Directionally Sensitive

Multivariate Statistical Process Control Methods

Ronald D. Fricker, Jr.Naval Postgraduate School

October 5, 2005

Abstract

In this paper we develop two directionally sen-sitive statistical process control procedures bymodifying Hotelling’s χ2 procedure and a mul-tivariate CUSUM procedure by Crosier. Wethen compare the performance of these pro-cedures via simulation to the original direc-tionally invariant procedures and to simultane-ous univariate directionally sensitive Shewhartand CUSUM procedures. The results show,not unexpectedly, that the modified multivari-ate procedures work better than their origi-nal counterparts in the problem for which theywere designed. Interestingly, the results ofcomparing simultaneous univariate Shewhartprocedures to the modified χ2 was mixed, withthe better procedure depending upon the co-variance structure of the distribution. In con-trast, the modified MCUSUM generally outper-formed the simultaneous univariate CUSUMsfor all covariance structures we considered.Furthermore, the modified MCUSUM also per-formed better than the univariate Shewhartsand the modified χ2. These results thus sug-gest that the modified MCUSUM procedure isthe preferred choice (from among the proce-dures considered here) for monitoring multi-variate processes in a particular direction.

1 Introduction

Many existing multivariate statistical processcontrol (SPC) methods are directionally in-variant, meaning they are designed to detectchanges in a mean vector in all directionsequally well. Examples of such procedures in-clude Hotelling’s χ2 [3], Crosier’s MultivariateCUSUM [1], and more recently the nonpara-metric method of Qui and Hawkins [8]. SeeLowry and Montgomery [5] for a more detaileddiscussion. The lack of directional sensitivityis often seen as a limitation of these methods,particularly when practitioners are interestedin detecting changes in some directions morethan others.

For example, the Centers for Disease Con-trol and Prevention (CDC) as well as manystate and local health departments around theUnited States have started to develop and fieldsyndromic surveillance systems. Making useof existing health care or other data, oftenalready in electronic form, these surveillancesystems are intended to give early warnings ofbioterrorist attacks or other emerging healthconditions. (See Stoto, Fricker, et al. [10] fora more detailed discussion.)

With such syndromic surveillance systems,it is important to quickly flag increases in therelevant measures because, in terms of signal-ing a terrorist event, decreases are generally ir-relevant. Current syndromic surveillance sys-tems tend to run multiple simultaneous uni-variate schemes, each focused on detecting anincrease in a single dimension.

Multiple simultaneous univariate schemeshave the advantages of ease of implementa-tion and interpretation, but they can be lesssensitive to some types of changes when com-pared to multivariate methods. Further, un-less the signal thresholds of the multiple si-multaneous procedures are properly set, theycan suffer from a higher than desired combinedfalse alarm rate.

In this paper, we present modifications totwo multivariate methods – Hotelling’s χ2

and a multivariate CUSUM (MCUSUM) byCrosier [1] – to make them directionally sensi-tive and then illustrate their performance viasimulation. The modifications are motivatedby the procedures’ univariate counterparts andhow those counterparts achieve directionality.

• The univariate counterpart to Hotelling’sχ2 is the Shewhart procedure [9] where di-rectionality is achieved by signaling onlywhen an observation falls far enough outin one particular tail of the distribution.

For the “modified Hotelling’s χ2,” di-rectionality is achieved by only signalingwhen an observation falls within a par-ticular region of space corresponding toa tail region of the multivariate distribu-tion.

• The CUSUM is, of course, the univari-ate counterpart to the MCUSUM, wheredirectionality (and signalling speed) isachieved by reflecting the CUSUM at zeroin either the positive or negative direc-tion combined with an appropriate sig-nal threshold in that direction. For the“modified MCUSUM,” directionality isachieved by reflecting each component ofthe cumulative sum vector at zero in thedesired direction combined with an appro-priate signal threshold.

In this paper we first describe the twostandard univariate procedures (Shewhart andCUSUM) followed by their multivariate coun-terparts (Hotelling’s χ2 and Crosier’s multi-variate CUSUM). We then describe how tomodify the multivariate procedures to makethem directionally sensitive and compare andcontrast the various procedures’ performancevia simulation.

2 Notation andTerminology

In the simple case of detecting a shift fromone specific distribution to another, let F0 de-note the in-control distribution, which is thedesired or preferred state of the system. Forsyndromic surveillance, for example, this couldbe the distribution of the daily counts of indi-viduals diagnosed with a particular complaintat a specific hospital or within a particulargeographic region under normal conditions.Let F1 denote the out-of-control distributionwhere, under the standard SPC paradigm,this would be a particular distribution rep-resenting a condition or state that is impor-tant to detect. Within the syndromic surveil-lance problem, F1 might represent an elevatedmean daily count resulting from the release ofa bioterrorism pathogen for example.

