Optimal Design of Second-Order Linear Filters for Control Charting

12 Pages·2006·0.44 MB·English

Optimal Design of Second-Order Linear
Filters for Control Charting
Chang-HoCHIN DanielW.APLEY
SchoolofMechanicalandIndustrialSystemsEngineering DepartmentofIndustrialEngineering
KyungHeeUniversity andManagementSciences
Yongin-si,Gyeonggi-do446-701 NorthwesternUniversity
RepublicofKorea Evanston,IL60208-3119
([email protected]) ([email protected])
Inmanycommoncontrolchartingsituations,thestatistictobechartedcanbeviewedastheoutputofa
linearfilterappliedtothesequenceofprocessmeasurementdata.Inrecentworkthathasgeneralizedthis
concept,thechartedstatisticistheoutputofagenerallinearfilterinimpulseresponseform,andthefilter
isdesignedbyselectingitsimpulseresponsecoefficientstooptimizeitsaveragerunlengthperformance.
Inthiswork,werestrictattentiontotheclassofallsecond-orderlinearfiltersappliedtotheresidualsof
atimeseriesmodeloftheprocessdata.Wepresentanalgorithmforoptimizingthedesignofthesecond-
orderfilterthatismorecomputationallyefficientandrobustthanthealgorithmforoptimizingthegeneral
linearfilter.Wedemonstratethattheoptimalsecond-orderfilterperformsalmostaswellastheoptimal
generallinearfilterinmanysituations.Bothmethodsshareanumberofinterestingcharacteristicsandare
tunedtodetectanydistinctfeaturesoftheprocessmeanshiftasitmanifestsitselfintheresiduals.
KEY WORDS: Autocorrelation; Control chart; Linear filtering; Markov chain method; Statistical
processcontrol;Timeseries.
1. INTRODUCTION Jiang,Tsui,andWoodall(2000)andJiang(2001)andthePro-
portional Integral Derivative (PID) chart of Jiang, Wu, Tsung,
Many common control charting methods are based on lin-
Nair, and Tsui (2002). A more complex filter structure, with
ear filtering in the following sense. The sequence of statistics
more filter design parameters, creates the potential for better
{y :t = 1,2,3,...} to be charted is calculated as the output
t controlchartperformance,especiallywhentheprocessdataare
of a linear filter applied to the sequence of process observa-
autocorrelated.Itmaybedifficulttotakeadvantageofthispo-
tions {x :t=1,2,3,...}. An alarm is sounded at observation
t tential,however,becauseofdifficultyinproperlyselectingthe
number t if y falls outside a set of control limits. Note that a
t design parameters. The only available guidelines are heuristic
general (stationary) linear filter applied to a sequence {x} is
t and rather anecdotal.An ARMA or PID chart that is not opti-
defined to be of the form yt =h0xt +h1xt−1 +h2xt−2 +···, mizedmayperformworsethanawell-designedEWMA.
where{h :j=0,1,2,...}areweightingcoefficients(Boxetal.
j Recently, Apley and Chin (2006) proposed a compete gen-
1994, chap. 1). In other words, the output of a linear filter is
eralization of the concept of a control chart based on linear
simply a weighted sum of past observations. If B denotes the
filtering. They considered a control chart statistic of the form
time-seriesbackshiftoperator,thenthelinearfiltercanbewrit- y = H(B)x, where H(B) is a general linear filter (GLF) in
tenasy =H(B)x,whereH(B)=h +h B+h B2+···.Alin- t t
t t 0 1 2 impulse response form. They treated this as an optimal fil-
earfilterH(B)expressedinthismannerissaidtobeinimpulse
ter design problem and developed a method for finding the
response form, and {hj:j=0,1,2,...} are called the impulse filter impulse response coefficients, {h :j = 0,1,2,...}, that
j
responsecoefficients.Linearfiltersareoftenexpressedintrans-
minimize the out-of-control average run length (ARL) for a
ferfunctionform,suchas(1).Alltransferfunctionformsofsta-
specified mean shift of interest, under the constraint that the
tionarylinearfiltershaveanequivalentimpulseresponseform.
in-controlARLequalssomedesiredvalue.Theydemonstrated
(SeeBoxetal.1994,chap.1,foranintroductorydiscussionof
that for step mean shifts in independently identically distrib-
linearfiltersandhowtoconvertbetweenimpulseresponseand
uted(iid)data,theoptimalGLF(OGLF)coincideswithasim-
transferfunctionforms.)
ple EWMA. For many autocorrelated processes, however, the
A classic example of control charts based on linear filters
OGLF has an intricate structure and can achieve much bet-
is the exponentially weighted moving average (EWMA) con-
terARLperformancethananoptimizedEWMA.Wenotethat
trol chart of Roberts (1959), in which y is an EWMA of x.
t t ApleyandChin(2006)directlyoptimizedthedesignofaGLF
The Shewhart individual chart is a trivial case with yt equal of the form y =H(B)e, where e denotes the residuals of an
t t t
tox.Whenx isanautocorrelatedprocess,EWMAchartsand
t t ARMAprocessmodel(seeSec.2).Thereisnolossofgeneral-
Shewhart individual charts on the residuals of an autoregres-
ityinoptimizingaGLFappliedto e versusoneappliedto x,
t t
sive movingaverage (ARMA) model of the process (see, e.g.,
andviceversa,iftheARMAmodelisassumedtobestableand
Montgomery and Mastrangelo 1991; Lu and Reynolds 1999a)
invertible.
constitute two more examples. This is because the residuals
themselvescanbeviewedastheoutputofalinearfilterapplied
©2006AmericanStatisticalAssociationand
tox.
t theAmericanSocietyforQuality
More recent examples, in which the linear filter has a more
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
complexstructurethananEWMA,includetheARMAchartof DOI10.1198/004017006000000020
337
338 CHANG–HOCHINANDDANIELW.APLEY
One disadvantage of the method of Apley and Chin (2006) one-step-ahead prediction errors) are generated by the linear
is that calculating the ARL for a GLF is so complex that cer- filteringoperation(ApleyandShi1999),
(cid:1) (cid:2)
tainapproximationsandMonteCarlosimulationsarerequired
(cid:4)(B) (cid:4)(B) (cid:5)(B)
intheGLFoptimizationalgorithm.Moreover,theGLFcanbe e = x = a +µ =a +µ˜ ,
t t t t t t
(cid:5)(B) (cid:5)(B) (cid:4)(B)
somewhatcumbersometoimplement,becauseitrequiresstor-
age of the entire set of impulse response coefficients (up to a where µ˜ =(cid:5)−1(B)(cid:4)(B)µ is a filtered version of the deter-
t t
suitably large truncation time, after which the coefficients are ministic mean shift µ. The residuals are an independent se-
t
essentially0).Toavoidthesedrawbacks,weproposeasacon- quence of Gaussian random variables with variance σ2 and
trolchartstatisticasecond-orderlinearfilter(SLF)oftheform time-varyingmean µ˜ . Apleyand Shi (1999) referred to µ˜ as
(cid:1) (cid:2) t t
1−βB thefaultsignature.
