.7 −1.2 0.2 0.5b −0.7 −0.8 0.2 0.5c −0.7 −0.4 0.2 0.5d −0.7 0.0 0.2 0.5e −0.7 0.4 0.2 0.5f −0.7 0.6 0.2 0.5g −0.7 0.8 0.2 0.5h −0.7 1.2 0.2 0.5i −0.7 1.6 0.2 0.5

the first three fault signatures steadily increase to a constantlevel higher than the original level by different amounts,and the last three have some upside-down patterns aroundtheir final constant levels. Figures 10 to 12 illustrate thefault signature of the mean shift vector (1, 1)T for differentVARMA(1, 1)T filters. It is common that the first set ofvalues of the fault signatures for both variables remain thesame as the original mean and then the rest of the faultsignatures display different shift patterns.

5. Performance evaluation

Many variables in the form of vectors and matrices areinvolved in an autocorrelated multivariate process fitted bya VARMA model, such as µ, �, �p, �q , and �tγ. Changesin any of their elements can lead to a significant change inthe whole process. The derivation of the MCuscore chartis based on some strong assumptions, especially priorknowledge on the signal’s size and pattern. Therefore, aperformance evaluation of the robustness or sensitivityof the MCuscore chart to the disturbances in processvariables is needed.

We use Monte Carlo simulation to evaluate the perfor-mance of the two-sided MCuscore chart in the detection ofthe steady mean shift vector in the bivariate VAR(1) and

Table 3. � matrix obtained by varying φ

φ1 φ12 φ21 φ2

a 0.7 0.1 0.2 0.5b −0.7 0.1 0.2 0.5c −0.7 0.6 0.2 0.5

Table 4. � matrix obtained by varying θ

θ1 θ12 θ21 θ2

1 0.3 0.4 0.6 0.32 0.3 0.2 0.6 0.33 0.3 0.0 0.6 0.34 0.3 −0.2 0.6 0.35 0.3 −0.4 0.6 0.36 0.3 −0.6 0.6 0.3

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Fig. 7. VAR(1) fault signatures of a bivariate transient unit mean shift with � in Table 1.

VARMA(1,1) processes. By varying some of the elementsin the �1 and �1 matrices, we briefly analyze the robustnessof the MCuscore chart.

First, we investigate the out-of-control ARL of theMCuscore chart in monitoring bivariate VAR(1) models.The white noise term in each inverse VAR(1) filter is as-

sumed to follow a bivariate normal distribution with con-stant µ = (0, 0)T and

� =(

1 0.50.5 1

).

Table 5. ARL properties of the MCuscore and the residual-based MCusum chart for VAR(l) models with mean shifts andvarying �s

γ = (1.0, 1.0)′γ = (1.0, 0.5)′

� =[

φ1 φ12φ21 φ2

]MCuscore

Res-MCusum(H = 3.70) MCuscore

Res-MCusum(H = 4.17)

φ1 φ12 φ21 φ2 H ARL1 ARL1 H ARL1 ARL1

0.7 0.1 0.2 0.5 9.25 46.31 67.94 9.92 55.25 75.120.4 0.1 0.2 0.5 6.96 23.95 34.56 6.20 18.57 23.28

−0.1 0.1 0.2 0.5 5.01 12.16 18.73 4.45 9.44 11.11−0.1 0.1 0.2 0.5 4.17 8.44 14.06 3.71 6.97 8.09−0.4 0.1 0.2 0.5 3.3 5.53 9.81 2.94 4.81 5.93−0.7 0.1 0.2 0.5 2.65 4.10 7.46 2.43 3.66 4.80−0.7 −1.2 0.2 0.5 1.30 2.03 4.05 1.69 2.47 3.57−0.7 −0.8 0.2 0.5 1.59 2.36 4.67 1.90 2.77 3.83−0.7 −0.4 0.2 0.5 1.95 2.82 5.49 2.09 3.07 4.19−0.7 0 0.2 0.5 2.43 3.70 7.06 2.36 3.52 4.67−0.7 0.4 0.2 0.5 3.22 5.61 9.89 2.64 4.15 5.25−0.7 0.8 0.2 0.5 2.62 4.04 15.88 3.43 6.06 6.25−0.7 1.2 0.2 0.5 6.82 23.52 33.67 3.57 6.46 7.58−0.7 1.6 0.2 0.5 9.13 45.24 82.20 4.25 8.84 10.26

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Fig. 8. VAR(1) fault signatures of a bivariate transient unit mean shift with � in Table 2.

We compare the performance of the MCuscore and theresidual-based MCusum control chart in monitoring themean shift faults. The � matrices are chosen from Tables 1and 2 and satisfy the stationarity condition of a VAR(1)time series.