Let ν be the actual (unknown) time whenthe process shifts from F0 to F1 and letT be the length of time from ν to whena procedure signals (referred to as the de-

lay). The notation IEν(T |T ≥ 0) is usedto indicate the expected delay, which is theaverage time it takes an procedure to sig-nal once the shift has occurred. The notationIE∞(T ) indicates the expected time to a falsealarm, where ν = ∞ means the process nevershifts to the out-of-control distribution.

In the SPC literature, procedures are com-pared in terms of the expected time to sig-nal, where IE∞(T ) is first set equally for twoprocedures and then the procedure with thesmallest IEν(T |T ≥ 0), for a particular F1, isdeemed better. Often when conducting sim-ulation comparisons, ν is set to be 0, so theconditioning in the expectation is automatic.

The term average run length (ARL) is fre-quently used for the expected time to signal,where it is understood that when ν = ∞ theARL denotes the expected time to false alarm.In simulation experiments, the performance ofvarious procedures is compared by setting theexpected time to false alarms to be equal andthen comparing ARLs when ν = 0, where itis then understood that the ARL is the meandelay time.

3 Univariate Procedures

3.1 Shewhart’s Procedure

Shewhart’s procedure [9] is probably the sim-plest and best known of all SPC methods. Thebasic idea is to sequentially evaluate one obser-vation (or statistic) at a time, signaling whenan observation that is rare under F0 occurs.The most common form of the procedure, of-ten known as the X chart, signals when theabsolute value of an observed sample mean ex-ceeds a pre-specified threshold c, often definedas the mean value plus some number of stan-dard deviations of the mean.

More sophisticated versions of the Shew-hart procedure exist that look for increasesin variation and other types of out-of-controlconditions. These versions are not consideredhere in order to keep the evaluations simpleand consistent with Hotelling’s χ2 procedure.

The Shewhart procedure can be made di-rectionally sensitive by only signaling for de-viations in one direction. For example, insyndromic surveillance, only deviations in thepositive direction that would indicate a poten-tial outbreak are assumed to be important to

detect. Thus, for a univariate random vari-able X, and for some desired probability p,the threshold c is chosen to satisfy

∫ ∞

x=c

f0(x) dx = p.

The algorithm proceeds by observing values ofXi; it stops and concludes Xi ∼ F1 at the firsttime i when Xi > c.

If the change to be detected is a one-timejump in the mean and the probability of anobservation exceeding the threshold is known,then simulation is not required as the delayis geometrically distributed and exact calcu-lations for the average run lengths can be di-rectly calculated as IE∞(T ) = 1/p and

IEν(T |T ≥ 0) = IE0(T ) =[∫ ∞

x>c

f1(x) dx

]−1

.

3.2 CUSUM Procedure

The CUSUM of Page [6] and Lorden [4] is asequential hypothesis test for a change from aknown in-control density f0 to a known alter-native density f1. The procedure monitors thestatistic Si, which satisfies the recursion

Si = max(0, Si−1 + Li), (1)

where the increment Li is the log likelihoodratio

Li = log(

f1(Xi)f0(Xi)

).

The procedure stops and concludes that Xi ∼F1 at the first time i for which Si > c, where cis some prespecified threshold that achieves adesired ARL under the in-control distribution.

If F0 and F1 are normal distributions withmeans µ and µ+δ, respectively, and unit vari-ances, then (1) reduces to

Si = max(0, Si−1 + (Xi − µ)− k), (2)

where k = δ/2. This is the form commonlyused, even when the underlying data is onlyapproximately normally distributed.

Note that, since the univariate CUSUM is“reflected” at zero, it is only capable of look-ing for departures in one direction. If it isnecessary to guard against both positive andnegative changes in the mean, then one mustsimultaneously run two CUSUMs, one of the

form in (2) to look for changes in the positivedirection, and one of the form

Si = max(0, Si−1 − (Xi + µ)− k),

to look for changes in the negative direction.When directional sensitivity is desired, say todetect only positive shifts in the mean, it isonly necessary to use (2).