y =γ e, (1)
t 1−α B−α B2 t TheobjectiveistofindtheSLFparametersthatminimizethe
1 2
out-of-control ARL for a specified mean shift µ (e.g., a step
t
whereα1,α2,β,andγ aretheSLFdesignparameterstobede- shiftofsizeµ,representedbyµ =0fort<τ andµ =µfor
t t
termined. We include the scaling constant γ, because we use t≥τ), while simultaneously constraining the in-control ARL
normalizedcontrollimits±1.Thefilterin(1)isaspecialcase
tosomedesiredvalue.Toaccomplishthis,weexpresstheARL
oftheGLFconsideredbyApleyandChin(2006),withthere-
asafunctionofthefilterparametersusingthefollowingvaria-
strictionsthatitstransferfunctionformissecondorder,andit
tionoftheMarkovchainapproachofBrookandEvans(1972).
isappliedtoet insteadofxt. DefinethevectorVt=(yt,zt)(cid:2),wherezt=α2yt−1−γβet,and
Following Apley and Chin (2006), we focus on optimizing
notethatV canbewrittenas
t
the design of the filter. Specifically, we develop an approach (cid:1) (cid:2) (cid:1) (cid:2)(cid:1) (cid:2) (cid:1) (cid:2)
forselectingtheSLFparametersα1,α2,β,andγ tominimize V = yt = α1 1 yt−1 + γ e
theout-of-controlARLundertheconstraintthatthein-control t zt α2 0 zt−1 −γβ t
ARL equals some desired value. In Section 2 and the Appen- =DVt−1+Wet, (2)
dix we describe our approach for calculating the ARL of the (cid:3) (cid:4) (cid:3) (cid:4)
SLF and its gradient with respect to the filter design parame- whereD= α1 1 andW= γ .
α2 0 −γβ
ters,whichisneededintheoptimizationalgorithm. Because V is a two-dimensional vector Markov process,
t
Our focus on filter design optimization is one aspect that two-dimensional Markov chain methods similar to those used
distinguishes this work from the work on ARMA and PID by Runger and Prabhu (1996), VanBrackle and Reynolds
charts.TheheuristicdesignproceduressuggestedbyJiangetal. (1997), and Jiang (2001) can be used to calculate the ARL of
(2000,2002)fortheARMAandPIDchartsaresomewhatam- thecontrolchartony.Readersfamiliarwithautomaticcontrol
t
biguousandmayresultincontrolchartsthatperformfarfrom theory might recognize (2) as the observable canonical form
optimally. We demonstrate this in Section 3.4 using the same (Åström and Wittenmark 1990) of the filter. Although Jiang
vibrationdataexampleconsideredbyJiangetal.(2000,2002). (2001) suggested using a three-dimensional vector Markov
Anotherdifferencebetweenthis workandtheARMAchart chain representation for such a process, invoking the canoni-
of Jiang et al. (2000) is that our SLF is applied to the residu- calformofthefilteryieldsthetwo-dimensionalrepresentation
als,whereastheARMAchartisappliedtotheoriginaldataxt. of (2). The reduction in dimensionality substantially reduces
Applying the SLF to the residuals has two advantages. First, the computational expense involved in calculating the ARL.
for reasons that become apparent in the next section, it al- ItalsoeliminatestheneedfortheMonteCarlosimulationused
lowsamorecomputationallyfeasibleapproachforcalculating byApleyandChin(2006)foroptimizingaGLF.Theresultis
the ARL. This is important when optimizing the performance thatthealgorithmforoptimizingtheSLFismuchmorecompu-
oftheSLF.TheapproachisapplicableforanyARMAprocess, tationallyefficientthanthealgorithmforoptimizingtheGLF.
regardlessofthemodelorder.Second,itappearsthatapplying The two-dimensionalMarkov chain approach used by Run-
theSLFtotheresidualsresultsinbetterARLperformancethan ger and Prabhu (1996), VanBrackle and Reynolds (1997), and
applying the SLF to the original data, evidence of which we Jiang (2001) can be applied to the present situation to calcu-
presentinSection3.Indeed,formanyoftheexamplesthatwe late the ARL, as described in the Appendix. The strategy for
considerinSection3,theperformanceofouroptimizedSLFis optimizing the vector of SLF parameters ζ =[α α β γ](cid:2)
1 2
almostequaltothatofthemostgenerallinearfilteroptimized is summarized in the remainder of this section. As inputs to
byApleyandChin(2006). theoptimizationroutine,theuserspecifiestheARMAprocess
model, a mean shift that is of particular interest (the type as
wellasthemagnitude),andadesiredin-controlARL.Theop-
2. AVERAGERUNLENGTHCALCULATIONAND
timizationalgorithmthen finds the filter parameters that mini-
FILTEROPTIMIZATIONSTRATEGY
mizetheout-of-controlARLforthespecifiedmeanshift,while
Throughoutthisarticle,weassumethatx followsanARMA providingthedesiredin-controlARL.Theefficiencyoftheop-
t
process model of the form x =(cid:4)−1(B)(cid:5)(B)a +µ, where timization routine is substantially improved by incorporating
t t t
µ represents the deterministic process mean, t is a time in- the gradient ∂ARL/∂ζ, an expression for which we derive in
t
dex, a is an iid Gaussian process with mean 0 and vari- theAppendix.
t
ance σ2, and (cid:4)(B) = (1 − φ B − φ B2 − ··· − φ Bp) and Using (A.2) and (A.3) of the Appendix for the ARL and
1 2 p
(cid:5)(B)=(1−θ B−θ B2 −···−θ Bq) are the AR and MA its gradient, we have coded in MATLAB a straightforward
1 2 q
polynomials of order p and q. The model residuals (i.e., the gradient-based algorithm for optimizing the SLF parameters,
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
SECOND–ORDERLINEARFILTERS 339
which is available on request. The Appendix provides some Sec.3.5),andthein-controlARLwas500inallcases.Although
additional details on the algorithm, and further details have theMarkovchainmethodwasusedtooptimizetheEWMAand
been given by Chin (2004). Note that numerical evaluation of the SLF, all ARL values shown in Table 1 were from Monte
∂ARL/∂ζ in(A.3)involvesroughlythesamecomputationalex- Carlo simulation with 250,000 replications. The standard er-
penseasevaluationoftheARLin(A.2). rors of the ARL estimates are shown in parentheses. The op-
timized parameters for the EWMA and SLF are also shown.