We simulated 5000 runs with 1000 observations in eachrun to determine the control limits of the MCuscore chart

and residual-based MCusum chart based on an in-controlARL0 = 200. Table 5 shows that the control limits areconstant for the residual-based MCusum chart at equalARL0; i.e., H = 3.70 and H = 4.17. This occurs becausethe mean shift vector in calculating the sequential MCusumstatistics is constant. On the other hand, for the MCuscorechart, the control limit is different for different � matrices

Table 6. ARL properties of the MCuscore and the residual-based MCusum chart for detecting mean shift (1,1)T inVARMA(1,1) models with varying �s or θs

� =[

θ1 θ12θ21 θ2

]� =

[0.7 0.10.2 0.5

]� =

[−0.4 0.10.2 0.5

]� =

[−0.7 1.60.2 0.5

]

MCuscoreRes-MCusum

(H = 3.7) MCuscoreRes-MCusum

(H = 3.7) MCuscoreRes-MCusum

(H = 3.7)θ1 θ12 θ21 θ2 H ARL1 ARL1 H ARL1 ARL1 H ARL1 ARL1

0.3 0.4 0.6 0.3 3.22 5.68 6.05 0.44 1.55 5.08 1.25 2.33 3.840.3 0.2 0.6 0.3 4.65 9.78 10.21 1.11 2.06 5.16 2.03 3.34 4.230.3 0.0 0.6 0.3 5.74 14.85 21.71 1.67 2.70 5.18 2.74 4.44 4.940.3 −0.2 0.6 0.3 6.49 19.38 42.81 2.14 3.26 5.16 3.46 6.03 6.270.3 −0.4 0.6 0.3 6.98 22.55 70.24 2.62 4.06 5.10 4.12 8.05 8.420.3 −0.6 0.6 0.3 7.24 23.93 92.39 3.07 5.02 5.11 4.64 10.13 11.96

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Fig. 9. VMA(1) fault signatures of a bivariate steady unit mean shift with � in Table 4.

Fig. 10. VARMA(1,1) fault signatures of a bivariate steady unit mean shift with � from “a” in Table 3 and � from 1–6 in Table 4.

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Fig. 11. VARMA(1,1) fault signatures of a bivariate steady unit mean shift with � from “b” in Table 3 and � from 1–6 in Table 4.

because the mean vectors and the reference values in thesequential MCuscore statistics are not constant. Therefore,we must reidentify the control limits for MCuscore whenthe � matrix has been changed.

Two mean shift vectors (1.0, 1.0)T and (1.0, 0.5)T are ex-amined for each VAR(1) model. (Note that we focus on a

moderate and a small shift; other techniques can perhapsbetter be used to detect large shifts.) It can be seen in Ta-ble 5 that the MCuscore chart consistently outperformsthe residual-based MCusum chart in terms of the out-of-control ARLs. The same conclusion can also be drawn fromthe simulation results in Table 6, which compares the two

Fig. 12. VARMA(1,1) fault signatures of a bivariate steady unit mean shift with � from “c” in Table 3 and � from 1–6 in Table 4.

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Fig. 13. Flowchart of process control.

control charts in detecting the mean shift vector (1.0, 1.0)T

in an VARMA(1,1) process whose parameters are from Ta-ble 4.

For model-based control charts such as the residual-based MCusum and MCuscore charts, the accuracy of theestimated parameters has an effect on performance. Ta-bles 5 and 6 briefly illustrate how different model parame-ters affect the performance of the control charts.

It can be observed in both Tables 5 and 6 that the per-turbation in the parameter of the � or � matrix, indicatedby the italic φs or θs, usually causes either a smaller out-of-control ARL at the cost of a higher false alarm rate or alarger out-of-control ARL with the benefit of a lower falsealarm rate. For an example of an MCuscore chart with the� matrix in Table 5, if the φ1 value perturbs from −0.1 to0.1 with the other three φ values being fixed, we can obtainan out-of-control ARL of 9.18 with the control limit 4.17.This ARL is higher than the expected value of 8.44 but hasa smaller false alarm rate than 1/200 because the actualcontrol limit is smaller than the simulated limit of 5.01.

6. Application impact of the MCuscore chart

The MCuscore chart can have a significant impact across arange of industries—from manufacturing to services. Thequality of many processes can be characterized by a mul-tivariate vector with components that autocorrelate andcross-correlate over time, such as in the example in Sec-tion 3. To improve in process quality, practioners usu-ally optimize the settings of the process adjustors throughexperimental designs and then apply control chart pro-cedures to monitor the response on a regular basis. Anexample of the work flow is illustrated in Figure 13.