4 Directionally InvariantMultivariate Procedures

4.1 Hotelling’s χ2

Hotelling [3] introduced the χ2 procedure (of-ten referred to as the T 2 procedure; we useχ2 to indicate that we are assuming the co-variance matrix is known). For multivariateobservations Xi ∈ IRd, i = 1, 2, . . ., the proce-dure computes

χ2i = X′

iΣ−1Xi,

where Σ−1 is the inverse of the covariance ma-trix. The procedure stops at the first time ifor which χi > c, where c is a pre-specifiedthreshold.

Like the original univariate Shewhart Xprocedure, because it only uses the most re-cent observation to decide when to stop, theχ2 can react quickly to large departures fromthe in-control distribution but will also be rel-atively insensitive to small shifts. Of course,it also requires that the covariance matrix isknown or well estimated.

4.2 Crosier’s MCUSUM

The abbreviation MCUSUM, for multivariateCUSUM, is used here to refer to the procedureproposed by Crosier [1] that at each time iconsiders the statistic

Si = (Si−1+Xi−µ)(1−k/Ci), if Ci > k, (3)

where k is a statistical distance based on a pre-determined vector k, k = {k′Σ−1k}1/2 andCi = {(Si−1+Xi−µ)′Σ−1(Si−1+Xi−µ)}1/2.If Ci ≤ k then reset Si = 0. The procedurestarts with S0 = 0 and sequentially calculates

Yi = (S′iΣ−1Si)1/2.

It concludes that Xi ∼ F1 at the first time iwhen Yi > c for some threshold c > 0.

Crosier proposed a number of other multi-variate CUSUM-like algorithms but generallypreferred (3) after extensive simulation com-parisons. Pignatiello and Runger [7] proposedother multivariate CUSUM-like algorithms aswell, but found that they performed similar to(3).

It is worth noting that Crosier derived hisprocedure in an ad hoc manner, not from the-ory, but found it to work well in simulationcomparisons. Healy [2] derived a sequentiallikelihood ratio test to detect a shift in a meanvector of a multivariate normal distributionthat is a true multivariate CUSUM. However,while Healy’s procedure is more effective (hasshorter ARLs) when the shift is to the pre-cise mean vector of F1, it is less effective thanCrosier’s for detecting other types of shifts, in-cluding mean shifts that were close to but notprecisely the specific mean vector of F1.

5 Directionally SensitiveMultivariate Procedures

5.1 Modified Hotelling’s χ2

To modify Hotelling’s χ2 procedure to achievedirectional sensitivity, we modify the stoppingrule so that it meets two conditions: (1) χi > cand (2) Xi ∈ S, where S is a particular sub-space of IRp that corresponds to, say, a positiveshift in one or more components of the meanvector. In syndromic surveillance, this wouldcorrespond to an increase in one or more dis-ease indicators, for example.

For the purposes of the simulations to fol-low, S was defined as follows. Choose valuess1, s2, . . . , sd such that

∫ ∞

x1=s1

∫ ∞

x2=s2

· · ·∫ ∞

xd=sd

f0(X)dx ≈ 0.99

and then define S = {x1 > s1, x2 >s2, . . . , xd > sd}.

For example, consider an in-control distri-bution following a bivariate normal distribu-tion with some positive correlation, so that theprobability contours for the density of F0 is anellipse with its main axis along 45-degree linein the plane. Then you can think about S asthe upper the upper right quadrant that al-most encompasses the 99 percent probabilityellipse.

The idea of using this region for S is thatif F1 represents an increase in one or morecomponents of the F0 mean vector, then themodified χ2 procedure will have an increasedprobability of signaling, which should result ina decreased expected time to signal. On theother hand, if F1 represents a condition witha mean vector that corresponds to a decreasein one or more of the F0 mean vector com-ponents, then the probability of signaling willdecrease and the procedure will have less of achance of producing an signal.

5.2 Modified MCUSUM

Unlike other multivariate CUSUMs (e.g.,Healy’s [2]), Crosier’s MCUSUM formulationis easy to modify to only look for positive in-creases. In particular, for detecting positiveincreases, such as in the syndromic surveil-lance problem, when Ci > k we limit Si tobe positive in each dimension by replacing (3)with Si = {Si,1, . . . , Si,d} where

Si,j = max[0, (Si−1,j + Xi,j − µj)(1− k/Ci)],

for j = 1, 2, . . . , d.

The motivation for this modification followsdirectly from the univariate CUSUM’s reflec-tion at 0. As in (2), the reflection helps ensurethat a large cumulative sum vector causing asignal is the result of a mean shift in one ormore positive directions.