Inthesubsequentdiscussion,whenofinterest,wegivetheim-
3. DISCUSSIONANDEXAMPLES
pulseresponsecoefficientsfortheOGLFs.TheOGLFimpulse
3.1 ComparisonWiththeOptimalEWMA response coefficients for all 28 examples have been given by
andtheOGLF ApleyandChin(2006).
As shown by Apley and Chin (2006), the OGLF reduces to
In this section we compare the optimal SLF (OSLF) with thesimple-structuredEWMAforthecaseofstepmeanshiftsin
the OGLF of Apley and Chin (2006) and with an optimized iid processes (Examples 1–4). Consequently, because the SLF
residual-based EWMA. The parameters of all three charts are is contained within the class of GLFs, the OSLF also reduces
optimizedtominimizetheout-of-controlARLforashiftoccur- to an EWMA. This is evident from α =β =0 in Table 1 for
2
ringatinitialtimeτ =1,whileconstrainingthein-controlARL Examples1–4.NotethattheoptimalvalueoftheEWMApara-
to equal 500. Although we must specify a shift time of occur- meter(λ=1−α )becomeslargerasthesizeofthemeanshift
1
rence,τ,toimplementtheoptimizationalgorithm,wedemon-
increases, which is well known (Lucas and Saccucci 1990).
strate in Section 3.5 that the ARL of an OSLF optimized for
It should not be surprising that for step shifts in iid data the
τ =1isnearlyindependentofwhentheshiftactuallyoccurs.In
OGLF and OSLF reduce to an EWMA. It is well known that
otherwords,ifweoptimizetheOSLFforzero-stateARLper-
aproperlydesignedEWMAcanapproximatelymatchtheper-
formance(i.e., τ =1),thenwecanbeassuredthatitisnearly
formanceofany(two-sided)CUSUM,andMoustakides(1986)
optimalforsteady-stateARLperformanceaswell.
has shown that (one-sided) CUSUMs are optimal in a sense
Theresidual-basedEWMAisdefinedas
similar to ours over the class of all possible control charting
yt=(1−λ)yt−1+get, procedures,linearornonlinear.
InmanyoftheexampleslistedinTable1,substantialperfor-
where 0<λ≤1 is the EWMA parameter and g is a scaling
mance improvement can be achieved by increasing the com-
constant. The chart signals when the EWMA statistic y falls
t plexity of the filter from the first-order EWMA to the SLF
outside the control limits ±1. Although the last term in the
and the GLF. In most of the examples where the EWMA and
EWMAequationisusuallywrittenasλe,weusetheadditional
t OGLF performance differs significantly, the OSLF also per-
constantgtoaccountforthefactthatourcontrollimitsarefixed
formsmuchbetterthantheoptimalEWMAandalmostasgood
at±1.NotethattheEWMAcanbewrittenasafirst-orderlin-
astheOGLF.TheexceptiontothisisExample28,forwhichthe
earfiltery =H(B)e,whereH(B)=(1−(1−λ)B)−1g,which
t t OSLF performs only slightly better than the optimal EWMA
hasimpulseresponsecoefficientsh =g(1−λ)j.Consequently,
j and substantially worse than the OGLF. We provide an expla-
theoptimalEWMAcanbeviewedasamorerestrictivecoun-
nationforthisinSection3.3.
terpart of the OSLF, whereas OGLF can be viewed as a more
Consider Example 8, which is a step mean shift of magni-
generalcounterpart. tude4σ inanAR(1)processwithφ=.9.Notethatthevariance
The performance of all three charts depends heavily on the ofx isσ2=(1−φ2)−1σ2,andameanshiftof4σ translatesto
form and magnitude of the residual mean and on the ARMA t x
only1.74σ .AsillustratedinFigure1(a),theresidualmeanin
x
model describing the process. Because of this, we compare
thiscaseexperiencesapronouncedinitialspikebeforedropping
performanceforthesamebroadcombinationofscenariosthat
down to a much smaller steady-state value. In situations like
Apley and Chin (2006) considered, which are represented by
this,LinandAdams(1996)andLuandReynolds(1999b)rec-
the 28 examples listed in Table 1. The process models are all
ommendedusingacombinedShewhart–EWMAscheme.(See
ARMA(1,1)modelsoftheformxt−φxt−1=at−θat−1,which LucasandSaccucci1990andReynoldsandStoumbos2001for
includes their special cases of first-order AR and iid. With-
more discussion of combined Shewhart–EWMA charts.) The
outlossofgenerality,weassumethatσ =1fortheremainder
shapesoftheOSLFandOGLFimpulseresponsefunctionsfor
of the article. We also consider three different types of mean
Example8showninFigure1(b)indicatethattheycloselycor-
shifts—step, spike, and sinusoidal—and a range of mean shift
respondtoacombinedShewhart–EWMAscheme.Notethatthe
sizes depending on the specific example. The step mean shift
plotsinFigure1(b)wereobtainedbyexpressingboththeOGLF
wasdefinedintheprevioussection,andthespikemeanshiftis
and the OSLF in their impulse response forms, y =H(B)e,
defined as µ =µ and µ =0 for t(cid:4)=1. The sinusoidal shifts t t
1 t withH(B)=h +h B+h B2+···.AlsonotethattheOSLF
aredenotedby S –S inTable1. S , S ,and S aresinusoidal 0 1 2
1 4 1 2 3 andOGLFarealmostidentical,inthesensethattheirimpulse
functionswithamplitude.75andperiodsoftwo,four,andeight
responsecoefficientsnearlycoincide.
observations.S hasamplitude1.5andaperiodofeightobser-
4 TheconnectiontoacombinedShewhart–EWMAschemebe-
vations. The phase of each sinusoid is such that it achieves its
comes more apparent if we write the OSLF in the following
maximumvalueontheinitialobservation.