For some established processes, historical data are avail-able for use in estimating the system dynamics and theparameters of the VARMA model. In practice, multi-ple standard univariate control charts, such as Shewhart,Cusum or EWMA charts, and standard multivariate con-trol charts, such as MCusum, or MEWMA charts, are ap-plied to monitoring such processes. In some special situa-tions, the size γ and pattern � of the system mean shift asa response to the adjustment change have been estimated apriori. The objective of the control chart is to capture the ex-pected system mean shift using the least observations with alow false alarm rate. In such cases, the MCuscore chart out-performs the MCusum chart as shown in the Section 3. Infact, the application of MCuscore chart can be viewed as atrade-off between the system and signal’s prior informationwith the detection power, which provide a supplementaryapproach to the battery of standard control charts.

7. Conclusions

In this article, the MCuscore approach based on the like-lihood ratio test and fault signature analysis is introducedfor monitoring the mean vector shift in an autocorrelatedmultivariate process when the parameters can be estimatedfrom historical data.

A VARMA(p,q) time series model is used to establishthe theory of the MCuscore chart. An example of onlinesearch engine marketing process illustrates the applica-tion of the MCuscore chart in monitoring an autocorre-lated multivariate process. Simulation is used to show thatthe MCuscore chart outperforms the traditional residual-based MCusum control chart in detecting a mean vectorshift signal.

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A set of MCuscore charts may be conceived in order toreduce the need for a priori information. For example, sepa-rate MCuscores may represent alternative pattern matricesfor the signal or starting times of the signal. It remains anopen research opportunity to relax the assumption of priorknowledge of the signals so that the MCuscore chart can bemore practically applied in both Phase I and Phase II op-erations. It may also be worthwhile to monitor the signalsor changes in matrices �, �p, or �q . In addition, researchon comparisons of the MCuscore with other multivariateattributes charts may be considered.

Acknowledgements

The authors would like to thank the editor, associate ed-itors, and two referees for their thoughtful and construc-tive comments, which were greatly helpful in improving thequality of this article.

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MCuscore chart for autocorrelated vectors 307

Appendix

The derivation of Equation (9)

Equation (8) can be written as

LRk =k∑

i=1

[( f ◦ at)(γ) − ( f ◦ at)(γ0)],

where ( f ◦ at)(·) denotes composite function f (at(·));f (x) = xT�x and f : D → � is a quadratic function de-fined on a subset D of �p; f (·) is differentiable at point x0;R → �p is a linear function defined on a subset R of �p,and that at(γ) is differentiable at point γ0.

The second-order vector Taylor expansion of compositefunction ( f ◦ at)(γ) at point γ0 = 0 takes the form:

( f ◦ at)(γ) = ( f ◦ at)(γ0) + γT( f ◦ at)′(γ)|γ=0

+ 12γT( f ◦ at)′′(γ)|γ=0γ. (A1)

Based on the third and fourth assumptions in Section 2.1,the second part of Equation (A1) can be put into the fol-lowing form using vector calculus and the chain rule

γT( f ◦ at)′(γ)|γ=0 = 2γT(

∂at(γ)∂γ

)T ∣∣∣∣γ=0

�−1at(0), (A2)

and the third part of Equation (A1) can be written as

γT( f ◦ at)′′(γ)γ|γ=0=γT ∂

∂γ

((∂at(γ)

∂γ

)T

�−1at(γ)

) ∣∣∣∣γ=0

γ

= γT ∂

∂γ

(∂at(γ)

∂γ

)T ∣∣∣∣γ=0

�−1at(0)γ

+γT(

∂at(γ)∂γ

)T ∣∣∣∣γ=0

∂

∂γ

(�−1at(γ)

) ∣∣∣∣γ=0

γ

= γT(

∂at(γ)∂γ

)T ∣∣∣∣γ=0

�−1 ∂at(γ)∂γ

∣∣∣∣γ=0

γ. (A3)

Substituting Equation (A2) and Equation (A3) into Equa-tion (A1) gives Equation (9).

Biographies

Shuohui Chen is a member of the American Society for Quality (ASQ),Institute for Operations Research and the Management Sciences (IN-FORMS), and American Statistical Association (ASA). He received hisPh.D. in Industrial Engineering from the Pennsylvania State Universityin 2006. He is now working in the IT industry as a statistician.

Harriet Black Nembhard is an Associate Professor of Industrial andManufacturing Engineering and Director of the Laboratory for Qual-ity Engineering and System Transitions at the Pennsylvania State Uni-versity. Her lab aims to develop tools for productivity improvementby bringing together methods from applied statistics, quality engi-neering, and financial engineering. She is a Fellow of the AmericanSociety for Quality, an Academician of the International Academyof Quality, and a Senior Member of the Institute for IndustrialEngineers.

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