6 Results: PerformanceComparisons viaSimulation

These simulations compare the performanceby average run length, first setting the ARLunder the in-control distribution (i.e., IE∞(T ),the expected time to false alarm) equally, andthen comparing the ARL performance undernumerous out-of-control distributions result-ing from various shifts in the mean vector attime 0 (i.e., IE0(T )).

The in-control distribution (F0) is a six-dimensional multivariate normal with a zeromean vector, ~µ0 = {0, 0, 0, 0, 0, 0} and a co-variance matrix Σ consisting of unit varianceson the diagonal and constant covariance ρ onthe off-diagonals. The out-of-control distribu-tions (F1s) have the same covariance structure

but with the mean vector shifted by some dis-tance d,

| ~µ0 − ~µ1 |=[

n∑

i=1

(µ1(i))2

]1/2

= d,

where the shift occurs in some number of di-mensions n, 1 < n ≤ 6. For those dimen-sions with a shift, the shifts were made equally:

µ1(1) = · · · = µ1(n) =√

d2

n .

The simulations were conducted in Math-ematica where the random observations weregenerated using the MultinormalDistributionfunction. The in-control ARLs were set to 100by empirically determining the threshold foreach procedure via simulation. For the multi-variate procedures this involved determining asingle threshold for each value of ρ (except forHotelling’s χ2 procedure for which one thresh-old applies to all values of ρ).

For the simultaneous univariate proceduresapproach, which requires a separate thresholdfor each individual procedure, there was noreason to favor one direction over another, soall the thresholds were set such that the prob-ability of false alarm was equal in all dimen-sions and so that the resulting expected timeto false alarm for the combined set of univari-ate procedures was equal to the expected timeto false alarm of the multivariate procedure.

In addition, for the univariate CUSUMswe set k = 0.5 in (2). For theMCUSUM and modified MCUSUM we setk = {0.2, 0.2, 0.2, 0.2, 0.2, 0.2} in (3).

Generally, it is quite simple to empiricallyestimate the ARLs via simulation. For a par-ticular F0, choose a c and run the particularprocedure m times, recording for each run thetime t when the first Xi > c (where each Xi isa random draw from F0, of course). Estimatethe in-control ARL as

IE∞(T ) = Σt/m

adjusting c and re-running as necessary toachieve the desired in-control ARL, where m ismade large enough to make the standard errorof IE∞(T ) acceptably small.

Having established the threshold c for thatF0 with sufficient precision, then for each F1

of interest re-run the algorithm n times (wheren is often smaller than m), drawing the Xisfrom F1 starting at time 1. As before, take theaverage of ts to estimate the expected delay.

In the simulations to follow, we first demon-strate the modified procedures’ performancecompared to their counterpart unmodifiedprocedures. This establishes the directionalsensitivity and effectiveness of the modifiedprocedures. We follow this with comparisonsof the modified procedures to procedures con-sisting of the application of simultaneous uni-variate procedures. The simultaneous univari-ate procedures are implemented to be direc-tionally sensitive in the same direction as themodified multivariate procedures and henceprovide some indication about the effective-ness of the modified multivariate methods. Fi-nally, we compare the best procedures fromthe previous comparisons in an effort to deter-mine whether a single procedure is generallybest.

6.1 Modified Procedures vs.Original Procedures

Figure 1 shows the improved performance ofthe modified χ2 procedure and the modifiedMCUSUM for almost all types of mean vectorshifts where, as previously described, the com-ponentwise shifts are in the positive direction.This is not surprising given that the modifiedprocedures were designed to look for positivemean shifts.

In Figure 1, the various lines correspond tothe number of dimensions in ~µ1 that shiftedand the horizontal axis is the distance of themean shift. For example, the n = 1 line showsthe results for ~µ1 = {d, 0, 0, 0, 0, 0}, where theARL was evaluated at d = 0.0, 0.2, 0.4, . . . , 3.4.Similarly, the n = 2 line shows the results for

~µ1 = {√

d2

2 ,√

d2

2 , 0, 0, 0, 0}.The vertical axis is the difference ∆ between

the ARL for the original procedure and themodified procedure for a given mean vectorshift. Positive values indicate the modifiedprocedure had a smaller ARL, so that for aparticular out-of-control condition the modi-fied procedure had a shorter time to signal. A∆ = 0 at d = 0.0 indicates that the false alarmrates (equivalently, the in-control ARLs) wereset equally for each procedure before compar-ing the expected time to signal for various~µ1s (within the bounds of experimental error,where a sufficient number of simulation runswere conducted to achieve a standard error of∆ of approximately 2 percent of the estimatedin-control ARLs.)