form,usingtheExample8parametersfromTable1:
Table1liststheout-of-controlARLvaluesforallthreecharts
(cid:1) (cid:2)
for the 28 examples. All ARL values listed are zero-state val- 1−.847B
y =.298 e
ues (corresponding steady-state ARL values are discussed in t 1−.863B−.105B2 t
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
340 CHANG–HOCHINANDDANIELW.APLEY
Table1.ARLComparisonfortheOSLF,theOptimalEWMA,andtheOGLF
Timeseriesmodel Shift OGLF OptimalEWMA OSLF
No. φ θ Type Sizeµ ARL λ g ARL α1 α2 β γ ARL
1 0 0 Step .5 28.82 .047 .1167 28.82 .953 .000 .000 .1167 28.82
(.03) (.03) (.03)
2 1.5 5.45 .242 .2179 5.45 .758 .000 .000 .2179 5.45
(.01) (.01) (.01)
3 3 1.86 .676 .3067 1.86 .324 .000 .000 .3067 1.86
(.00) (.00) (.00)
4 4 1.21 .887 .3216 1.21 .113 .000 .000 .3216 1.21
(.00) (.00) (.00)
5 .9 0 Step .5 355.31 .002 .0527 355.31 .998 .000 .000 .0527 355.31
(.57) (.57) (.57)
6 1.5 130.64 .007 .0654 130.64 .993 .000 .000 .0654 130.64
(.18) (.18) (.18)
7 3 46.91 .021 .0887 49.43 .863 .105 .784 .2754 47.26
(.10) (.07) (.10)
8 4 13.72 .038 .1080 29.78 .863 .105 .847 .2983 13.72
(.06) (.05) (.06)
9 .9 0 Spike .5 495.39 1.000 .3236 497.12 −.070 .046 .869 .2368 496.83
(.98) (1.00) (1.00)
10 1.5 422.01 1.000 .3236 454.46 −.071 .037 .872 .2364 427.08
(.98) (.99) (.98)
11 3 82.72 1.000 .3236 177.83 −.103 .001 .844 .2360 85.12
(.54) (.76) (.55)
12 4 6.72 1.000 .3236 28.70 −.069 .035 .872 .2367 7.12
(.14) (.32) (.15)
13 0 0 Sinusoid S1 15.79 1.000 .3236 103.6 −.558 .322 .326 .1506 15.79
(.02) (.51) (.02)
14 S2 30.69 1.000 .3236 170.8 −.026 −.903 −.243 .1494 30.69
(.04) (.81) (.04)
15 S3 32.90 .608 .2986 137.6 1.160 −.716 −1.208 .0849 43.30
(.04) (.57) (.08)
16 S4 10.61 .616 .2997 26.31 1.024 −.636 −1.070 .1068 11.46
(.01) (.05) (.01)
17 .9 −.9 Step .5 447.66 .002 .0527 447.66 .998 .000 .000 .0527 447.66
(.75) (.75) (.75)
18 1.5 139.26 .003 .0557 255.72 −.924 .007 −.039 .1399 163.10
(.54) (.39) (.71)
19 2 41.54 .004 .0584 194.09 −.924 .007 −.039 .1399 43.31
(.36) (.28) (.37)
20 3 3.12 1.000 .3236 76.23 −.861 −.045 −.084 .2051 3.21
(.03) (.49) (.04)
21 .9 .5 Step .5 205.04 .004 .0584 205.58 .996 .000 .000 .0584 205.58
(.30) (.30) (.30)
22 1.5 50.28 .021 .0887 50.28 .979 .000 .000 .0887 50.28
(.07) (.07) (.07)
23 3 10.77 .120 .1662 10.80 .879 .000 −.020 .1639 10.77
(.03) (.03) (.03)
24 4 2.74 .304 .2374 2.88 .696 .000 .000 .2374 2.88
(.01) (.01) (.01)
25 .9 .5 Spike .5 497.47 1.000 .3236 497.61 −.238 −.001 −.038 .3172 497.47
(.99) (.99) (1.00)
26 1.5 461.86 1.000 .3236 469.74 −.222 −.005 −.163 .3231 469.23
(.99) (.99) (.99)
27 3 208.77 1.000 .3236 259.67 −.220 −.006 −.185 .3234 259.77
(.80) (.87) (.88)
28 4 50.75 1.000 .3236 86.10 −.230 −.004 −.156 .3227 83.72
(.41) (.56) (.55)
NOTE:Thesimulationstandarderrorsareshowninparentheses.
(cid:1) (cid:2) (cid:1) (cid:2)
= .264 e + .034 e self represents a scaled version of an EWMA with a small
1+.108B t 1−.971B t value λ=1−.971=.029 for the EWMA parameter. Conse-
∼=.264et+ 1−.0.39471Bet. qwueeingthltye,dthceomOSbiLnFatiaonndoOfGaLSFhfeowrhEaxrtaminpdlievi8duaarlecehsasretntainadllyana
EWMAchart,ascanbeseeninFigure1(b).Whereasthetyp-
The first term by itself represents a scaled version of a Shew- icalcombinedShewhart–EWMAschemechartsthetwostatis-
hart individual chart on the residuals. The second term by it- tics separately but simultaneously, the OSLF combines them
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
SECOND–ORDERLINEARFILTERS 341
(a) (b)
Figure1.ResidualMean(a)andImpulseResponsesoftheOSLF( )andtheOGLF( )(b)forExample8.
together into a single statistic y. Despite this difference, we coefficients also oscillate. A more precise explanation of this
t
wouldexpectthetwochartstobehavesimilarly;theShewhart phenomenonisprovidedinSection3.2.
componentiseffectiveindetectingtheinitialspikeintheresid- Figures 3 and 4 show the residual mean and the OGLF and
uals, and the EWMA component with small λ is effective in OSLF impulse responses for Examples 13 and 15 (sinusoidal
detectingthesmallbutsustainedsteady-stateshiftintheresid- meanshiftsiniidprocesses).Notethatthesinusoidalmeanin
ualmean.OneattractivefeatureoftheOSLFisthattherelative Example13,withaperiodoftwoobservations,ischaracteristic
weightingofthetwocomponentsisselectedoptimally,tomin- ofmixturedistributionsinwhichsuccessiveobservationsalter-
imizetheARL. natebetweentwodifferentdistributions.Inbothexamples,the
TheOSLFsinExamples5–7canbeviewedsimilartocom-
OSLFimpulseresponsesaredamped(i.e.,exponentiallydecay-
bined Shewhart–EWMA charts, where the relative weighting
ing)sinusoidswithperiodequaltotheperiodofthesinusoidal
of the Shewhart component decreases as the size of the mean
shift.WecanviewboththeOSLFandtheOGLFasbeingtuned
shiftdecreases.Whenthemeanshiftsizedecreases,sodoesthe
to detect the sinusoidal dynamics of the residual mean. The
prominence of the initial spike in the residuals, and one must
first-order EWMA obviously does not have a structure that is
relymoreheavilyontheEWMAcomponenttodetectthesmall
sufficientlyflexibletobethustuned,andconsequently,itsARL
butsustainedshiftintheresidualmean.
ismuchlarger.
TheOSLFandtheOGLFarealsoquitesimilartoeachother
for the AR(1) processes with φ = .9 and spike mean shift
(Examples 9–12 in Table 1), and both outperform the optimal 3.2 ApplyinganSLFtoetversusxt
EWMA for large mean shifts. Figure 2(a) shows the residual
mean for Example 12, and Figure 2(b) shows the correspond- LetHe(B)et denotetheOGLFappliedtoet,andsupposethat
ingimpulseresponsecoefficientsfortheOSLFandtheOGLF. we also optimize a GLF applied to xt, denoted by Hx(B)xt.
The reason why the OSLF and OGLF outperform the optimal Because et =(cid:5)−1(B)(cid:4)(B)xt, if the ARMA process model is
EWMA in this case is apparent from Figure 2. The residual stableandinvertible,thenitmustbethecasethatthetwoopti-
mean oscillates above and below 0 on the first two observa- mized GLFs satisfy the relationship Hx(B)=He(B)(cid:5)−1(B)×
tionsaftertheshift.BoththeOSLFandtheOGLFaretunedto (cid:4)(B) or, equivalently, He(B)=Hx(B)(cid:4)−1(B)(cid:5)(B). The per-
detect this oscillation, in the sense that their impulse response formance of the two charts is identical,and there is no loss of
(a) (b)
Figure2.ResidualMean(a)andImpulseResponsesoftheOSLF( )andtheOGLF( )(b)forExample12.
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
342 CHANG–HOCHINANDDANIELW.APLEY
(a) (b)
Figure3.ResidualMean(a)andImpulseResponsesoftheOSLF( )andtheOGLF( )(b)forExample13.
generalityinoptimizingaGLFappliedtoe,versusoneapplied mechanisms by which the OGLF or OSLF may achieve sub-
t
tox. stantiallybetterperformancethantheoptimalEWMA.Wecan
t
ThesamecannotbesaidabouttheSLFs.Ifweoptimizean write each of the control chart statistics in terms of their filter
SLF applied to e, then its performance may be very differ- impulseresponsecoefficientsthrough
t
ent from that of an optimized SLF applied to xt. The results (cid:5)t−1 (cid:5)t−1 (cid:5)t−1
in the previous section provide some evidence that better per- yt= hjet−j= hjµ˜t−j+ hjat−j.
formance can be achieved by applying the SLF to the residu- j=0 j=0 j=0
als.Specifically,weseethattheOSLFsarealmostidenticalto
t2h(eb)O,aGnLdF3s(bfo)]r,manadnytheexOamGpLleFsi[set.hge.,othpotismeaslhloiwnenarinfiFltiegrso.f1a(bn)y, dTihsutrsi,buatteadnwyittihmmeeta,nth(cid:6)e ctj=−o01nthrjoµ˜lt−chjaarntdstvaatirsitainccyetσis2(cid:6)norjt=−m01ahll2jy.
Comparing the residual mean functions and the OGLF im-
order,appliedtoeithere orx.
t t
pulseresponsecoefficientsshowninFigures1–4showsthatthe
Another advantage of applying the SLF to the residuals is
OGLF is tuned so that its impulse response function is highly
thattheARLcomputationismoretractable.Becausetheresid-
“(cid:6)correlated”withtheresidualmean,inthesensethatthemean
ureaplsreasreentaasneqSuLenFceapopfliienddetpoentdheenrtersaindduoalms tvharroiaubglhest,hweetwcaon- tj=−01hjµ˜t−j ofyt islargeinmagnitudeafterameanshift.
Althoughthestructureofthesimplefirst-orderEWMAfilter
dimensional vector Markov process described in Section 2. In
is not sufficiently flexible to allow it to be tuned to correlate
contrast,itcanbeshownthatanSLFappliedtoanARMA(p,q)
closely with a complex residual mean function, the increased
processx requiresavectorMarkovchainrepresentationofdi-
t flexibility of the second-order filter evidently does allow this.
mensionmax{p+2,q+2}.
ComparingtheOGLFsandOSLFsinFigures1–4clearlyshows
thattheOSLFiscapableofcapturingthedynamicsofavariety
3.3 Second-OrderversusHigher-OrderFilters ofimpulseresponsefunctions.
One mild exception to this occurs in Example 15 (Fig. 4),
ThesimilarityoftheOGLFandOSLFinmanyoftheexam- in which the OSLF impulse response is a damped sinusoid of
plesalsoimpliesthatitisoftenunnecessarytoconsiderfilters periodeight.Althoughthisapproximatesthegeneralcharacter-
of order higher than 2. One explanation for this relates to the istics of the OGLF, the differences are not negligible, and the
(a) (b)
Figure4.ResidualMean(a)andImpulseResponsesoftheOSLF( )andtheOGLF( )(b)forExample15.
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
SECOND–ORDERLINEARFILTERS 343
(a) (b)
Figure5.ResidualMean(a)andImpulseResponsesoftheOSLF( )andtheOGLF( )(b)forExample28.
OGLFperformssubstantiallybetterthantheOSLF.TheARLs The mechanical vibration data, originally considered by
forthisexampleare32.9fortheOGLFand43.3fortheOSLF. PanditandWu(1983),werefittedbytheARMA(2,1)model,
Example28,theresidualmeanandfilterimpulseresponsecoef- xt−1.439xt−1+.600xt−2=at+.519at−1, (3)
ficientsforwhichareshowninFigure5,isamoreextremecase.
The OSLF cannot adequately approximate the OGLF, which withσ =2.21(σx=9.13).Fornotationalsimplicity,wehence-
closely mimics the oddly shaped residual mean. The result is forthassumethatthedataarescaledsothatσ =1(σx=4.13).
thattheARLfortheOSLFisalmost70%largerthantheARL Wealsoignoreanymodelingerrors.
fortheOGLF(83.7vs.50.8). Table2comparesthezero-stateout-of-controlARLsforthe
OSLFandthePIDchartswithdesignparameterstakendirectly
3.4 ComparisonWiththePIDChart fromtable1ofJiangetal.(2002),whoselectedthesechartpa-
rameters according to their heuristic design guidelines. More
The PID chart statistic of Jiang et al. (2002) can be written specifically,theychoseeachofthethreesetsofPIDparameters
asyt=[1−(1−kP−kD−kI)B−(kP+2kD)B2+kDB3]−1× tooptimizeapairofcapabilityindicesforoneofthemeanshifts
[1−B]xt,wherekP,kD,and kI aredesignparameters.Wecan in Table 2, which, they argued, should translate to reasonably
view this as a third-order filter on the original data xt, with good ARL performance for that size mean shift. For purposes
therestrictionthatthenumeratorpolynomialofthefilteristhe ofcomparisonwiththeOSLF,thecolumnlabeled“PID*”lists
differencing operator [1−B]. The PID chart signals when yt theARLforthemosteffectiveofthethreePIDchartsforeach
falls outside a set of symmetric control limits ±L. One diffi- mean shift. All ARLs were evaluated based on Monte Carlo
culty with the PID chart is that it is not straightforward how simulation with 250,000 replicates. The out-of-control condi-
to select the filter design parameters. For computational rea- tionwassimulatedbygeneratingdatathatfollowthemodel(3)
sons, it would be quite difficult to optimize the chart using a withastepmeanshiftofsizeµaddedattheinitialobservation.