Figure 1: Performance comparison of the χ2 and MCUSUM procedures vs. their modifiedcounterpart procedures for ρ = 0.3. A positive value of ∆ indicates that the ARL for themodified procedure is shorter than the ARL of its unmodified counterpart.

Figure 2: Performance comparison of the modified χ2 procedure vs. multiple simultaneousunivariate Shewhart procedures. The figure on the left shows that for ρ = 0.3 the multiplesimultaneous Shewhart procedures give smaller ARLs for all n. However, the figure on the rightwith n = 3 shows that either procedure can be significantly better than the other dependingon the value of ρ.

Figure 3: Performance comparison of the modified MCUSUM procedure vs. multiple simul-taneous univariate CUSUM procedures. The figure on the left shows that for ρ = 0.3 theMCUSUM does better for small values of d and marginally worse for large d. However, unlikethe modified χ2 in Figure 2, the figure on the right for the modified MCUSUM with n = 3 isgenerally better than simultaneous univariate CUSUMs for all values of ρ.

Figure 4: Performance comparison of the modified MCUSUM procedure to multiple simultane-ous univariate Shewhart procedures (left graph) for ρ = 0.3 and to the modified χ2 procedure(right graph) for ρ = 0.9. In both cases, the modified MCUSUM procedure performs generallybetter than the preferred Shewhart-type procedure.

As previously mentioned, the expected re-sult that the modified procedures generallyoutperform the original procedures at detect-ing positive shifts. Figure 1 shows this forthe case of ρ = 0.3. Not shown, the resultsfor other values of ρ we tried, from ρ = 0 toρ = 0.9, are very similar.

In particular, the modified χ2 outperformsHotelling’s χ2 for all combinations of 1 <n ≤ 6, 0.0 < d ≤ 3.4, and 0 ≤ ρ ≤0.9. The modified MCUSUM outperformsCrosier’s MCUSUM except for larger valuesof ρ with small d and small n.

For example, in Figure 1, Crosier’sMCUSUM slightly outperforms the modifiedMCUSUM for n = 1 with 0 < d < 0.6 or so.For ρ = 0.6, Crosier’s MCUSUM outperformsthe modified MCUSUM with −6 < ∆ < 0 orso for n = 1, 2 with 0 < d < 0.6. And, forρ = 0.9, Crosier’s MCUSUM outperforms themodified MCUSUM with −9 < ∆ < 0 or sofor n = 1, . . . , 5 with 0 < d < 1.

In this work, we are interested in moderatevalues of ρ, since the syndromic surveillancedata we have seen has only had moderate cor-relations, roughly on the order of 0 < r < 0.6.In addition, we are interested in shifts in themean vector that exhibit themselves as smallchanges in multiple dimensions. (Indeed, ifthe shift is expected only a small number (n)of dimensions or that the covariance ρ will behigh, then it’s likely that univariate methodswould be more appropriate anyway.)

With this in mind, what is most notable inFigure 1 is that as n increases the modifiedprocedures do considerably better than theircounterparts, particularly for moderate ds.

6.2 Modified Procedures vs.Univariate Procedures

Given that the modified χ2 performs betterthan the original χ2 for this problem, Figure 2focuses on comparing the performance of themodified χ2 to six one-sided Shewhart proce-dures operating simultaneously. The left-sidegraph of Figure 2, constructed just like Fig-ure 1, shows that six simultaneous univari-ate Shewharts are more effective (have shorterARLs) than the modified χ2 for ρ = 0.3. Atbest, for large shifts, the ARL of the modifiedχ2 approaches the performance of the multipleunivariate Shewharts, and for small to mod-

Figure 5: Modified MCUSUM ARLs for n = 3and various values of ρ.

erate shifts the multiple univariate Shewhartsare clearly better.

The graph on the right side of Figure 2shows the performance comparison for n = 3and for various values of ρ (0.0, 0.3, 0.6, and0.9). Here we see that the better procedure de-pends on ρ, where the modified χ2 is better forvalues of ρ near 0.0 or 0.9 while the simultane-ous univariate Shewharts are better for mod-erate values of ρ. Most notably, and a bit sur-prisingly, the modified χ2 significantly outper-forms the simultaneous univariate Shewhartswhen there is no correlation (ρ = 0) and whenthe shift is only in 3 of the 6 dimensions.