Markovchainapproach.If xt isanARMA(p,q)process,then Thein-controlARLwas370forallcharts.Table2alsoincludes
thiswouldrequireavectorMarkovchainrepresentationofdi- theOGLFandaShewhartindividualchartontheresidualswith
mensionmax{p+3,q+2}.Forcomparisonpurposes,however, controllimits±3.
weconsiderthesameexampleofamechanicalvibrationsystem Asinmanyofthepreviousexamples,theOSLFandOGLF
studiedbyJiangetal.(2002)usingthevaluesfork ,k ,andk performcomparably.TheOSLFandOGLFbothperformsub-
P D I
thattheyrecommendedforthisexample. stantially better than the best of the three PID charts for each
Table2.ARLsoftheOSLF,theOGLF,theResidual-BasedShewhartChart,andthePIDChartsfortheMechanicalVibrationExample
PID PID PID
Rebsaisdeudal- {kP,kI,kD}= {kP,kI,kD}= {kP,kI,kD}=
Shift OSLF Shewhart {−.8,0,0} {−.3,1.8,0} {−.8,0,.5}
(∆=µ/σx) α1 α2 β γ ARL OGLF chart PID∗ (L=2.596) (L=2.978) (L=2.531)
0 370 370 370 370 370 370 370
(.68) (.68) (.74) (.73) (.73) (.73) (.72)
.5 .986 .000 .000 .0843 76.88 61.26 200.02 118.14 141.43 351.12 118.14
(.10) (.15) (.56) (.22) (.27) (.72) (.22)
1 −.529 .000 .000 .2855 1.59 1.40 3.56 37.25 44.89 118.24 37.25
(.03) (.01) (.06) (.06) (.08) (.53) (.06)
2 .000 .000 .000 .3334 1.00 1.00 1.00 1.00 11.57 1.00 10.94
(.00) (.00) (.00) (.00) (.02) (.00) (.01)
3 .000 .000 .000 .3334 1.00 1.00 1.00 1.00 5.44 1.00 5.60
(.00) (.00) (.00) (.00) (.01) (.00) (.00)
∗
NOTE:Thesimulationstandarderrorsareinparentheses.ThePID columnliststhebestofthethreePIDchartsforeachmeanshiftsize.
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
344 CHANG–HOCHINANDDANIELW.APLEY
(a) (b)
Figure6.ResidualMean(a)andImpulseResponsesoftheOSLF( )andthePIDChart( )(b)fortheMechanicalVibrationExample.
mean shift (except for the large shifts, which each chart de- the CUSCORE chart can be potentially very effective in de-
tects immediately). The reason for this is clear from Figure 6. tecting a dynamically varying residual mean, because it takes
Figure 6(a) shows the residual mean, and Figure 6(b) shows into consideration the dynamics of the fault signature. But the
theimpulseresponsecoefficientsfortheOSLFandthebestof CUSCORE chart is tuned to detect signals occurring at a spe-
thethreePIDcharts ({kP,kI,kD}={−.8,0,.5})for the (cid:13)=1 cific point in time and typically will have radically different
case.Tohaveacommonbasisforcomparison,weplottheim- zero-stateandsteady-stateARLperformance(Shu,Apley,and
pulse response coefficients for the equivalent form of the PID Tsung2002).Forexample,considertheversionofCUSCORE
chart filter when we view it as being applied to the residu- chartfordetectingmeanshiftsinautocorrelatedprocessesthat
als.Inotherwords,weshowtheimpulseresponsecoefficients was proposed by Luceño (1999) and analyzed by Shu et al.
for the filter yt =H(B)et with H(B)=[1−(1−kP −kD − (2002), Runger and Testik (2003), and Luceño (2004), for
kI)B−(kP+2kD)B2+kDB3]−1[1−B](cid:4)−1(B)(cid:5)(B).Clearly, whichtheCUSCOREstatisticisdefinedrecursivelythrough
theOSLFistunedtodetecttheoscillatorynatureoftheresidual
mean,whereasthePIDchartisnot. St=max{0,St−1+µ˜t(et−µ˜t/2)}, t=1,2,3,....
The CUSCORE chart signals when S exceeds a threshold.
t
3.5 RobustnessProperties Luceño (1999) considered two different versions of the CU-
SCOREchart.Inthefirstversion,thefearedresidualmeanµ˜ is
The design and analysis of the OSLF was based on a num- t
takentobethatresultingfromameanshiftoccurringatτ =1.
berofassumptions,includingthefollowing:(a)Theshifttime
In the second version, µ˜ is reinitialized every time the CU-
of occurrence τ coincides with the initial observation, as op- t
SCOREstatisticisresetto0.
posedtosomelaterpointintime(zero-stateARLs,asopposed
Tables 3 and 4 compare the zero-state versus steady-state
tosteady-stateARLs);(b)weareabletospecifyasingleshift
performance of the two CUSCORE charts (with and without
size of particular interest, for which the OSLF is optimized;
reinitialization),theOSLF,andtheoptimalEWMAforExam-
and(c)forsituationsinwhichtheprocessisautocorrelated,the
ples15and20.Fortheseexamplestheresidualmeanfunctions
ARMAmodelperfectlydescribestheprocessandthereareno
havepronounceddynamics,sothatmismatchesbetweentheas-
modelingerrors.Inthissectionweinvestigatetherobustnessof
sumed and actual τ potentially will have the greatest effect.
theOSLFtoviolationsoftheseassumptionsanddiscusssteps
All charts were designed to have a zero-state in-control ARL
toensurethattheOSLFiseffectiveatdetectingarangeofshift
of500.TheOSLFsandoptimalEWMAsinTables3and4are
sizes.
exactly those for Examples 15 and 20 of Table 1, optimized
First, consider the shift time of occurrence. Although the
OSLFisoptimizedforτ =1,oneisgenerallymoreinterested
in the steady-state ARL (i.e., the expected detection lag for Table3.ComparisonofSteady-StateversusZero-StateARLs
shifts that occur after the control chart has been in operation forExample15
for a while) than the zero-state ARL (the expected detection
µ=0 µ=.75
lagforτ =1).Onemightconsideroptimizingthesteady-state (nosinusoid) (sinusoidadded)
ARLoftheOSLF,ratherthanitszero-stateARL;however,this Zero-state Steady-state Zero-state Steady-state
wouldbesubstantiallymorecomputationallyexpensive,aswell ARL ARL ARL ARL
as largely unnecessary. Apley and Chin (2006) demonstrated CUSCOREwithout 500.9 491.7 23.22 ≈∞
that the OGLF optimized for shifts occurring at the initial ob- reinitialization (.99) (1.09) (.06)
CUSCOREwith 500.9 490.8 37.30 43.12
servation is almost equally effective at detecting shifts occur-
reinitialization (.98) (1.12) (.14) (.15)
ringinthesteadystate,and,aswedemonstrateinthefollowing OSLF 500.1 496.1 43.30 41.78
paragraphs,thesameistruefortheOSLF. (.98) (1.14) (.08) (.15)
OptimalEWMA 500.3 494.2 137.6 136.6
NotethatthisisinsharpcontrasttotheCUSCOREchartsof
(.95) (1.21) (.57) (.61)
Box and Ramirez (1992) and Luceño (1999). Like the OSLF,
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
SECOND–ORDERLINEARFILTERS 345
Table4.ComparisonofSteady-StateversusZero-StateARLs Table5.ARLsoftheOSLFsforaRangeofShiftSizesfortheProcess
forExample20 DescribedbyEquation(3)
µ=0 µ=3 Shift OSLF OSLF OSLF OSLF
(noshift) (shiftadded) (∆=µ/σX) (∆=.5) (∆=1) (∆=2) (∆=3)
Zero-state Steady-state Zero-state Steady-state 0 370 370 370 370
ARL ARL ARL ARL (1.10) (1.17) (1.16) (1.16)
CUSCOREwithout 500.1 652.6 1.33 98.2 .5 77.03 163.1 199.2 199.2
reinitialization (1.02) (1.24) (.00) (.39) (.16) (.92) (.56) (.56)
CUSCOREwith 498.8 497.9 4.16 6.09 1 32.67 1.67 3.56 3.56
reinitialization (1.16) (1.18) (.07) (.11) (.05) (.05) (.06) (.06)
OSLF 500.4 495.8 3.21 3.41 2 14.07 1.00 1.00 1.00
(.99) (1.0) (.04) (.07) (.02) (0) (0) (0)
OptimalEWMA 500.7 500.2 76.23 75.97 3 3.52 1.00 1.00 1.00
(.01) (0) (0) (0)
(.99) (1.00) (.49) (.49)
under the assumption that τ = 1. The control limits for the (small) µ. In other situations, an OSLF optimized for one µ
may perform reasonably well for a range of µ. Each column
CUSCORE charts with and without reinitialization in Table 3
of Table 5 shows the ARLs over a range of µ for an OSLF
were3.78and3.60.ThecontrollimitsfortheCUSCOREcharts
optimized for one of the four shift sizes (.5, 1.0, 2.0, or 3.0)
withandwithoutreinitializationinTable4were4.59and2.32.
consideredinthemechanicalvibrationsystemdescribedby(3).
MonteCarlosimulationwasusedtoevaluatetheARLs.Forthe
steady-stateARLs,meanshiftswereintroducedatτ =100. TheparametersoftheOSLFsareasshowninTable2.
Ifoneisinterestedinarangeofµ,thenwerecommendcal-
Not surprisingly, the CUSCORE chart without reinitializa-
tionisthemosteffectiveatdetectingshiftsoccurringat τ =1 culating the OSLF for a few different µ values that span the
rangeofinterest.IfoneoftheOSLFsperformsadequatelyover
buttheleasteffectiveatdetectingshiftsoccurringinthesteady
the range, then that OSLF can be used by itself. For example,
state.Thereasonisthatitistunedtodetectshiftsoccurringex-
clusively at τ =1 and loses nearly all of its power if the shift the OSLF optimized for (cid:13)=µ/σx =1.0 in Table 5 also per-
forms quite well for the larger shifts. Conversely, if no single
occurs in the steady state (its steady-state ARL is worse than
OSLF performs well over the full range of µ, then we rec-
thatoftheEWMA).Forthisreason,wedonotrecommendus-
ommend using two or three OSLFs simultaneously, each one
ingtheCUSCOREchartwithoutreinitialization.Notethatthe
optimized for a different µ. This is similar to the practice of
out-of-controlsteady-stateARLfortheCUSCOREchartwith-
out reinitialization is ≈∞ in Table 3. The reason is that if the using a Shewhart chart in conjunction with a CUSUM (e.g.,
Lucas 1982) or, more generally, using multiple simultaneous
sinusoidisintroducedatobservation100,thenthefearedsignal
CUSUMS,eachoptimizedforadifferentµ(e.g.,Sparks2000).
and actual signal are exactly 180 degrees out of phase. Hence
thetermµ˜ (e −µ˜ /2)tendstoassumenegativevalues,sothat ApleyandChin(2006)discussedsimpleguidelinesfordesign-
t t t
ingmultiplesimultaneousOGLFs,whichalsoapplytothede-
theCUSCOREstatisticisresetto0onmostobservations.
signofmultiplesimultaneousOSLFs.
WemightviewtheCUSCOREwithreinitializationasbeing
The final robustness issue that we consider is robustness
tunedtodetectmeanshiftsthathaveoccurredimmediatelyaf-
to estimation errors in the ARMA model parameters. Tables
ter the CUSCORE statistic was last reset to 0. Consequently,
6 and 7 show the effects of ARMA parameter errors on the
the differences between its zero-state and steady-state ARLs,
in-controlandout-of-controlARLsforExamples8and19.Ex-
while still substantial, are much less than for the CUSCORE
ample19waschosenbecausetheresidualmeanoscillateswith
withoutreinitialization.Thedifferencesbetweenthezero-state
quite pronounceddynamics,due to the particularcombination
andsteady-stateARLsoftheOSLFareevensmaller,sothatthe
ofARMAparameters.Henceonemightspeculatethaterrorsin
OSLFperformsbetterthantheCUSCOREwithreinitialization
theARMAparameterswouldhavealargeeffectontheARLof
for shifts that occur in the steady state. We conclude that the
the OSLF. The residual mean for Example 8 is shown in Fig-
OSLF is almost equally effective at detecting shifts occurring
ure1.Tables6and7alsoshowtherobustnessoftheShewhart
inthesteadystate,eventhoughitwasexplicitlyoptimizedfor
individualchartandtheoptimalEWMA(OEWMA).Asarefer-
shiftsoccurringatτ =1.ApleyandChin(2006)attributedthis
ence,thein-controlandout-of-controlARLswithnoparameter
tothefactthattheOGLFandOSLFfiltersaremovingfunctions
errors(takenfromTable1)areshownintheleftmostcolumn.
oftheresidualsthatmoveacrosstheresidualsoneobservation
at a time. In this respect, the OSLF and OGLF look for mean
Table6.EffectsofARMAParameterErrorsontheARLofVarious
shiftsoccurringatallpointsintime,notanyonespecificpoint
ChartsforExample8
intime.