The results for the modified MCUSUM ver-sus simultaneous univariate CUSUMs are pre-sented in Figure 3. These results differ fromthose for the Shewhart-type procedures in Fig-ure 2 in that the modified MCUSUM is gen-erally better than the simultaneous univariateCUSUMs regardless of the value of ρ. In par-ticular, in the left graph of Figure 3 the mod-ified MCUSUM performance when ρ = 0.3is somewhat better for small shifts (roughly> 0.0 to 0.6 or so), and slightly worse thanmultiple univariate CUSUMs for moderate tolarge shifts. Yet, in the figure at the right wesee that the modified MCUSUM is better forsmall shifts for all values of ρ and only per-forms slightly worse for moderate values of ρcombined with moderate to large values of d.

6.3 Modified MCUSUM vs.Best Other Procedures

What the previous simulations have shownis that the modified MCUSUM is gener-ally better than the simultaneous univariateCUSUMs. However, whether the modified χ2

is better than simultaneous univariate Shew-harts depends on ρ. So, here we compare themodified MCUSUM to both the simultaneousunivariate Shewharts when it is better thanthe modified χ2 (ρ = 0.3) and the modified χ2

when it is better then the simultaneous uni-variate Shewharts (ρ = 0.9). The results areshown in Figure 4. In both comparisons, themodified MCUSUM procedures performanceis better. The obvious conclusion, then, is apreference for the modified MCUSUM that, atleast in these simulations for a jump change inthe mean vector of multivariate normal distri-butions.

Now, all the figures up to this point havedisplayed differences in ARL performance be-tween two procedures. Figure 5 shows theARLs for the modified MCUSUM for n = 3.Results for n = 1, 2, 4, 5, 6 were similar, thoughthe individual ρ curves moved around butlargely stayed within the same band/region.For example, for n = 1, the lowest ARLs wereachieved with ρ = 0.9 while for n = 6 thelowest ARLs were achieved with ρ = 0.0.

7 Discussion andConclusions

In this paper we have demonstrated how tomodify two directionally invariant multivari-ate procedures to make them directionally sen-sitive. The results of these and other sim-ulations not included here show, not unex-pectedly, that the modified multivariate pro-cedures work better than their original coun-terparts in the problem for which they weredesigned. It is not unexpected since the mod-ified procedures specifically look for positivechanges so that, when given such changes,should outperform their counterparts that arenot so designed.

In the comparison of simultaneous univari-ate Shewhart procedures to the modified χ2

the outcomes were mixed, with the better pro-cedure depending upon the distribution co-variance structure (i.e., ρ). In contrast, the

modified MCUSUM generally performed bet-ter than the simultaneous univariate CUSUMsfor all values of ρ that were considered. Fur-thermore, it also performed better than theShewharts and the modified χ2 except in thosecases where the shift d was moderate to verylarge (in which case a statistical detection pro-cedure may not even be required). These re-sults thus suggest that, among the proceduresconsidered here, the modified MCUSUM pro-cedure is the preferred choice for monitoringmultivariate processes for positive shifts in themean.

It is important to recognize that the use ofthe modified MCUSUM does come with somecosts. First, unlike Hotelling’s χ2 procedure,the choice of threshold, and hence the ARLperformance of the procedure, depends on thecovariance structure of the data. This limi-tation is also true of the multiple simultane-ous univariate procedures, though if practi-tioners treat each dimension separately it isless apparent. Second, practitioners are oftenless comfortable using multivariate proceduresbecause they tend to feel such procedures donot provide sufficient information about thecause(s) of a signal. The modified MCUSUMis no different in this regard, though becauseit is directional, the practitioner is at least as-sured that the signal is related to an out-of-control condition of interest.

7.1 Directions forFuture Research

Future work should consider the effects of esti-mation on the performance of the procedures.In particular, the multivariate procedures re-quire the estimation of the entire covariancematrix while the simultaneous univariate pro-cedures only require estimation of the diagonalelements. Whether and how this estimationaffects the performance of the procedures isnot known. In addition, this work assumed anon-negative, constant ρ covariance structure.However, it is conceivable that some problemsmight involve both positive and negative co-variances of varying magnitudes. In addition,while the effects of changing k in the CUSUMare well known, the effects of changes in k inthe MCUSUM are not as well known and thoseeffects were not explored in this work. Finally,the method we applied to make the multivari-ate procedures directionally sensitive can be

applied to other directionally invariant proce-dures, such as the nonparametric method ofQui and Hawkins [8]. How the performanceof those new methods compares to the per-formance of the modified MCUSUM requiresfurther research.

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