Next, consider the shift size µ. The OSLF is optimized for Chart φˆ=φ=.9 φˆ=.9,φ=.85 φˆ=.9,φ=.95
one specific value of µ, whereas in practice we are generally
OSLF 13.7 12.8 16.6
interestedinarangeofshiftsizes.Inmanysituations,anOSLF 500 831 182
optimized for one µ may perform poorly in detecting a much OEWMA 29.8 27.9 35.9
largerormuchsmallerµ.Forexample,theOSLFfordetecting 500 3,656 122
Shewhart 48.5 46.5 47.0
stepshiftsiniiddatawaspreviouslyshowntobequitesimilar 500 476 442
to an EWMA, and it is well known that an EWMA optimized
NOTE:Thetopnumbersaretheout-of-controlARLsforashiftofsizeµ=4.0,andthebottom
forasmall(large) µ willperformfar fromoptimalfor alarge numbersarethein-controlARLs.
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3
346 CHANG–HOCHINANDDANIELW.APLEY
Table7.EffectsofARMAParameterErrorsontheARLofVariousChartsforExample19
φˆ=φ=.9 φˆ=.9,φ=.85 φˆ=.9,φ=.95 φˆ=φ=.9 φˆ=φ=.9
Chart θˆ=θ=−.9 θˆ=θ=−.9 θˆ=θ=−.9 θˆ=−.9,θ=−.85 θˆ=−.9,θ=−.95
OSLF 43.3 37.1 44.9 26.1 170
500 417 571 158 4,613
OEWMA 194 202 137 197 193
500 1,653 161 540 465
Shewhart 299 285 263 285 276
500 477 442 469 468
NOTE:Thetopnumbersaretheout-of-controlARLsforashiftofsizeµ= 2.0,andthebottomnumbersarethein-controlARLs.
A few points are worth making. First, we cannot general- In situations where the OGLF performs substantially better
ize that the OSLF is any more affected or any less affected thantheOSLF(e.g., Examples15 and28),theOSLFstill has
by ARMA parameter errors than the OEWMA. Which chart someutility.AsdiscussedbyApleyandChin(2006),theopti-
ismoreaffecteddependsentirelyonthespecificsoftheexam- mizationalgorithmfortheOGLFcanbesensitivetotheinitial
ple. For the AR(1) process of Example 8, the in-control ARL guessforthedesignparameters.Areasonablestrategyforop-
oftheOEWMAismuchmoreaffectedthanthatoftheOSLF, timizing the OGLF is to first find the OSLF (the optimization
whereastheout-of-controlARLsofthetwochartsareroughly algorithmforwhichismorestableandrobust),andthenusethe
equallyaffected.(Theout-of-controlARLsforbothchartsare OSLFimpulseresponsecoefficientsastheinitialguessforthe
relativelyrobustforthisexample.)FortheARMA(1,1)process OGLF.
ofExample19,theOSLFisquiterobusttoerrorsinφ,whereas One drawback of the OSLF, as well as the OGLF, is that it
theOEWMA is affectedrather severely.Theexactoppositeis cannot be designed without the use of a numerical optimiza-
true regarding errors in θ, for which the OEWMA is quite ro- tionalgorithm(availableonrequest).Inotherwords,thereare
bust,whereastheOSLFisseverelyaffected. no simple tables, graphs, or heuristic rules that would allow a
The second point is that the Shewhart individual chart is practitionertoeasilydesigntheOSLFwithouttheuseofsoft-
by far the most robust of any of the charts. This is consistent ware. One could argue that this drawback is shared by all but
with the findings of Apley and Lee (2003). The third point the simplest of control charting methods, however. For exam-
is that even in the situations in Tables 6 and 7 in which the ple, for detecting step mean shifts of known magnitude in iid
OSLFisstronglyaffectedbyparametererrors,itstillperforms data,wehavefamiliartablesforselectingtheoptimalparame-
muchbetterthaneithertheOEWMAortheShewhartindivid- tersofEWMAandCUSUMcharts(LucasandSaccucci1990;
ualchart.Forexample,forθ =−.95(asopposedtoθˆ=−.90) Montgomery 2005). But no such tables are available for au-
tocorrelated data. The difficulty is that the optimal chart para-
inExample19,theout-of-controlARLoftheOSLFincreases
meters depend strongly on numerous quantities, including the
dramatically, to 170.0 from a value of 43.3 with no parameter
orderandparametersoftheARMAmodelusedtorepresentthe
errors (see Table 7). This is still better than the corresponding
autocorrelation, the desired in-control ARL, and the size and
out-of-controlARLsoftheOEWMA(192.7)andtheShewhart
natureofthemeanshiftforwhichthechartistobeoptimized.
chart (275.5), however. Moreover, the in-control ARL of the
Even a simple EWMA is very difficult to optimize for auto-
OSLFalsoincreasesto4,613,whichisdesirable.
correlateddata.Tothebestofourknowledge,ifwerestrictat-
tentiontomethodsthataresufficientlyversatiletobeeffective
4. CONCLUSIONS in a variety of control charting situations with autocorrelated
data, no methods can be optimally designed, or even approxi-
Inthisarticlewehaveproposedacontrolchartingprocedure
matelyoptimallydesigned,withouttheuseofsoftware.Atthe
basedonasecond-orderlinearfilterappliedtotheresidualsof
very least, even if the structure of the control chart is deter-
an ARMA process model. We used a two-dimensional vector
mined,software(e.g.,MonteCarlosimulationorMarkovchain
MarkovchainmethodtocalculatetheARLasafunctionofthe
algorithms)isneededtospecifycontrollimitsthatachieveade-
filter parameters and derived an expression for the derivative
sired in-control ARL. The ineffectiveness of simple heuristics
oftheARL,whichwasusedinagradient-basedalgorithmfor
for guaranteeing even close to optimal designs was illustrated
optimizingthefilterparameters.
withthePIDchartinTable2.
We have demonstrated through a number of examples that
the OSLF can perform substantially better than an optimized
ACKNOWLEDGMENT
EWMA and almost as good as the most general linear filter
with optimized impulse response coefficients. One advantage
ThisworkwassupportedbytheNationalScienceFoundation
of the OSLF over the OGLF is that it can be optimized us- (grantDMI-0093580).
ing a more computationally efficient algorithm that avoids the
needforMonteCarlosimulation.Anotheradvantageisthatthe
APPENDIX:SOMEDETAILSONCALCULATINGTHE
OSLF can be more easily implemented recursively using its
AVERAGERUNLENGTHANDITSGRADIENT
transferfunctionform.TheOGLFhasnoequivalentrecursive
implementationandmustbeimplementedinimpulseresponse In this appendix we provide some details of the approach
form, which requires storage of the entire set of impulse re- for calculating the ARL and its gradient (derivative) with re-
sponsecoefficients. specttothefilterparametervectorζ.Thetwo-dimensionalstate
TECHNOMETRICS,AUGUST2006,VOL.48,NO.3